:: LATTICE5 semantic presentation

begin

theorem Th1: :: LATTICE5:1
for f being Function
for F being Function-yielding Function st f = union (rng F) holds
dom f = union (rng (doms F))
proof
let f be Function; ::_thesis: for F being Function-yielding Function st f = union (rng F) holds
dom f = union (rng (doms F))
let F be Function-yielding Function; ::_thesis: ( f = union (rng F) implies dom f = union (rng (doms F)) )
assume A1: f = union (rng F) ; ::_thesis: dom f = union (rng (doms F))
thus  dom f c= union (rng (doms F)) :: according to XBOOLE_0:def_10 ::_thesis: union (rng (doms F)) c= dom f
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in dom f or x in union (rng (doms F)) )
assume  x in dom f ; ::_thesis: x in union (rng (doms F))
then  [x,(f . x)] in union (rng F) by A1, FUNCT_1:def_2;
then consider g being set  such that
A2: [x,(f . x)] in g and
A3: g in rng F by TARSKI:def_4;
consider u being set  such that
A4: u in dom F and
A5: g = F . u by A3, FUNCT_1:def_3;
 u in dom (doms F) by A4, A5, FUNCT_6:22;
then A6: (doms F) . u in rng (doms F) by FUNCT_1:def_3;
 x in dom (F . u) by A2, A5, FUNCT_1:1;
then  x in (doms F) . u by A4, FUNCT_6:22;
hence  x in union (rng (doms F)) by A6, TARSKI:def_4; ::_thesis: verum
end;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in union (rng (doms F)) or x in dom f )
assume  x in union (rng (doms F)) ; ::_thesis: x in dom f
then consider A being set  such that
A7: x in A and
A8: A in rng (doms F) by TARSKI:def_4;
consider u being set  such that
A9: u in dom (doms F) and
A10: A = (doms F) . u by A8, FUNCT_1:def_3;
A11: u in dom F by A9, FUNCT_6:59;
then A12: F . u in rng F by FUNCT_1:def_3;
consider g being Function such that
A13: g = F . u ;
 A = dom (F . u) by A10, A11, FUNCT_6:22;
then  [x,(g . x)] in F . u by A7, A13, FUNCT_1:def_2;
then  [x,(g . x)] in f by A1, A12, TARSKI:def_4;
hence  x in dom f by FUNCT_1:1; ::_thesis: verum
end;

theorem Th2: :: LATTICE5:2
for A, B being non empty set holds [:(union A),(union B):] = union  {  [:a,b:] where a is Element of A, b is Element of B : ( a in A & b in B )  }
proof
let A, B be non empty set ; ::_thesis: [:(union A),(union B):] = union  {  [:a,b:] where a is Element of A, b is Element of B : ( a in A & b in B )  }
set Y =  {  [:a,b:] where a is Element of A, b is Element of B : ( a in A & b in B )  }  ;
thus  [:(union A),(union B):] c= union  {  [:a,b:] where a is Element of A, b is Element of B : ( a in A & b in B )  }  :: according to XBOOLE_0:def_10 ::_thesis: union  {  [:a,b:] where a is Element of A, b is Element of B : ( a in A & b in B )  }  c= [:(union A),(union B):]
proof
let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in [:(union A),(union B):] or z in union  {  [:a,b:] where a is Element of A, b is Element of B : ( a in A & b in B )  }  )
assume A1: z in [:(union A),(union B):] ; ::_thesis: z in union  {  [:a,b:] where a is Element of A, b is Element of B : ( a in A & b in B )  }
then consider x, y being set  such that
A2: z = [x,y] by RELAT_1:def_1;
 y in union B by A1, A2, ZFMISC_1:87;
then consider b9 being set  such that
A3: y in b9 and
A4: b9 in B by TARSKI:def_4;
 x in union A by A1, A2, ZFMISC_1:87;
then consider a9 being set  such that
A5: x in a9 and
A6: a9 in A by TARSKI:def_4;
reconsider b9 = b9 as Element of B by A4;
reconsider a9 = a9 as Element of A by A6;
A7: [:a9,b9:] in  {  [:a,b:] where a is Element of A, b is Element of B : ( a in A & b in B )  }  ;
 z in [:a9,b9:] by A2, A5, A3, ZFMISC_1:def_2;
hence  z in union  {  [:a,b:] where a is Element of A, b is Element of B : ( a in A & b in B )  }  by A7, TARSKI:def_4; ::_thesis: verum
end;
let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in union  {  [:a,b:] where a is Element of A, b is Element of B : ( a in A & b in B )  }  or z in [:(union A),(union B):] )
assume  z in union  {  [:a,b:] where a is Element of A, b is Element of B : ( a in A & b in B )  }  ; ::_thesis: z in [:(union A),(union B):]
then consider e being set  such that
A8: z in e and
A9: e in  {  [:a,b:] where a is Element of A, b is Element of B : ( a in A & b in B )  }  by TARSKI:def_4;
consider a9 being Element of A, b9 being Element of B such that
A10: [:a9,b9:] = e and
 a9 in A and
 b9 in B by A9;
consider x, y being set  such that
A11: ( x in a9 & y in b9 ) and
A12: z = [x,y] by A8, A10, ZFMISC_1:def_2;
 ( x in union A & y in union B ) by A11, TARSKI:def_4;
hence  z in [:(union A),(union B):] by A12, ZFMISC_1:def_2; ::_thesis: verum
end;

theorem Th3: :: LATTICE5:3
for A being non empty set st A is c=-linear holds
[:(union A),(union A):] = union  {  [:a,a:] where a is Element of A : a in A  }
proof
let A be non empty set ; ::_thesis: ( A is c=-linear implies [:(union A),(union A):] = union  {  [:a,a:] where a is Element of A : a in A  }  )
set X =  {  [:a,a:] where a is Element of A : a in A  }  ;
set Y =  {  [:a,b:] where a, b is Element of A : ( a in A & b in A )  }  ;
assume A1: A is c=-linear ; ::_thesis: [:(union A),(union A):] = union  {  [:a,a:] where a is Element of A : a in A  }
A2: union  {  [:a,b:] where a, b is Element of A : ( a in A & b in A )  }  c= union  {  [:a,a:] where a is Element of A : a in A  }
proof
let Z be set ; :: according to TARSKI:def_3 ::_thesis: ( not Z in union  {  [:a,b:] where a, b is Element of A : ( a in A & b in A )  }  or Z in union  {  [:a,a:] where a is Element of A : a in A  }  )
assume  Z in union  {  [:a,b:] where a, b is Element of A : ( a in A & b in A )  }  ; ::_thesis: Z in union  {  [:a,a:] where a is Element of A : a in A  }
then consider z being set  such that
A3: Z in z and
A4: z in  {  [:a,b:] where a, b is Element of A : ( a in A & b in A )  }  by TARSKI:def_4;
consider a, b being Element of A such that
A5: z = [:a,b:] and
 a in A and
 b in A by A4;
A6: a,b are_c=-comparable by A1, ORDINAL1:def_8;
percases ( a c= b or b c= a ) by A6, XBOOLE_0:def_9;
supposeA7: a c= b ; ::_thesis: Z in union  {  [:a,a:] where a is Element of A : a in A  }
A8: [:b,b:] in  {  [:a,a:] where a is Element of A : a in A  }  ;
 [:a,b:] c= [:b,b:] by A7, ZFMISC_1:95;
hence  Z in union  {  [:a,a:] where a is Element of A : a in A  }  by A3, A5, A8, TARSKI:def_4; ::_thesis: verum
end;
supposeA9: b c= a ; ::_thesis: Z in union  {  [:a,a:] where a is Element of A : a in A  }
A10: [:a,a:] in  {  [:a,a:] where a is Element of A : a in A  }  ;
 [:a,b:] c= [:a,a:] by A9, ZFMISC_1:95;
hence  Z in union  {  [:a,a:] where a is Element of A : a in A  }  by A3, A5, A10, TARSKI:def_4; ::_thesis: verum
end;
end;
end;
  {  [:a,a:] where a is Element of A : a in A  }  c=  {  [:a,b:] where a, b is Element of A : ( a in A & b in A )  }
proof
let Z be set ; :: according to TARSKI:def_3 ::_thesis: ( not Z in  {  [:a,a:] where a is Element of A : a in A  }  or Z in  {  [:a,b:] where a, b is Element of A : ( a in A & b in A )  }  )
assume  Z in  {  [:a,a:] where a is Element of A : a in A  }  ; ::_thesis: Z in  {  [:a,b:] where a, b is Element of A : ( a in A & b in A )  }
then  ex a being Element of A st
( Z = [:a,a:] & a in A ) ;
hence  Z in  {  [:a,b:] where a, b is Element of A : ( a in A & b in A )  }  ; ::_thesis: verum
end;
then  union  {  [:a,a:] where a is Element of A : a in A  }  c= union  {  [:a,b:] where a, b is Element of A : ( a in A & b in A )  }  by ZFMISC_1:77;
then  union  {  [:a,a:] where a is Element of A : a in A  }  = union  {  [:a,b:] where a, b is Element of A : ( a in A & b in A )  }  by A2, XBOOLE_0:def_10;
hence  [:(union A),(union A):] = union  {  [:a,a:] where a is Element of A : a in A  }  by Th2; ::_thesis: verum
end;

begin

definition
let A be set ;
func EqRelLATT A ->  Poset equals :: LATTICE5:def 1
 LattPOSet (EqRelLatt A);
correctness
coherence
LattPOSet (EqRelLatt A) is Poset;
;
end;

:: deftheorem  defines EqRelLATT LATTICE5:def_1_:_
 for A being set holds EqRelLATT A = LattPOSet (EqRelLatt A);

registration
let A be set ;
cluster EqRelLATT A ->  with_suprema with_infima ;
coherence
 ( EqRelLATT A is with_infima & EqRelLATT A is with_suprema ) ;
end;

theorem Th4: :: LATTICE5:4
for A, x being set holds
( x in the carrier of (EqRelLATT A) iff x is Equivalence_Relation of A )
proof
let A, x be set ; ::_thesis: ( x in the carrier of (EqRelLATT A) iff x is Equivalence_Relation of A )
hereby  ::_thesis: ( x is Equivalence_Relation of A implies x in the carrier of (EqRelLATT A) )
assume  x in the carrier of (EqRelLATT A) ; ::_thesis: x is Equivalence_Relation of A
then reconsider e = x as Element of (LattPOSet (EqRelLatt A)) ;
 % e = e ;
then A1: x in the carrier of (EqRelLatt A) ;
 the carrier of (EqRelLatt A) =  {  r where r is Relation of A,A : r is Equivalence_Relation of A  }  by MSUALG_5:def_2;
then  ex x9 being Relation of A,A st
( x9 = x & x9 is Equivalence_Relation of A ) by A1;
hence  x is Equivalence_Relation of A ; ::_thesis: verum
end;
A2: the carrier of (EqRelLatt A) =  {  r where r is Relation of A,A : r is Equivalence_Relation of A  }  by MSUALG_5:def_2;
assume  x is Equivalence_Relation of A ; ::_thesis: x in the carrier of (EqRelLATT A)
then  x in the carrier of (EqRelLatt A) by A2;
then reconsider e = x as Element of (EqRelLatt A) ;
reconsider e = e as Element of (EqRelLATT A) ;
 e in the carrier of (EqRelLATT A) ;
hence  x in the carrier of (EqRelLATT A) ; ::_thesis: verum
end;

theorem Th5: :: LATTICE5:5
for A being set
for x, y being Element of (EqRelLatt A) holds
( x [= y iff x c= y )
proof
let A be set ; ::_thesis: for x, y being Element of (EqRelLatt A) holds
( x [= y iff x c= y )
let x, y be Element of (EqRelLatt A); ::_thesis: ( x [= y iff x c= y )
reconsider x9 = x, y9 = y as Equivalence_Relation of A by MSUALG_5:21;
A1: ( x9 /\ y9 = x9 iff x9 c= y9 ) by XBOOLE_1:17, XBOOLE_1:28;
x "/\" y = the L_meet of (EqRelLatt A) . (x9,y9) by LATTICES:def_2
.= x9 /\ y9 by MSUALG_5:def_2 ;
hence  ( x [= y iff x c= y ) by A1, LATTICES:4; ::_thesis: verum
end;

theorem Th6: :: LATTICE5:6
for A being set
for a, b being Element of (EqRelLATT A) holds
( a <= b iff a c= b )
proof
let A be set ; ::_thesis: for a, b being Element of (EqRelLATT A) holds
( a <= b iff a c= b )
let a, b be Element of (EqRelLATT A); ::_thesis: ( a <= b iff a c= b )
set El = EqRelLatt A;
reconsider a9 = a as Element of (EqRelLatt A) ;
reconsider b9 = b as Element of (EqRelLatt A) ;
thus  ( a <= b implies a c= b ) ::_thesis: ( a c= b implies a <= b )
proof
assume  a <= b ; ::_thesis: a c= b
then  a9 % <= b9 % ;
then  a9 [= b9 by LATTICE3:7;
hence  a c= b by Th5; ::_thesis: verum
end;
thus  ( a c= b implies a <= b ) ::_thesis: verum
proof
assume  a c= b ; ::_thesis: a <= b
then  a9 [= b9 by Th5;
then  a9 % <= b9 % by LATTICE3:7;
hence  a <= b ; ::_thesis: verum
end;
end;

theorem Th7: :: LATTICE5:7
for L being Lattice
for a, b being Element of (LattPOSet L) holds a "/\" b = (% a) "/\" (% b)
proof
let L be Lattice; ::_thesis: for a, b being Element of (LattPOSet L) holds a "/\" b = (% a) "/\" (% b)
let a, b be Element of (LattPOSet L); ::_thesis: a "/\" b = (% a) "/\" (% b)
reconsider x = a, y = b as Element of L ;
set c = x "/\" y;
A1: x "/\" y [= x by LATTICES:6;
A2: x "/\" y [= y by LATTICES:6;
A3: (x "/\" y) % = x "/\" y ;
reconsider c = x "/\" y as Element of (LattPOSet L) ;
A4: y % = y ;
then A5: c <= b by A2, A3, LATTICE3:7;
A6: x % = x ;
A7: for d being Element of (LattPOSet L) st d <= a & d <= b holds
d <= c
proof
let d be Element of (LattPOSet L); ::_thesis: ( d <= a & d <= b implies d <= c )
reconsider z = d as Element of L ;
A8: z % = z ;
assume  ( d <= a & d <= b ) ; ::_thesis: d <= c
then  ( z [= x & z [= y ) by A6, A4, A8, LATTICE3:7;
then  z [= x "/\" y by FILTER_0:7;
hence  d <= c by A3, A8, LATTICE3:7; ::_thesis: verum
end;
 c <= a by A1, A3, A6, LATTICE3:7;
hence  a "/\" b = (% a) "/\" (% b) by A5, A7, YELLOW_0:23; ::_thesis: verum
end;

theorem Th8: :: LATTICE5:8
for A being set
for a, b being Element of (EqRelLATT A) holds a "/\" b = a /\ b
proof
let A be set ; ::_thesis: for a, b being Element of (EqRelLATT A) holds a "/\" b = a /\ b
let a, b be Element of (EqRelLATT A); ::_thesis: a "/\" b = a /\ b
A1: now__::_thesis:_for_x,_y_being_Element_of_(EqRelLatt_A)_holds_x_"/\"_y_=_x_/\_y
let x, y be Element of (EqRelLatt A); ::_thesis: x "/\" y = x /\ y
reconsider e1 = x as Equivalence_Relation of A by MSUALG_5:21;
reconsider e2 = y as Equivalence_Relation of A by MSUALG_5:21;
thus x "/\" y = the L_meet of (EqRelLatt A) . (e1,e2) by LATTICES:def_2
.= x /\ y by MSUALG_5:def_2 ; ::_thesis: verum
end;
reconsider y = b as Element of (LattPOSet (EqRelLatt A)) ;
reconsider x = a as Element of (LattPOSet (EqRelLatt A)) ;
reconsider x = x as Element of (EqRelLatt A) ;
reconsider y = y as Element of (EqRelLatt A) ;
 ( % (x %) = x % & % (y %) = y % ) ;
hence a "/\" b = x "/\" y by Th7
.= a /\ b by A1 ;
::_thesis: verum
end;

theorem Th9: :: LATTICE5:9
for L being Lattice
for a, b being Element of (LattPOSet L) holds a "\/" b = (% a) "\/" (% b)
proof
let L be Lattice; ::_thesis: for a, b being Element of (LattPOSet L) holds a "\/" b = (% a) "\/" (% b)
let a, b be Element of (LattPOSet L); ::_thesis: a "\/" b = (% a) "\/" (% b)
reconsider x = a, y = b as Element of L ;
set c = x "\/" y;
A1: (x "\/" y) % = x "\/" y ;
A2: ( y [= x "\/" y & y % = y ) by LATTICES:5;
A3: ( x [= x "\/" y & x % = x ) by LATTICES:5;
reconsider c = x "\/" y as Element of (LattPOSet L) ;
A4: b <= c by A1, A2, LATTICE3:7;
A5: for d being Element of (LattPOSet L) st a <= d & b <= d holds
c <= d
proof
let d be Element of (LattPOSet L); ::_thesis: ( a <= d & b <= d implies c <= d )
assume that
A6: a <= d and
A7: b <= d ; ::_thesis: c <= d
reconsider z = d as Element of L ;
 y % <= z % by A7;
then A8: y [= z by LATTICE3:7;
 x % <= z % by A6;
then  x [= z by LATTICE3:7;
then  x "\/" y [= z by A8, FILTER_0:6;
then  (x "\/" y) % <= z % by LATTICE3:7;
hence  c <= d ; ::_thesis: verum
end;
 a <= c by A1, A3, LATTICE3:7;
hence  a "\/" b = (% a) "\/" (% b) by A4, A5, YELLOW_0:22; ::_thesis: verum
end;

theorem Th10: :: LATTICE5:10
for A being set
for a, b being Element of (EqRelLATT A)
for E1, E2 being Equivalence_Relation of A st a = E1 & b = E2 holds
a "\/" b = E1 "\/" E2
proof
let A be set ; ::_thesis: for a, b being Element of (EqRelLATT A)
for E1, E2 being Equivalence_Relation of A st a = E1 & b = E2 holds
a "\/" b = E1 "\/" E2
let a, b be Element of (EqRelLATT A); ::_thesis: for E1, E2 being Equivalence_Relation of A st a = E1 & b = E2 holds
a "\/" b = E1 "\/" E2
let E1, E2 be Equivalence_Relation of A; ::_thesis: ( a = E1 & b = E2 implies a "\/" b = E1 "\/" E2 )
assume A1: ( a = E1 & b = E2 ) ; ::_thesis: a "\/" b = E1 "\/" E2
reconsider y = b as Element of (LattPOSet (EqRelLatt A)) ;
reconsider x = a as Element of (LattPOSet (EqRelLatt A)) ;
reconsider x = x as Element of (EqRelLatt A) ;
reconsider y = y as Element of (EqRelLatt A) ;
 ( % (x %) = x % & % (y %) = y % ) ;
hence a "\/" b = x "\/" y by Th9
.= the L_join of (EqRelLatt A) . (x,y) by LATTICES:def_1
.= E1 "\/" E2 by A1, MSUALG_5:def_2 ;
::_thesis: verum
end;

definition
let L be non empty RelStr ;
redefine attr L is complete  means :: LATTICE5:def 2
 for X being Subset of L ex a being Element of L st
( a is_<=_than X & ( for b being Element of L st b is_<=_than X holds
b <= a ) );
compatibility
 ( L is complete iff for X being Subset of L ex a being Element of L st
( a is_<=_than X & ( for b being Element of L st b is_<=_than X holds
b <= a ) ) )
proof
hereby  ::_thesis: ( ( for X being Subset of L ex a being Element of L st
( a is_<=_than X & ( for b being Element of L st b is_<=_than X holds
b <= a ) ) ) implies L is complete )
assume A1: L is complete ; ::_thesis: for X being Subset of L ex p being Element of L st
( p is_<=_than X & ( for b being Element of L st b is_<=_than X holds
b <= p ) )
let X be Subset of L; ::_thesis: ex p being Element of L st
( p is_<=_than X & ( for b being Element of L st b is_<=_than X holds
b <= p ) )
set Y =  {  c where c is Element of L : c is_<=_than X  }  ;
consider p being Element of L such that
A2:  {  c where c is Element of L : c is_<=_than X  }  is_<=_than p and
A3: for r being Element of L st  {  c where c is Element of L : c is_<=_than X  }  is_<=_than r holds
p <= r by A1, LATTICE3:def_12;
take p = p; ::_thesis: ( p is_<=_than X & ( for b being Element of L st b is_<=_than X holds
b <= p ) )
thus  p is_<=_than X ::_thesis: for b being Element of L st b is_<=_than X holds
b <= p
proof
let q be Element of L; :: according to LATTICE3:def_8 ::_thesis: ( not q in X or p <= q )
assume A4: q in X ; ::_thesis: p <= q
  {  c where c is Element of L : c is_<=_than X  }  is_<=_than q
proof
let s be Element of L; :: according to LATTICE3:def_9 ::_thesis: ( not s in  {  c where c is Element of L : c is_<=_than X  }  or s <= q )
assume  s in  {  c where c is Element of L : c is_<=_than X  }  ; ::_thesis: s <= q
then  ex t being Element of L st
( s = t & t is_<=_than X ) ;
hence  s <= q by A4, LATTICE3:def_8; ::_thesis: verum
end;
hence  p <= q by A3; ::_thesis: verum
end;
let b be Element of L; ::_thesis: ( b is_<=_than X implies b <= p )
assume  b is_<=_than X ; ::_thesis: b <= p
then  b in  {  c where c is Element of L : c is_<=_than X  }  ;
hence  b <= p by A2, LATTICE3:def_9; ::_thesis: verum
end;
assume A5: for X being Subset of L ex a being Element of L st
( a is_<=_than X & ( for b being Element of L st b is_<=_than X holds
b <= a ) ) ; ::_thesis: L is complete
let X be set ; :: according to LATTICE3:def_12 ::_thesis: ex b1 being Element of the carrier of L st
( X is_<=_than b1 & ( for b2 being Element of the carrier of L holds
( not X is_<=_than b2 or b1 <= b2 ) ) )
set Y =  {  c where c is Element of L : X is_<=_than c  }  ;
  {  c where c is Element of L : X is_<=_than c  }  c= the carrier of L
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in  {  c where c is Element of L : X is_<=_than c  }  or x in the carrier of L )
assume  x in  {  c where c is Element of L : X is_<=_than c  }  ; ::_thesis: x in the carrier of L
then  ex c being Element of L st
( x = c & X is_<=_than c ) ;
hence  x in the carrier of L ; ::_thesis: verum
end;
then consider p being Element of L such that
A6: p is_<=_than  {  c where c is Element of L : X is_<=_than c  }  and
A7: for r being Element of L st r is_<=_than  {  c where c is Element of L : X is_<=_than c  }  holds
r <= p by A5;
take  p ; ::_thesis: ( X is_<=_than p & ( for b1 being Element of the carrier of L holds
( not X is_<=_than b1 or p <= b1 ) ) )
thus  X is_<=_than p ::_thesis: for b1 being Element of the carrier of L holds
( not X is_<=_than b1 or p <= b1 )
proof
let q be Element of L; :: according to LATTICE3:def_9 ::_thesis: ( not q in X or q <= p )
assume A8: q in X ; ::_thesis: q <= p
 q is_<=_than  {  c where c is Element of L : X is_<=_than c  }
proof
let s be Element of L; :: according to LATTICE3:def_8 ::_thesis: ( not s in  {  c where c is Element of L : X is_<=_than c  }  or q <= s )
assume  s in  {  c where c is Element of L : X is_<=_than c  }  ; ::_thesis: q <= s
then  ex t being Element of L st
( s = t & X is_<=_than t ) ;
hence  q <= s by A8, LATTICE3:def_9; ::_thesis: verum
end;
hence  q <= p by A7; ::_thesis: verum
end;
let r be Element of L; ::_thesis: ( not X is_<=_than r or p <= r )
assume  X is_<=_than r ; ::_thesis: p <= r
then  r in  {  c where c is Element of L : X is_<=_than c  }  ;
hence  p <= r by A6, LATTICE3:def_8; ::_thesis: verum
end;
end;

:: deftheorem  defines complete LATTICE5:def_2_:_
 for L being non empty RelStr holds
( L is complete iff for X being Subset of L ex a being Element of L st
( a is_<=_than X & ( for b being Element of L st b is_<=_than X holds
b <= a ) ) );

registration
let A be set ;
cluster EqRelLATT A ->  complete ;
coherence
 EqRelLATT A is complete
proof
let X be Subset of (EqRelLATT A); :: according to LATTICE5:def_2 ::_thesis: ex a being Element of (EqRelLATT A) st
( a is_<=_than X & ( for b being Element of (EqRelLATT A) st b is_<=_than X holds
b <= a ) )
set B = X /\ the carrier of (EqRelLATT A);
 X /\ the carrier of (EqRelLATT A) c= bool [:A,A:]
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in X /\ the carrier of (EqRelLATT A) or x in bool [:A,A:] )
assume  x in X /\ the carrier of (EqRelLATT A) ; ::_thesis: x in bool [:A,A:]
then  x is Equivalence_Relation of A by Th4;
hence  x in bool [:A,A:] ; ::_thesis: verum
end;
then reconsider B = X /\ the carrier of (EqRelLATT A) as Subset-Family of [:A,A:] ;
consider b being Subset of [:A,A:] such that
A1: b = Intersect B ;
 for x being set st x in A holds
[x,x] in b
proof
let x be set ; ::_thesis: ( x in A implies [x,x] in b )
assume A2: x in A ; ::_thesis: [x,x] in b
A3: for Y being set st Y in B holds
[x,x] in Y
proof
let Y be set ; ::_thesis: ( Y in B implies [x,x] in Y )
assume  Y in B ; ::_thesis: [x,x] in Y
then  Y is Equivalence_Relation of A by Th4;
hence  [x,x] in Y by A2, EQREL_1:5; ::_thesis: verum
end;
 [x,x] in [:A,A:] by A2, ZFMISC_1:def_2;
hence  [x,x] in b by A1, A3, SETFAM_1:43; ::_thesis: verum
end;
then A4: b is_reflexive_in A by RELAT_2:def_1;
reconsider b = b as Relation of A ;
A5: ( dom b = A & field b = A ) by A4, ORDERS_1:13;
 for x, y, z being set st x in A & y in A & z in A & [x,y] in b & [y,z] in b holds
[x,z] in b
proof
let x, y, z be set ; ::_thesis: ( x in A & y in A & z in A & [x,y] in b & [y,z] in b implies [x,z] in b )
assume that
A6: x in A and
 y in A and
A7: z in A and
A8: ( [x,y] in b & [y,z] in b ) ; ::_thesis: [x,z] in b
A9: for Y being set st Y in B holds
[x,z] in Y
proof
let Y be set ; ::_thesis: ( Y in B implies [x,z] in Y )
assume A10: Y in B ; ::_thesis: [x,z] in Y
then A11: Y is Equivalence_Relation of A by Th4;
 ( [x,y] in Y & [y,z] in Y ) by A1, A8, A10, SETFAM_1:43;
hence  [x,z] in Y by A11, EQREL_1:7; ::_thesis: verum
end;
 [x,z] in [:A,A:] by A6, A7, ZFMISC_1:def_2;
hence  [x,z] in b by A1, A9, SETFAM_1:43; ::_thesis: verum
end;
then A12: b is_transitive_in A by RELAT_2:def_8;
 for x, y being set st x in A & y in A & [x,y] in b holds
[y,x] in b
proof
let x, y be set ; ::_thesis: ( x in A & y in A & [x,y] in b implies [y,x] in b )
assume that
A13: ( x in A & y in A ) and
A14: [x,y] in b ; ::_thesis: [y,x] in b
A15: for Y being set st Y in B holds
[y,x] in Y
proof
let Y be set ; ::_thesis: ( Y in B implies [y,x] in Y )
assume  Y in B ; ::_thesis: [y,x] in Y
then  ( [x,y] in Y & Y is Equivalence_Relation of A ) by A1, A14, Th4, SETFAM_1:43;
hence  [y,x] in Y by EQREL_1:6; ::_thesis: verum
end;
 [y,x] in [:A,A:] by A13, ZFMISC_1:def_2;
hence  [y,x] in b by A1, A15, SETFAM_1:43; ::_thesis: verum
end;
then  b is_symmetric_in A by RELAT_2:def_3;
then reconsider b = b as Equivalence_Relation of A by A5, A12, PARTFUN1:def_2, RELAT_2:def_11, RELAT_2:def_16;
reconsider b = b as Element of (EqRelLATT A) by Th4;
take  b ; ::_thesis: ( b is_<=_than X & ( for b being Element of (EqRelLATT A) st b is_<=_than X holds
b <= b ) )
now__::_thesis:_for_a_being_Element_of_(EqRelLATT_A)_st_a_in_X_/\_the_carrier_of_(EqRelLATT_A)_holds_
b_<=_a
let a be Element of (EqRelLATT A); ::_thesis: ( a in X /\ the carrier of (EqRelLATT A) implies b <= a )
reconsider a9 = a as Equivalence_Relation of A by Th4;
reconsider b9 = b as Equivalence_Relation of A ;
assume  a in X /\ the carrier of (EqRelLATT A) ; ::_thesis: b <= a
then  for x, y being set st [x,y] in b9 holds
[x,y] in a9 by A1, SETFAM_1:43;
then  b9 c= a9 by RELAT_1:def_3;
hence  b <= a by Th6; ::_thesis: verum
end;
then  b is_<=_than X /\ the carrier of (EqRelLATT A) by LATTICE3:def_8;
hence  b is_<=_than X by YELLOW_0:5; ::_thesis: for b being Element of (EqRelLATT A) st b is_<=_than X holds
b <= b
let a be Element of (EqRelLATT A); ::_thesis: ( a is_<=_than X implies a <= b )
reconsider a9 = a as Equivalence_Relation of A by Th4;
assume  a is_<=_than X ; ::_thesis: a <= b
then A16: a is_<=_than X /\ the carrier of (EqRelLATT A) by YELLOW_0:5;
A17: for d being Element of (EqRelLATT A) st d in B holds
a9 c= d
proof
let d be Element of (EqRelLATT A); ::_thesis: ( d in B implies a9 c= d )
assume  d in B ; ::_thesis: a9 c= d
then  a <= d by A16, LATTICE3:def_8;
hence  a9 c= d by Th6; ::_thesis: verum
end;
 a9 c= Intersect B
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in a9 or x in Intersect B )
assume A18: x in a9 ; ::_thesis: x in Intersect B
 for Y being set st Y in B holds
x in Y
proof
let Y be set ; ::_thesis: ( Y in B implies x in Y )
assume  Y in B ; ::_thesis: x in Y
then  a9 c= Y by A17;
hence  x in Y by A18; ::_thesis: verum
end;
hence  x in Intersect B by A18, SETFAM_1:43; ::_thesis: verum
end;
hence  a <= b by A1, Th6; ::_thesis: verum
end;
end;

begin

registration
let L1, L2 be LATTICE;
cluster Relation-like the carrier of L1 -defined the carrier of L2 -valued Function-like quasi_total meet-preserving join-preserving for Element of bool [: the carrier of L1, the carrier of L2:];
existence
 ex b1 being Function of L1,L2 st
( b1 is meet-preserving & b1 is join-preserving )
proof
set z = the Element of L2;
reconsider f = the carrier of L1 --> the Element of L2 as Function of L1,L2 ;
take  f ; ::_thesis: ( f is meet-preserving & f is join-preserving )
 for x, y being Element of L1 holds f . (x "/\" y) = (f . x) "/\" (f . y)
proof
let x, y be Element of L1; ::_thesis: f . (x "/\" y) = (f . x) "/\" (f . y)
thus f . (x "/\" y) = the Element of L2 by FUNCOP_1:7
.= the Element of L2 "/\" the Element of L2 by YELLOW_5:2
.= (f . x) "/\" the Element of L2 by FUNCOP_1:7
.= (f . x) "/\" (f . y) by FUNCOP_1:7 ; ::_thesis: verum
end;
hence  f is meet-preserving by WAYBEL_6:1; ::_thesis: f is join-preserving
 for x, y being Element of L1 holds f . (x "\/" y) = (f . x) "\/" (f . y)
proof
let x, y be Element of L1; ::_thesis: f . (x "\/" y) = (f . x) "\/" (f . y)
thus f . (x "\/" y) = the Element of L2 by FUNCOP_1:7
.= the Element of L2 "\/" the Element of L2 by YELLOW_5:1
.= (f . x) "\/" the Element of L2 by FUNCOP_1:7
.= (f . x) "\/" (f . y) by FUNCOP_1:7 ; ::_thesis: verum
end;
hence  f is join-preserving by WAYBEL_6:2; ::_thesis: verum
end;
end;

definition
let L1, L2 be LATTICE;
mode Homomorphism of L1,L2 is meet-preserving join-preserving Function of L1,L2;
end;

registration
let L be LATTICE;
cluster strict meet-inheriting join-inheriting for SubRelStr of L;
existence
 ex b1 being SubRelStr of L st
( b1 is meet-inheriting & b1 is join-inheriting & b1 is strict )
proof
set a = the Element of L;
set r = the Relation of { the Element of L};
A1: for x, y being Element of L st x in { the Element of L} & y in { the Element of L} & ex_sup_of {x,y},L holds
sup {x,y} in { the Element of L}
proof
let x, y be Element of L; ::_thesis: ( x in { the Element of L} & y in { the Element of L} & ex_sup_of {x,y},L implies sup {x,y} in { the Element of L} )
assume that
A2: ( x in { the Element of L} & y in { the Element of L} ) and
 ex_sup_of {x,y},L ; ::_thesis: sup {x,y} in { the Element of L}
 ( x = the Element of L & y = the Element of L ) by A2, TARSKI:def_1;
then  sup {x,y} = the Element of L "\/" the Element of L by YELLOW_0:41
.= the Element of L by YELLOW_5:1 ;
hence  sup {x,y} in { the Element of L} by TARSKI:def_1; ::_thesis: verum
end;
 the Relation of { the Element of L} c= the InternalRel of L
proof
let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in the Relation of { the Element of L} or z in the InternalRel of L )
assume  z in the Relation of { the Element of L} ; ::_thesis: z in the InternalRel of L
then consider x, y being set  such that
A3: z = [x,y] and
A4: x in { the Element of L} and
A5: y in { the Element of L} by RELSET_1:2;
 x = the Element of L by A4, TARSKI:def_1;
then A6: z = [ the Element of L, the Element of L] by A3, A5, TARSKI:def_1;
 the Element of L <= the Element of L ;
hence  z in the InternalRel of L by A6, ORDERS_2:def_5; ::_thesis: verum
end;
then reconsider S = RelStr(# { the Element of L}, the Relation of { the Element of L} #) as strict SubRelStr of L by YELLOW_0:def_13;
take  S ; ::_thesis: ( S is meet-inheriting & S is join-inheriting & S is strict )
 for x, y being Element of L st x in { the Element of L} & y in { the Element of L} & ex_inf_of {x,y},L holds
inf {x,y} in { the Element of L}
proof
let x, y be Element of L; ::_thesis: ( x in { the Element of L} & y in { the Element of L} & ex_inf_of {x,y},L implies inf {x,y} in { the Element of L} )
assume that
A7: ( x in { the Element of L} & y in { the Element of L} ) and
 ex_inf_of {x,y},L ; ::_thesis: inf {x,y} in { the Element of L}
 ( x = the Element of L & y = the Element of L ) by A7, TARSKI:def_1;
then  inf {x,y} = the Element of L "/\" the Element of L by YELLOW_0:40
.= the Element of L by YELLOW_5:2 ;
hence  inf {x,y} in { the Element of L} by TARSKI:def_1; ::_thesis: verum
end;
hence  ( S is meet-inheriting & S is join-inheriting & S is strict ) by A1, YELLOW_0:def_16, YELLOW_0:def_17; ::_thesis: verum
end;
end;

definition
let L be non empty RelStr ;
mode Sublattice of L is meet-inheriting join-inheriting SubRelStr of L;
end;

registration
let L1, L2 be LATTICE;
let f be Homomorphism of L1,L2;
cluster Image f ->  meet-inheriting join-inheriting ;
coherence
 ( Image f is meet-inheriting & Image f is join-inheriting )
proof
set S = subrelstr (rng f);
A1: the carrier of (subrelstr (rng f)) = rng f by YELLOW_0:def_15;
A2: dom f = the carrier of L1 by FUNCT_2:def_1;
 for x, y being Element of L2 st x in the carrier of (subrelstr (rng f)) & y in the carrier of (subrelstr (rng f)) & ex_sup_of {x,y},L2 holds
sup {x,y} in the carrier of (subrelstr (rng f))
proof
let x, y be Element of L2; ::_thesis: ( x in the carrier of (subrelstr (rng f)) & y in the carrier of (subrelstr (rng f)) & ex_sup_of {x,y},L2 implies sup {x,y} in the carrier of (subrelstr (rng f)) )
assume that
A3: x in the carrier of (subrelstr (rng f)) and
A4: y in the carrier of (subrelstr (rng f)) and
 ex_sup_of {x,y},L2 ; ::_thesis: sup {x,y} in the carrier of (subrelstr (rng f))
consider a being set  such that
A5: a in dom f and
A6: x = f . a by A1, A3, FUNCT_1:def_3;
consider b being set  such that
A7: b in dom f and
A8: y = f . b by A1, A4, FUNCT_1:def_3;
reconsider a9 = a, b9 = b as Element of L1 by A5, A7;
A9: ( f preserves_sup_of {a9,b9} & ex_sup_of {a9,b9},L1 ) by WAYBEL_0:def_35, YELLOW_0:20;
sup {x,y} = sup (f .: {a9,b9}) by A5, A6, A7, A8, FUNCT_1:60
.= f . (sup {a9,b9}) by A9, WAYBEL_0:def_31 ;
hence  sup {x,y} in the carrier of (subrelstr (rng f)) by A1, A2, FUNCT_1:def_3; ::_thesis: verum
end;
then A10: subrelstr (rng f) is join-inheriting by YELLOW_0:def_17;
 for x, y being Element of L2 st x in the carrier of (subrelstr (rng f)) & y in the carrier of (subrelstr (rng f)) & ex_inf_of {x,y},L2 holds
inf {x,y} in the carrier of (subrelstr (rng f))
proof
let x, y be Element of L2; ::_thesis: ( x in the carrier of (subrelstr (rng f)) & y in the carrier of (subrelstr (rng f)) & ex_inf_of {x,y},L2 implies inf {x,y} in the carrier of (subrelstr (rng f)) )
assume that
A11: x in the carrier of (subrelstr (rng f)) and
A12: y in the carrier of (subrelstr (rng f)) and
 ex_inf_of {x,y},L2 ; ::_thesis: inf {x,y} in the carrier of (subrelstr (rng f))
consider a being set  such that
A13: a in dom f and
A14: x = f . a by A1, A11, FUNCT_1:def_3;
consider b being set  such that
A15: b in dom f and
A16: y = f . b by A1, A12, FUNCT_1:def_3;
reconsider a9 = a, b9 = b as Element of L1 by A13, A15;
A17: ( f preserves_inf_of {a9,b9} & ex_inf_of {a9,b9},L1 ) by WAYBEL_0:def_34, YELLOW_0:21;
inf {x,y} = inf (f .: {a9,b9}) by A13, A14, A15, A16, FUNCT_1:60
.= f . (inf {a9,b9}) by A17, WAYBEL_0:def_30 ;
hence  inf {x,y} in the carrier of (subrelstr (rng f)) by A1, A2, FUNCT_1:def_3; ::_thesis: verum
end;
then  subrelstr (rng f) is meet-inheriting by YELLOW_0:def_16;
hence  ( Image f is meet-inheriting & Image f is join-inheriting ) by A10, YELLOW_2:def_2; ::_thesis: verum
end;
end;


definition
let X be non empty set ;
let f be non empty FinSequence of X;
let x, y be set ;
let R1, R2 be Relation;
predx,y are_joint_by f,R1,R2 means :Def3: :: LATTICE5:def 3
( f . 1 = x & f . (len f) = y & ( for i being Element of NAT st 1 <= i & i < len f holds
( ( i is odd implies [(f . i),(f . (i + 1))] in R1 ) & ( i is even implies [(f . i),(f . (i + 1))] in R2 ) ) ) );
end;

:: deftheorem Def3 defines are_joint_by LATTICE5:def_3_:_
 for X being non empty set
for f being non empty FinSequence of X
for x, y being set
for R1, R2 being Relation holds
( x,y are_joint_by f,R1,R2 iff ( f . 1 = x & f . (len f) = y & ( for i being Element of NAT st 1 <= i & i < len f holds
( ( i is odd implies [(f . i),(f . (i + 1))] in R1 ) & ( i is even implies [(f . i),(f . (i + 1))] in R2 ) ) ) ) );

theorem Th11: :: LATTICE5:11
for X being non empty set
for x being set
for o being Element of NAT
for R1, R2 being Relation
for f being non empty FinSequence of X st R1 is_reflexive_in X & R2 is_reflexive_in X & f = o |-> x holds
x,x are_joint_by f,R1,R2
proof
let X be non empty set ; ::_thesis: for x being set
for o being Element of NAT
for R1, R2 being Relation
for f being non empty FinSequence of X st R1 is_reflexive_in X & R2 is_reflexive_in X & f = o |-> x holds
x,x are_joint_by f,R1,R2
let x be set ; ::_thesis: for o being Element of NAT
for R1, R2 being Relation
for f being non empty FinSequence of X st R1 is_reflexive_in X & R2 is_reflexive_in X & f = o |-> x holds
x,x are_joint_by f,R1,R2
let o be Element of NAT ; ::_thesis: for R1, R2 being Relation
for f being non empty FinSequence of X st R1 is_reflexive_in X & R2 is_reflexive_in X & f = o |-> x holds
x,x are_joint_by f,R1,R2
let R1, R2 be Relation; ::_thesis: for f being non empty FinSequence of X st R1 is_reflexive_in X & R2 is_reflexive_in X & f = o |-> x holds
x,x are_joint_by f,R1,R2
let f be non empty FinSequence of X; ::_thesis: ( R1 is_reflexive_in X & R2 is_reflexive_in X & f = o |-> x implies x,x are_joint_by f,R1,R2 )
assume that
A1: R1 is_reflexive_in X and
A2: R2 is_reflexive_in X and
A3: f = o |-> x ; ::_thesis: x,x are_joint_by f,R1,R2
A4: dom f = Seg o by A3, FUNCOP_1:13;
then A5: f . 1 = x by A3, FINSEQ_5:6, FUNCOP_1:7;
A6: for i being Element of NAT st 1 <= i & i < len f holds
( ( i is odd implies [(f . i),(f . (i + 1))] in R1 ) & ( i is even implies [(f . i),(f . (i + 1))] in R2 ) )
proof
let i be Element of NAT ; ::_thesis: ( 1 <= i & i < len f implies ( ( i is odd implies [(f . i),(f . (i + 1))] in R1 ) & ( i is even implies [(f . i),(f . (i + 1))] in R2 ) ) )
assume that
A7: 1 <= i and
A8: i < len f ; ::_thesis: ( ( i is odd implies [(f . i),(f . (i + 1))] in R1 ) & ( i is even implies [(f . i),(f . (i + 1))] in R2 ) )
A9: ( i is even implies [(f . i),(f . (i + 1))] in R2 )
proof
 ( 1 <= i + 1 & i + 1 <= len f ) by A7, A8, NAT_1:13;
then  i + 1 in Seg (len f) ;
then  i + 1 in Seg o by A3, CARD_1:def_7;
then A10: f . (i + 1) = x by A3, FUNCOP_1:7;
assume  i is even ; ::_thesis: [(f . i),(f . (i + 1))] in R2
 i <= o by A3, A8, CARD_1:def_7;
then  i in Seg o by A7;
then A11: f . i = x by A3, FUNCOP_1:7;
 x in X by A4, A5, FINSEQ_2:11, FINSEQ_5:6;
hence  [(f . i),(f . (i + 1))] in R2 by A2, A10, A11, RELAT_2:def_1; ::_thesis: verum
end;
 ( i is odd implies [(f . i),(f . (i + 1))] in R1 )
proof
 ( 1 <= i + 1 & i + 1 <= len f ) by A7, A8, NAT_1:13;
then  i + 1 in Seg (len f) ;
then  i + 1 in Seg o by A3, CARD_1:def_7;
then A12: f . (i + 1) = x by A3, FUNCOP_1:7;
assume  i is odd ; ::_thesis: [(f . i),(f . (i + 1))] in R1
 i <= o by A3, A8, CARD_1:def_7;
then  i in Seg o by A7;
then A13: f . i = x by A3, FUNCOP_1:7;
 x in X by A4, A5, FINSEQ_2:11, FINSEQ_5:6;
hence  [(f . i),(f . (i + 1))] in R1 by A1, A12, A13, RELAT_2:def_1; ::_thesis: verum
end;
hence  ( ( i is odd implies [(f . i),(f . (i + 1))] in R1 ) & ( i is even implies [(f . i),(f . (i + 1))] in R2 ) ) by A9; ::_thesis: verum
end;
 len f in Seg o by A4, FINSEQ_5:6;
then  f . (len f) = x by A3, FUNCOP_1:7;
hence  x,x are_joint_by f,R1,R2 by A5, A6, Def3; ::_thesis: verum
end;

Lm1: now__::_thesis:_for_i,_n,_m_being_Element_of_NAT_st_1_<=_i_&_i_<_n_+_m_&_not_(_1_<=_i_&_i_<_n_)_&_not_(_n_=_i_&_i_<_n_+_m_)_holds_
(_n_+_1_<=_i_&_i_<_n_+_m_)
let i, n, m be Element of NAT ; ::_thesis: ( 1 <= i & i < n + m & not ( 1 <= i & i < n ) & not ( n = i & i < n + m ) implies ( n + 1 <= i & i < n + m ) )
assume  ( 1 <= i & i < n + m ) ; ::_thesis: ( ( 1 <= i & i < n ) or ( n = i & i < n + m ) or ( n + 1 <= i & i < n + m ) )
then  ( ( 1 <= i & i < n ) or ( n = i & i < n + m ) or ( n < i & i < n + m ) ) by XXREAL_0:1;
hence  ( ( 1 <= i & i < n ) or ( n = i & i < n + m ) or ( n + 1 <= i & i < n + m ) ) by NAT_1:13; ::_thesis: verum
end;

theorem Th12: :: LATTICE5:12
for X being non empty set
for x, y being set
for R1, R2 being Relation
for n, m being Element of NAT st n <= m & R1 is_reflexive_in X & R2 is_reflexive_in X & ex f being non empty FinSequence of X st
( len f = n & x,y are_joint_by f,R1,R2 ) holds
ex h being non empty FinSequence of X st
( len h = m & x,y are_joint_by h,R1,R2 )
proof
let X be non empty set ; ::_thesis: for x, y being set
for R1, R2 being Relation
for n, m being Element of NAT st n <= m & R1 is_reflexive_in X & R2 is_reflexive_in X & ex f being non empty FinSequence of X st
( len f = n & x,y are_joint_by f,R1,R2 ) holds
ex h being non empty FinSequence of X st
( len h = m & x,y are_joint_by h,R1,R2 )
let x, y be set ; ::_thesis: for R1, R2 being Relation
for n, m being Element of NAT st n <= m & R1 is_reflexive_in X & R2 is_reflexive_in X & ex f being non empty FinSequence of X st
( len f = n & x,y are_joint_by f,R1,R2 ) holds
ex h being non empty FinSequence of X st
( len h = m & x,y are_joint_by h,R1,R2 )
let R1, R2 be Relation; ::_thesis: for n, m being Element of NAT st n <= m & R1 is_reflexive_in X & R2 is_reflexive_in X & ex f being non empty FinSequence of X st
( len f = n & x,y are_joint_by f,R1,R2 ) holds
ex h being non empty FinSequence of X st
( len h = m & x,y are_joint_by h,R1,R2 )
let n, m be Element of NAT ; ::_thesis: ( n <= m & R1 is_reflexive_in X & R2 is_reflexive_in X & ex f being non empty FinSequence of X st
( len f = n & x,y are_joint_by f,R1,R2 ) implies ex h being non empty FinSequence of X st
( len h = m & x,y are_joint_by h,R1,R2 ) )
assume that
A1: n <= m and
A2: R1 is_reflexive_in X and
A3: R2 is_reflexive_in X ; ::_thesis: ( for f being non empty FinSequence of X holds
( not len f = n or not x,y are_joint_by f,R1,R2 ) or ex h being non empty FinSequence of X st
( len h = m & x,y are_joint_by h,R1,R2 ) )
given f being non empty FinSequence of X such that A4: len f = n and
A5: x,y are_joint_by f,R1,R2 ; ::_thesis: ex h being non empty FinSequence of X st
( len h = m & x,y are_joint_by h,R1,R2 )
A6: f . (len f) = y by A5, Def3;
percases ( n < m or n = m ) by A1, XXREAL_0:1;
supposeA7: n < m ; ::_thesis: ex h being non empty FinSequence of X st
( len h = m & x,y are_joint_by h,R1,R2 )
 len f in dom f by FINSEQ_5:6;
then  y in rng f by A6, FUNCT_1:def_3;
then reconsider y9 = y as Element of X ;
reconsider i = m - n as Element of NAT by A1, INT_1:5;
reconsider g = i |-> y9 as FinSequence of X ;
 i > 0 by A7, XREAL_1:50;
then reconsider g = g as non empty FinSequence of X ;
A8: 1 in dom g by FINSEQ_5:6;
reconsider h = f ^ g as non empty FinSequence of X ;
take  h ; ::_thesis: ( len h = m & x,y are_joint_by h,R1,R2 )
A9: len g = m - n by CARD_1:def_7;
A10: y,y are_joint_by g,R1,R2 by A2, A3, Th11;
thus  len h = (len f) + (len g) by FINSEQ_1:22
.= n + (m - n) by A4, CARD_1:def_7
.= m ; ::_thesis: x,y are_joint_by h,R1,R2
A11: len g in dom g by FINSEQ_5:6;
thus  x,y are_joint_by h,R1,R2 ::_thesis: verum
proof
 rng f <> {} ;
then  1 in dom f by FINSEQ_3:32;
hence h . 1 = f . 1 by FINSEQ_1:def_7
.= x by A5, Def3 ;
:: according to LATTICE5:def_3 ::_thesis: ( h . (len h) = y & ( for i being Element of NAT st 1 <= i & i < len h holds
( ( i is odd implies [(h . i),(h . (i + 1))] in R1 ) & ( i is even implies [(h . i),(h . (i + 1))] in R2 ) ) ) )
thus h . (len h) = h . ((len f) + (len g)) by FINSEQ_1:22
.= g . (len g) by A11, FINSEQ_1:def_7
.= y by A10, Def3 ; ::_thesis: for i being Element of NAT st 1 <= i & i < len h holds
( ( i is odd implies [(h . i),(h . (i + 1))] in R1 ) & ( i is even implies [(h . i),(h . (i + 1))] in R2 ) )
let j be Element of NAT ; ::_thesis: ( 1 <= j & j < len h implies ( ( j is odd implies [(h . j),(h . (j + 1))] in R1 ) & ( j is even implies [(h . j),(h . (j + 1))] in R2 ) ) )
A12: dom f = Seg (len f) by FINSEQ_1:def_3;
assume A13: ( 1 <= j & j < len h ) ; ::_thesis: ( ( j is odd implies [(h . j),(h . (j + 1))] in R1 ) & ( j is even implies [(h . j),(h . (j + 1))] in R2 ) )
thus  ( j is odd implies [(h . j),(h . (j + 1))] in R1 ) ::_thesis: ( j is even implies [(h . j),(h . (j + 1))] in R2 )
proof
assume A14: j is odd ; ::_thesis: [(h . j),(h . (j + 1))] in R1
percases ( ( 1 <= j & j < len f ) or j = len f or ( (len f) + 1 <= j & j < (len f) + (len g) ) ) by A13, Lm1, FINSEQ_1:22;
supposeA15: ( 1 <= j & j < len f ) ; ::_thesis: [(h . j),(h . (j + 1))] in R1
then  ( 1 <= j + 1 & j + 1 <= len f ) by NAT_1:13;
then  j + 1 in dom f by A12;
then A16: f . (j + 1) = h . (j + 1) by FINSEQ_1:def_7;
 j in dom f by A12, A15;
then  f . j = h . j by FINSEQ_1:def_7;
hence  [(h . j),(h . (j + 1))] in R1 by A5, A14, A15, A16, Def3; ::_thesis: verum
end;
supposeA17: j = len f ; ::_thesis: [(h . j),(h . (j + 1))] in R1
then  j in dom f by FINSEQ_5:6;
then A18: h . j = y by A6, A17, FINSEQ_1:def_7;
h . (j + 1) = g . 1 by A8, A17, FINSEQ_1:def_7
.= y by A10, Def3 ;
hence  [(h . j),(h . (j + 1))] in R1 by A2, A18, RELAT_2:def_1; ::_thesis: verum
end;
supposeA19: ( (len f) + 1 <= j & j < (len f) + (len g) ) ; ::_thesis: [(h . j),(h . (j + 1))] in R1
then  j + 1 <= (len f) + (len g) by NAT_1:13;
then A20: j + 1 <= len h by FINSEQ_1:22;
A21: 1 <= j - (len f) by A19, XREAL_1:19;
then  0 < j - (len f) by XXREAL_0:2;
then A22: 0 + (len f) < (j - (len f)) + (len f) by XREAL_1:6;
then reconsider k = j - (len f) as Element of NAT by INT_1:5;
A23: j - (len f) < ((len f) + (len g)) - (len f) by A19, XREAL_1:9;
then A24: k + 1 <= len g by NAT_1:13;
 j < j + 1 by XREAL_1:29;
then  len f < j + 1 by A22, XXREAL_0:2;
then A25: h . (j + 1) = g . ((j + 1) - (len f)) by A20, FINSEQ_1:24
.= g . (k + 1) ;
 1 <= k + 1 by A21, NAT_1:13;
then  k + 1 in Seg (len g) by A24;
then A26: g . (k + 1) = y by A9, FUNCOP_1:7;
 k in Seg (len g) by A21, A23;
then  g . k = y by A9, FUNCOP_1:7;
then  h . j = y by A19, FINSEQ_1:23;
hence  [(h . j),(h . (j + 1))] in R1 by A2, A26, A25, RELAT_2:def_1; ::_thesis: verum
end;
end;
end;
assume A27: j is even ; ::_thesis: [(h . j),(h . (j + 1))] in R2
percases ( ( 1 <= j & j < len f ) or j = len f or ( (len f) + 1 <= j & j < (len f) + (len g) ) ) by A13, Lm1, FINSEQ_1:22;
supposeA28: ( 1 <= j & j < len f ) ; ::_thesis: [(h . j),(h . (j + 1))] in R2
then  ( 1 <= j + 1 & j + 1 <= len f ) by NAT_1:13;
then  j + 1 in dom f by A12;
then A29: f . (j + 1) = h . (j + 1) by FINSEQ_1:def_7;
 j in dom f by A12, A28;
then  f . j = h . j by FINSEQ_1:def_7;
hence  [(h . j),(h . (j + 1))] in R2 by A5, A27, A28, A29, Def3; ::_thesis: verum
end;
supposeA30: j = len f ; ::_thesis: [(h . j),(h . (j + 1))] in R2
then  j in dom f by FINSEQ_5:6;
then A31: h . j = y by A6, A30, FINSEQ_1:def_7;
h . (j + 1) = g . 1 by A8, A30, FINSEQ_1:def_7
.= y by A10, Def3 ;
hence  [(h . j),(h . (j + 1))] in R2 by A3, A31, RELAT_2:def_1; ::_thesis: verum
end;
supposeA32: ( (len f) + 1 <= j & j < (len f) + (len g) ) ; ::_thesis: [(h . j),(h . (j + 1))] in R2
then  j + 1 <= (len f) + (len g) by NAT_1:13;
then A33: j + 1 <= len h by FINSEQ_1:22;
A34: 1 <= j - (len f) by A32, XREAL_1:19;
then  0 < j - (len f) by XXREAL_0:2;
then A35: 0 + (len f) < (j - (len f)) + (len f) by XREAL_1:6;
then reconsider k = j - (len f) as Element of NAT by INT_1:5;
A36: j - (len f) < ((len f) + (len g)) - (len f) by A32, XREAL_1:9;
then A37: k + 1 <= len g by NAT_1:13;
 j < j + 1 by XREAL_1:29;
then  len f < j + 1 by A35, XXREAL_0:2;
then A38: h . (j + 1) = g . ((j + 1) - (len f)) by A33, FINSEQ_1:24
.= g . (k + 1) ;
 1 <= k + 1 by A34, NAT_1:13;
then  k + 1 in Seg (len g) by A37;
then A39: g . (k + 1) = y by A9, FUNCOP_1:7;
 k in Seg (len g) by A34, A36;
then  g . k = y by A9, FUNCOP_1:7;
then  h . j = y by A32, FINSEQ_1:23;
hence  [(h . j),(h . (j + 1))] in R2 by A3, A39, A38, RELAT_2:def_1; ::_thesis: verum
end;
end;
end;
end;
suppose n = m ; ::_thesis: ex h being non empty FinSequence of X st
( len h = m & x,y are_joint_by h,R1,R2 )
hence  ex h being non empty FinSequence of X st
( len h = m & x,y are_joint_by h,R1,R2 ) by A4, A5; ::_thesis: verum
end;
end;
end;

definition
let X be non empty set ;
let Y be Sublattice of EqRelLATT X;
given e being Equivalence_Relation of X such that A1: e in the carrier of Y and
A2: e <> id X ;
given o being Element of NAT  such that A3: for e1, e2 being Equivalence_Relation of X
for x, y being set st e1 in the carrier of Y & e2 in the carrier of Y & [x,y] in e1 "\/" e2 holds
ex F being non empty FinSequence of X st
( len F = o & x,y are_joint_by F,e1,e2 ) ;
func type_of Y ->  Element of NAT  means :Def4: :: LATTICE5:def 4
( ( for e1, e2 being Equivalence_Relation of X
for x, y being set st e1 in the carrier of Y & e2 in the carrier of Y & [x,y] in e1 "\/" e2 holds
ex F being non empty FinSequence of X st
( len F = it + 2 & x,y are_joint_by F,e1,e2 ) ) & ex e1, e2 being Equivalence_Relation of X ex x, y being set st
( e1 in the carrier of Y & e2 in the carrier of Y & [x,y] in e1 "\/" e2 & ( for F being non empty FinSequence of X holds
( not len F = it + 1 or not x,y are_joint_by F,e1,e2 ) ) ) );
existence
 ex b1 being Element of NAT st
( ( for e1, e2 being Equivalence_Relation of X
for x, y being set st e1 in the carrier of Y & e2 in the carrier of Y & [x,y] in e1 "\/" e2 holds
ex F being non empty FinSequence of X st
( len F = b1 + 2 & x,y are_joint_by F,e1,e2 ) ) & ex e1, e2 being Equivalence_Relation of X ex x, y being set st
( e1 in the carrier of Y & e2 in the carrier of Y & [x,y] in e1 "\/" e2 & ( for F being non empty FinSequence of X holds
( not len F = b1 + 1 or not x,y are_joint_by F,e1,e2 ) ) ) )
proof
defpred S1[ Element of NAT ] means  for e1, e2 being Equivalence_Relation of X
for x, y being set st e1 in the carrier of Y & e2 in the carrier of Y & [x,y] in e1 "\/" e2 holds
ex F being non empty FinSequence of X st
( len F = $1 + 2 & x,y are_joint_by F,e1,e2 );
set A =  {  n where n is Element of NAT : S1[n]  }  ;
consider e1, e2 being Equivalence_Relation of X such that
A4: ( e1 = e & e2 = e ) ;
A5: field e = X by EQREL_1:9;
then  id X c= e by RELAT_2:1;
then  not e c= id X by A2, XBOOLE_0:def_10;
then consider x, y being set  such that
A6: [x,y] in e and
A7: not [x,y] in id X by RELAT_1:def_3;
A8: ( not x in X or x <> y ) by A7, RELAT_1:def_10;
A9: [x,y] in e1 "\/" e2 by A6, A4;
then consider F being non empty FinSequence of X such that
A10: len F = o and
A11: x,y are_joint_by F,e1,e2 by A1, A3, A4;
A12: ( F . 1 = x & F . (len F) = y ) by A11, Def3;
 o >= 2
proof
assume  not o >= 2 ; ::_thesis: contradiction
then  len F < 1 + 1 by A10;
then  ( 0 <= len F & len F <= 0 + 1 ) by NAT_1:2, NAT_1:13;
hence  contradiction by A5, A6, A8, A12, NAT_1:9, RELAT_1:15; ::_thesis: verum
end;
then consider o9 being Nat such that
A13: o = 2 + o9 by NAT_1:10;
A14:  {  n where n is Element of NAT : S1[n]  }  is Subset of NAT from DOMAIN_1:sch_7();
 o9 in NAT by ORDINAL1:def_12;
then consider k being Element of NAT  such that
 k = o9 and
A15: for e1, e2 being Equivalence_Relation of X
for x, y being set st e1 in the carrier of Y & e2 in the carrier of Y & [x,y] in e1 "\/" e2 holds
ex F being non empty FinSequence of X st
( len F = k + 2 & x,y are_joint_by F,e1,e2 ) by A3, A13;
 k in  {  n where n is Element of NAT : S1[n]  }  by A15;
then reconsider A =  {  n where n is Element of NAT : S1[n]  }  as non empty Subset of NAT by A14;
set m = min A;
A16: ex e1, e2 being Equivalence_Relation of X ex x, y being set st
( e1 in the carrier of Y & e2 in the carrier of Y & [x,y] in e1 "\/" e2 & ( for F being non empty FinSequence of X holds
( not len F = (min A) + 1 or not x,y are_joint_by F,e1,e2 ) ) )
proof
assume A17: for e1, e2 being Equivalence_Relation of X
for x, y being set holds
( not e1 in the carrier of Y or not e2 in the carrier of Y or not [x,y] in e1 "\/" e2 or ex F being non empty FinSequence of X st
( len F = (min A) + 1 & x,y are_joint_by F,e1,e2 ) ) ; ::_thesis: contradiction
then consider F being non empty FinSequence of X such that
A18: len F = (min A) + 1 and
A19: x,y are_joint_by F,e1,e2 by A1, A4, A9;
A20: ( F . 1 = x & F . (len F) = y ) by A19, Def3;
 len F >= 2
proof
assume  not len F >= 2 ; ::_thesis: contradiction
then  len F < 1 + 1 ;
then  ( 0 <= len F & len F <= 0 + 1 ) by NAT_1:2, NAT_1:13;
hence  contradiction by A5, A6, A8, A20, NAT_1:9, RELAT_1:15; ::_thesis: verum
end;
then  (min A) + 1 >= 1 + 1 by A18;
then A21: min A >= 1 by XREAL_1:6;
then  ( (min A) + 1 = ((min A) - 1) + 2 & (min A) - 1 = (min A) -' 1 ) by XREAL_1:233;
then A22: (min A) -' 1 in A by A17;
 min A < (min A) + 1 by XREAL_1:29;
then A23: (min A) - 1 < ((min A) + 1) - 1 by XREAL_1:9;
 (min A) - 1 >= 0 by A21, XREAL_1:48;
then  (min A) -' 1 < min A by A23, XREAL_0:def_2;
hence  contradiction by A22, XXREAL_2:def_7; ::_thesis: verum
end;
 min A in A by XXREAL_2:def_7;
then  ex m9 being Element of NAT st
( m9 = min A & ( for e1, e2 being Equivalence_Relation of X
for x, y being set st e1 in the carrier of Y & e2 in the carrier of Y & [x,y] in e1 "\/" e2 holds
ex F being non empty FinSequence of X st
( len F = m9 + 2 & x,y are_joint_by F,e1,e2 ) ) ) ;
hence  ex b1 being Element of NAT st
( ( for e1, e2 being Equivalence_Relation of X
for x, y being set st e1 in the carrier of Y & e2 in the carrier of Y & [x,y] in e1 "\/" e2 holds
ex F being non empty FinSequence of X st
( len F = b1 + 2 & x,y are_joint_by F,e1,e2 ) ) & ex e1, e2 being Equivalence_Relation of X ex x, y being set st
( e1 in the carrier of Y & e2 in the carrier of Y & [x,y] in e1 "\/" e2 & ( for F being non empty FinSequence of X holds
( not len F = b1 + 1 or not x,y are_joint_by F,e1,e2 ) ) ) ) by A16; ::_thesis: verum
end;
uniqueness
 for b1, b2 being Element of NAT st ( for e1, e2 being Equivalence_Relation of X
for x, y being set st e1 in the carrier of Y & e2 in the carrier of Y & [x,y] in e1 "\/" e2 holds
ex F being non empty FinSequence of X st
( len F = b1 + 2 & x,y are_joint_by F,e1,e2 ) ) & ex e1, e2 being Equivalence_Relation of X ex x, y being set st
( e1 in the carrier of Y & e2 in the carrier of Y & [x,y] in e1 "\/" e2 & ( for F being non empty FinSequence of X holds
( not len F = b1 + 1 or not x,y are_joint_by F,e1,e2 ) ) ) & ( for e1, e2 being Equivalence_Relation of X
for x, y being set st e1 in the carrier of Y & e2 in the carrier of Y & [x,y] in e1 "\/" e2 holds
ex F being non empty FinSequence of X st
( len F = b2 + 2 & x,y are_joint_by F,e1,e2 ) ) & ex e1, e2 being Equivalence_Relation of X ex x, y being set st
( e1 in the carrier of Y & e2 in the carrier of Y & [x,y] in e1 "\/" e2 & ( for F being non empty FinSequence of X holds
( not len F = b2 + 1 or not x,y are_joint_by F,e1,e2 ) ) ) holds
b1 = b2
proof
let n1, n2 be Element of NAT ; ::_thesis: ( ( for e1, e2 being Equivalence_Relation of X
for x, y being set st e1 in the carrier of Y & e2 in the carrier of Y & [x,y] in e1 "\/" e2 holds
ex F being non empty FinSequence of X st
( len F = n1 + 2 & x,y are_joint_by F,e1,e2 ) ) & ex e1, e2 being Equivalence_Relation of X ex x, y being set st
( e1 in the carrier of Y & e2 in the carrier of Y & [x,y] in e1 "\/" e2 & ( for F being non empty FinSequence of X holds
( not len F = n1 + 1 or not x,y are_joint_by F,e1,e2 ) ) ) & ( for e1, e2 being Equivalence_Relation of X
for x, y being set st e1 in the carrier of Y & e2 in the carrier of Y & [x,y] in e1 "\/" e2 holds
ex F being non empty FinSequence of X st
( len F = n2 + 2 & x,y are_joint_by F,e1,e2 ) ) & ex e1, e2 being Equivalence_Relation of X ex x, y being set st
( e1 in the carrier of Y & e2 in the carrier of Y & [x,y] in e1 "\/" e2 & ( for F being non empty FinSequence of X holds
( not len F = n2 + 1 or not x,y are_joint_by F,e1,e2 ) ) ) implies n1 = n2 )
assume A24: for e1, e2 being Equivalence_Relation of X
for x, y being set st e1 in the carrier of Y & e2 in the carrier of Y & [x,y] in e1 "\/" e2 holds
ex F being non empty FinSequence of X st
( len F = n1 + 2 & x,y are_joint_by F,e1,e2 ) ; ::_thesis: ( for e1, e2 being Equivalence_Relation of X
for x, y being set holds
( not e1 in the carrier of Y or not e2 in the carrier of Y or not [x,y] in e1 "\/" e2 or ex F being non empty FinSequence of X st
( len F = n1 + 1 & x,y are_joint_by F,e1,e2 ) ) or ex e1, e2 being Equivalence_Relation of X ex x, y being set st
( e1 in the carrier of Y & e2 in the carrier of Y & [x,y] in e1 "\/" e2 & ( for F being non empty FinSequence of X holds
( not len F = n2 + 2 or not x,y are_joint_by F,e1,e2 ) ) ) or for e1, e2 being Equivalence_Relation of X
for x, y being set holds
( not e1 in the carrier of Y or not e2 in the carrier of Y or not [x,y] in e1 "\/" e2 or ex F being non empty FinSequence of X st
( len F = n2 + 1 & x,y are_joint_by F,e1,e2 ) ) or n1 = n2 )
given e19, e29 being Equivalence_Relation of X, x9, y9 being set  such that A25: ( e19 in the carrier of Y & e29 in the carrier of Y & [x9,y9] in e19 "\/" e29 ) and
A26: for F being non empty FinSequence of X holds
( not len F = n1 + 1 or not x9,y9 are_joint_by F,e19,e29 ) ; ::_thesis: ( ex e1, e2 being Equivalence_Relation of X ex x, y being set st
( e1 in the carrier of Y & e2 in the carrier of Y & [x,y] in e1 "\/" e2 & ( for F being non empty FinSequence of X holds
( not len F = n2 + 2 or not x,y are_joint_by F,e1,e2 ) ) ) or for e1, e2 being Equivalence_Relation of X
for x, y being set holds
( not e1 in the carrier of Y or not e2 in the carrier of Y or not [x,y] in e1 "\/" e2 or ex F being non empty FinSequence of X st
( len F = n2 + 1 & x,y are_joint_by F,e1,e2 ) ) or n1 = n2 )
assume  for e1, e2 being Equivalence_Relation of X
for x, y being set st e1 in the carrier of Y & e2 in the carrier of Y & [x,y] in e1 "\/" e2 holds
ex F being non empty FinSequence of X st
( len F = n2 + 2 & x,y are_joint_by F,e1,e2 ) ; ::_thesis: ( for e1, e2 being Equivalence_Relation of X
for x, y being set holds
( not e1 in the carrier of Y or not e2 in the carrier of Y or not [x,y] in e1 "\/" e2 or ex F being non empty FinSequence of X st
( len F = n2 + 1 & x,y are_joint_by F,e1,e2 ) ) or n1 = n2 )
then A27: ex F2 being non empty FinSequence of X st
( len F2 = n2 + 2 & x9,y9 are_joint_by F2,e19,e29 ) by A25;
 field e29 = X by EQREL_1:9;
then A28: e29 is_reflexive_in X by RELAT_2:def_9;
 field e19 = X by EQREL_1:9;
then A29: e19 is_reflexive_in X by RELAT_2:def_9;
given e1, e2 being Equivalence_Relation of X, x, y being set  such that A30: ( e1 in the carrier of Y & e2 in the carrier of Y & [x,y] in e1 "\/" e2 ) and
A31: for F being non empty FinSequence of X holds
( not len F = n2 + 1 or not x,y are_joint_by F,e1,e2 ) ; ::_thesis: n1 = n2
A32: ex F1 being non empty FinSequence of X st
( len F1 = n1 + 2 & x,y are_joint_by F1,e1,e2 ) by A24, A30;
 field e2 = X by EQREL_1:9;
then A33: e2 is_reflexive_in X by RELAT_2:def_9;
 field e1 = X by EQREL_1:9;
then A34: e1 is_reflexive_in X by RELAT_2:def_9;
assume A35: not n1 = n2 ; ::_thesis: contradiction
percases ( n1 < n2 or n2 < n1 ) by A35, XXREAL_0:1;
suppose n1 < n2 ; ::_thesis: contradiction
then  n1 + 2 < n2 + (1 + 1) by XREAL_1:6;
then  n1 + 2 < (n2 + 1) + 1 ;
then  n1 + 2 <= n2 + 1 by NAT_1:13;
hence  contradiction by A31, A32, A34, A33, Th12; ::_thesis: verum
end;
suppose n2 < n1 ; ::_thesis: contradiction
then  n2 + 2 < n1 + (1 + 1) by XREAL_1:6;
then  n2 + 2 < (n1 + 1) + 1 ;
then  n2 + 2 <= n1 + 1 by NAT_1:13;
hence  contradiction by A26, A27, A29, A28, Th12; ::_thesis: verum
end;
end;
end;
end;

:: deftheorem Def4 defines type_of LATTICE5:def_4_:_
 for X being non empty set
for Y being Sublattice of EqRelLATT X st ex e being Equivalence_Relation of X st
( e in the carrier of Y & e <> id X ) & ex o being Element of NAT st
for e1, e2 being Equivalence_Relation of X
for x, y being set st e1 in the carrier of Y & e2 in the carrier of Y & [x,y] in e1 "\/" e2 holds
ex F being non empty FinSequence of X st
( len F = o & x,y are_joint_by F,e1,e2 ) holds
for b3 being Element of NAT holds
( b3 = type_of Y iff ( ( for e1, e2 being Equivalence_Relation of X
for x, y being set st e1 in the carrier of Y & e2 in the carrier of Y & [x,y] in e1 "\/" e2 holds
ex F being non empty FinSequence of X st
( len F = b3 + 2 & x,y are_joint_by F,e1,e2 ) ) & ex e1, e2 being Equivalence_Relation of X ex x, y being set st
( e1 in the carrier of Y & e2 in the carrier of Y & [x,y] in e1 "\/" e2 & ( for F being non empty FinSequence of X holds
( not len F = b3 + 1 or not x,y are_joint_by F,e1,e2 ) ) ) ) );

theorem Th13: :: LATTICE5:13
for X being non empty set
for Y being Sublattice of EqRelLATT X
for n being Element of NAT st ex e being Equivalence_Relation of X st
( e in the carrier of Y & e <> id X ) & ( for e1, e2 being Equivalence_Relation of X
for x, y being set st e1 in the carrier of Y & e2 in the carrier of Y & [x,y] in e1 "\/" e2 holds
ex F being non empty FinSequence of X st
( len F = n + 2 & x,y are_joint_by F,e1,e2 ) ) holds
type_of Y <= n
proof
let X be non empty set ; ::_thesis: for Y being Sublattice of EqRelLATT X
for n being Element of NAT st ex e being Equivalence_Relation of X st
( e in the carrier of Y & e <> id X ) & ( for e1, e2 being Equivalence_Relation of X
for x, y being set st e1 in the carrier of Y & e2 in the carrier of Y & [x,y] in e1 "\/" e2 holds
ex F being non empty FinSequence of X st
( len F = n + 2 & x,y are_joint_by F,e1,e2 ) ) holds
type_of Y <= n
let Y be Sublattice of EqRelLATT X; ::_thesis: for n being Element of NAT st ex e being Equivalence_Relation of X st
( e in the carrier of Y & e <> id X ) & ( for e1, e2 being Equivalence_Relation of X
for x, y being set st e1 in the carrier of Y & e2 in the carrier of Y & [x,y] in e1 "\/" e2 holds
ex F being non empty FinSequence of X st
( len F = n + 2 & x,y are_joint_by F,e1,e2 ) ) holds
type_of Y <= n
let n be Element of NAT ; ::_thesis: ( ex e being Equivalence_Relation of X st
( e in the carrier of Y & e <> id X ) & ( for e1, e2 being Equivalence_Relation of X
for x, y being set st e1 in the carrier of Y & e2 in the carrier of Y & [x,y] in e1 "\/" e2 holds
ex F being non empty FinSequence of X st
( len F = n + 2 & x,y are_joint_by F,e1,e2 ) ) implies type_of Y <= n )
assume that
A1: ex e being Equivalence_Relation of X st
( e in the carrier of Y & e <> id X ) and
A2: for e1, e2 being Equivalence_Relation of X
for x, y being set st e1 in the carrier of Y & e2 in the carrier of Y & [x,y] in e1 "\/" e2 holds
ex F being non empty FinSequence of X st
( len F = n + 2 & x,y are_joint_by F,e1,e2 ) and
A3: n < type_of Y ; ::_thesis: contradiction
 n + 1 <= type_of Y by A3, NAT_1:13;
then consider m being Nat such that
A4: type_of Y = (n + 1) + m by NAT_1:10;
reconsider m = m as Element of NAT by ORDINAL1:def_12;
 ((n + 1) + m) + 1 = (n + m) + 2 ;
then consider e1, e2 being Equivalence_Relation of X, x, y being set  such that
A5: ( e1 in the carrier of Y & e2 in the carrier of Y & [x,y] in e1 "\/" e2 ) and
A6: for F being non empty FinSequence of X holds
( not len F = (n + m) + 2 or not x,y are_joint_by F,e1,e2 ) by A1, A4, Def4;
A7: (n + 2) + m = (n + m) + 2 ;
 field e2 = X by EQREL_1:9;
then A8: e2 is_reflexive_in X by RELAT_2:def_9;
 field e1 = X by EQREL_1:9;
then A9: e1 is_reflexive_in X by RELAT_2:def_9;
 ex F1 being non empty FinSequence of X st
( len F1 = n + 2 & x,y are_joint_by F1,e1,e2 ) by A2, A5;
hence  contradiction by A6, A9, A8, A7, Th12, NAT_1:11; ::_thesis: verum
end;

begin


definition
let A be set ;
let L be 1-sorted ;
mode BiFunction of A,L is Function of [:A,A:], the carrier of L;
end;

definition
let A be non empty set ;
let L be 1-sorted ;
let f be BiFunction of A,L;
let x, y be Element of A;
:: original: .
redefine funcf . (x,y) ->  Element of L;
coherence
 f . (x,y) is Element of L
proof
reconsider xy = [x,y] as Element of [:A,A:] ;
 f . xy is Element of L ;
hence  f . (x,y) is Element of L ; ::_thesis: verum
end;
end;

definition
let A be non empty set ;
let L be 1-sorted ;
let f be BiFunction of A,L;
attrf is symmetric  means :Def5: :: LATTICE5:def 5
 for x, y being Element of A holds f . (x,y) = f . (y,x);
end;

:: deftheorem Def5 defines symmetric LATTICE5:def_5_:_
 for A being non empty set
for L being 1-sorted
for f being BiFunction of A,L holds
( f is symmetric iff for x, y being Element of A holds f . (x,y) = f . (y,x) );

definition
let A be non empty set ;
let L be lower-bounded LATTICE;
let f be BiFunction of A,L;
attrf is zeroed  means :Def6: :: LATTICE5:def 6
 for x being Element of A holds f . (x,x) = Bottom L;
attrf is u.t.i.  means :Def7: :: LATTICE5:def 7
 for x, y, z being Element of A holds (f . (x,y)) "\/" (f . (y,z)) >= f . (x,z);
end;

:: deftheorem Def6 defines zeroed LATTICE5:def_6_:_
 for A being non empty set
for L being lower-bounded LATTICE
for f being BiFunction of A,L holds
( f is zeroed iff for x being Element of A holds f . (x,x) = Bottom L );

:: deftheorem Def7 defines u.t.i. LATTICE5:def_7_:_
 for A being non empty set
for L being lower-bounded LATTICE
for f being BiFunction of A,L holds
( f is u.t.i. iff for x, y, z being Element of A holds (f . (x,y)) "\/" (f . (y,z)) >= f . (x,z) );

registration
let A be non empty set ;
let L be lower-bounded LATTICE;
cluster Relation-like [:A,A:] -defined the carrier of L -valued Function-like quasi_total symmetric zeroed u.t.i. for Element of bool [:[:A,A:], the carrier of L:];
existence
 ex b1 being BiFunction of A,L st
( b1 is symmetric & b1 is zeroed & b1 is u.t.i. )
proof
reconsider f = [:A,A:] --> (Bottom L) as Function of [:A,A:], the carrier of L ;
A1: for x, y being Element of A holds f . [x,y] = Bottom L by FUNCOP_1:7;
reconsider f = f as BiFunction of A,L ;
 for x, y being Element of A holds f . (x,y) = f . (y,x)
proof
let x, y be Element of A; ::_thesis: f . (x,y) = f . (y,x)
thus f . (x,y) = Bottom L by A1
.= f . (y,x) by A1 ; ::_thesis: verum
end;
then A2: f is symmetric by Def5;
 for x, y, z being Element of A holds (f . (x,y)) "\/" (f . (y,z)) >= f . (x,z)
proof
let x, y, z be Element of A; ::_thesis: (f . (x,y)) "\/" (f . (y,z)) >= f . (x,z)
A3: f . (x,z) <= Bottom L by A1;
 ( f . (x,y) = Bottom L & f . (y,z) = Bottom L ) by A1;
hence  (f . (x,y)) "\/" (f . (y,z)) >= f . (x,z) by A3, YELLOW_5:1; ::_thesis: verum
end;
then A4: f is u.t.i. by Def7;
 for x being Element of A holds f . (x,x) = Bottom L by A1;
then  f is zeroed by Def6;
hence  ex b1 being BiFunction of A,L st
( b1 is symmetric & b1 is zeroed & b1 is u.t.i. ) by A2, A4; ::_thesis: verum
end;
end;

definition
let A be non empty set ;
let L be lower-bounded LATTICE;
mode distance_function of A,L is symmetric zeroed u.t.i. BiFunction of A,L;
end;

definition
let A be non empty set ;
let L be lower-bounded LATTICE;
let d be distance_function of A,L;
func alpha d ->  Function of L,(EqRelLATT A) means :Def8: :: LATTICE5:def 8
 for e being Element of L ex E being Equivalence_Relation of A st
( E = it . e & ( for x, y being Element of A holds
( [x,y] in E iff d . (x,y) <= e ) ) );
existence
 ex b1 being Function of L,(EqRelLATT A) st
for e being Element of L ex E being Equivalence_Relation of A st
( E = b1 . e & ( for x, y being Element of A holds
( [x,y] in E iff d . (x,y) <= e ) ) )
proof
defpred S1[ Element of L, Element of (EqRelLATT A)] means  ex E being Equivalence_Relation of A st
( E = $2 & ( for x, y being Element of A holds
( [x,y] in E iff d . (x,y) <= $1 ) ) );
A1: for e being Element of L ex r being Element of (EqRelLATT A) st S1[e,r]
proof
let e be Element of L; ::_thesis: ex r being Element of (EqRelLATT A) st S1[e,r]
defpred S2[ Element of A, Element of A] means d . ($1,$2) <= e;
consider E being Relation of A,A such that
A2: for x, y being Element of A holds
( [x,y] in E iff S2[x,y] ) from RELSET_1:sch_2();
 for x, y being set st x in A & y in A & [x,y] in E holds
[y,x] in E
proof
let x, y be set ; ::_thesis: ( x in A & y in A & [x,y] in E implies [y,x] in E )
assume that
A3: x in A and
A4: y in A and
A5: [x,y] in E ; ::_thesis: [y,x] in E
reconsider y9 = y as Element of A by A4;
reconsider x9 = x as Element of A by A3;
 d . (x9,y9) <= e by A2, A5;
then  d . (y9,x9) <= e by Def5;
hence  [y,x] in E by A2; ::_thesis: verum
end;
then A6: E is_symmetric_in A by RELAT_2:def_3;
 for x being set st x in A holds
[x,x] in E
proof
let x be set ; ::_thesis: ( x in A implies [x,x] in E )
assume  x in A ; ::_thesis: [x,x] in E
then reconsider x9 = x as Element of A ;
 Bottom L <= e by YELLOW_0:44;
then  d . (x9,x9) <= e by Def6;
hence  [x,x] in E by A2; ::_thesis: verum
end;
then  E is_reflexive_in A by RELAT_2:def_1;
then A7: ( dom E = A & field E = A ) by ORDERS_1:13;
 for x, y, z being set st x in A & y in A & z in A & [x,y] in E & [y,z] in E holds
[x,z] in E
proof
let x, y, z be set ; ::_thesis: ( x in A & y in A & z in A & [x,y] in E & [y,z] in E implies [x,z] in E )
assume that
A8: ( x in A & y in A & z in A ) and
A9: ( [x,y] in E & [y,z] in E ) ; ::_thesis: [x,z] in E
reconsider x9 = x, y9 = y, z9 = z as Element of A by A8;
 ( d . (x9,y9) <= e & d . (y9,z9) <= e ) by A2, A9;
then A10: (d . (x9,y9)) "\/" (d . (y9,z9)) <= e by YELLOW_0:22;
 d . (x9,z9) <= (d . (x9,y9)) "\/" (d . (y9,z9)) by Def7;
then  d . (x9,z9) <= e by A10, ORDERS_2:3;
hence  [x,z] in E by A2; ::_thesis: verum
end;
then  E is_transitive_in A by RELAT_2:def_8;
then reconsider E = E as Equivalence_Relation of A by A7, A6, PARTFUN1:def_2, RELAT_2:def_11, RELAT_2:def_16;
reconsider E = E as Element of (EqRelLATT A) by Th4;
 ex r being Element of (EqRelLATT A) st r = E ;
hence  ex r being Element of (EqRelLATT A) st S1[e,r] by A2; ::_thesis: verum
end;
 ex f being Function of L,(EqRelLATT A) st
for e being Element of L holds S1[e,f . e] from FUNCT_2:sch_3(A1);
hence  ex b1 being Function of L,(EqRelLATT A) st
for e being Element of L ex E being Equivalence_Relation of A st
( E = b1 . e & ( for x, y being Element of A holds
( [x,y] in E iff d . (x,y) <= e ) ) ) ; ::_thesis: verum
end;
uniqueness
 for b1, b2 being Function of L,(EqRelLATT A) st ( for e being Element of L ex E being Equivalence_Relation of A st
( E = b1 . e & ( for x, y being Element of A holds
( [x,y] in E iff d . (x,y) <= e ) ) ) ) & ( for e being Element of L ex E being Equivalence_Relation of A st
( E = b2 . e & ( for x, y being Element of A holds
( [x,y] in E iff d . (x,y) <= e ) ) ) ) holds
b1 = b2
proof
let f1, f2 be Function of L,(EqRelLATT A); ::_thesis: ( ( for e being Element of L ex E being Equivalence_Relation of A st
( E = f1 . e & ( for x, y being Element of A holds
( [x,y] in E iff d . (x,y) <= e ) ) ) ) & ( for e being Element of L ex E being Equivalence_Relation of A st
( E = f2 . e & ( for x, y being Element of A holds
( [x,y] in E iff d . (x,y) <= e ) ) ) ) implies f1 = f2 )
assume that
A11: for e being Element of L ex E being Equivalence_Relation of A st
( E = f1 . e & ( for x, y being Element of A holds
( [x,y] in E iff d . (x,y) <= e ) ) ) and
A12: for e being Element of L ex E being Equivalence_Relation of A st
( E = f2 . e & ( for x, y being Element of A holds
( [x,y] in E iff d . (x,y) <= e ) ) ) ; ::_thesis: f1 = f2
reconsider f19 = f1, f29 = f2 as Function of the carrier of L, the carrier of (EqRelLATT A) ;
 for e being Element of L holds f1 . e = f2 . e
proof
let e be Element of L; ::_thesis: f1 . e = f2 . e
consider E1 being Equivalence_Relation of A such that
A13: E1 = f1 . e and
A14: for x, y being Element of A holds
( [x,y] in E1 iff d . (x,y) <= e ) by A11;
consider E2 being Equivalence_Relation of A such that
A15: E2 = f2 . e and
A16: for x, y being Element of A holds
( [x,y] in E2 iff d . (x,y) <= e ) by A12;
A17: for x, y being Element of A holds
( [x,y] in E1 iff [x,y] in E2 )
proof
let x, y be Element of A; ::_thesis: ( [x,y] in E1 iff [x,y] in E2 )
 ( [x,y] in E1 iff d . (x,y) <= e ) by A14;
hence  ( [x,y] in E1 iff [x,y] in E2 ) by A16; ::_thesis: verum
end;
 for x, y being set holds
( [x,y] in E1 iff [x,y] in E2 )
proof
let x, y be set ; ::_thesis: ( [x,y] in E1 iff [x,y] in E2 )
A18: field E1 = A by EQREL_1:9;
hereby  ::_thesis: ( [x,y] in E2 implies [x,y] in E1 )
assume A19: [x,y] in E1 ; ::_thesis: [x,y] in E2
then reconsider x9 = x, y9 = y as Element of A by A18, RELAT_1:15;
 [x9,y9] in E2 by A17, A19;
hence  [x,y] in E2 ; ::_thesis: verum
end;
assume A20: [x,y] in E2 ; ::_thesis: [x,y] in E1
 field E2 = A by EQREL_1:9;
then reconsider x9 = x, y9 = y as Element of A by A20, RELAT_1:15;
 [x9,y9] in E1 by A17, A20;
hence  [x,y] in E1 ; ::_thesis: verum
end;
hence  f1 . e = f2 . e by A13, A15, RELAT_1:def_2; ::_thesis: verum
end;
then  for e being set st e in the carrier of L holds
f19 . e = f29 . e ;
hence  f1 = f2 by FUNCT_2:12; ::_thesis: verum
end;
end;

:: deftheorem Def8 defines alpha LATTICE5:def_8_:_
 for A being non empty set
for L being lower-bounded LATTICE
for d being distance_function of A,L
for b4 being Function of L,(EqRelLATT A) holds
( b4 = alpha d iff for e being Element of L ex E being Equivalence_Relation of A st
( E = b4 . e & ( for x, y being Element of A holds
( [x,y] in E iff d . (x,y) <= e ) ) ) );

theorem Th14: :: LATTICE5:14
for A being non empty set
for L being lower-bounded LATTICE
for d being distance_function of A,L holds alpha d is meet-preserving
proof
let A be non empty set ; ::_thesis: for L being lower-bounded LATTICE
for d being distance_function of A,L holds alpha d is meet-preserving
let L be lower-bounded LATTICE; ::_thesis: for d being distance_function of A,L holds alpha d is meet-preserving
let d be distance_function of A,L; ::_thesis: alpha d is meet-preserving
let a, b be Element of L; :: according to WAYBEL_0:def_34 ::_thesis: alpha d preserves_inf_of {a,b}
set f = alpha d;
A1: ex_inf_of (alpha d) .: {a,b}, EqRelLATT A by YELLOW_0:17;
consider E3 being Equivalence_Relation of A such that
A2: E3 = (alpha d) . (a "/\" b) and
A3: for x, y being Element of A holds
( [x,y] in E3 iff d . (x,y) <= a "/\" b ) by Def8;
consider E2 being Equivalence_Relation of A such that
A4: E2 = (alpha d) . b and
A5: for x, y being Element of A holds
( [x,y] in E2 iff d . (x,y) <= b ) by Def8;
consider E1 being Equivalence_Relation of A such that
A6: E1 = (alpha d) . a and
A7: for x, y being Element of A holds
( [x,y] in E1 iff d . (x,y) <= a ) by Def8;
A8: for x, y being Element of A holds
( [x,y] in E1 /\ E2 iff [x,y] in E3 )
proof
let x, y be Element of A; ::_thesis: ( [x,y] in E1 /\ E2 iff [x,y] in E3 )
hereby  ::_thesis: ( [x,y] in E3 implies [x,y] in E1 /\ E2 )
assume A9: [x,y] in E1 /\ E2 ; ::_thesis: [x,y] in E3
then  [x,y] in E2 by XBOOLE_0:def_4;
then A10: d . (x,y) <= b by A5;
 [x,y] in E1 by A9, XBOOLE_0:def_4;
then  d . (x,y) <= a by A7;
then  d . (x,y) <= a "/\" b by A10, YELLOW_0:23;
hence  [x,y] in E3 by A3; ::_thesis: verum
end;
assume  [x,y] in E3 ; ::_thesis: [x,y] in E1 /\ E2
then A11: d . (x,y) <= a "/\" b by A3;
 a "/\" b <= b by YELLOW_0:23;
then  d . (x,y) <= b by A11, ORDERS_2:3;
then A12: [x,y] in E2 by A5;
 a "/\" b <= a by YELLOW_0:23;
then  d . (x,y) <= a by A11, ORDERS_2:3;
then  [x,y] in E1 by A7;
hence  [x,y] in E1 /\ E2 by A12, XBOOLE_0:def_4; ::_thesis: verum
end;
A13: for x, y being set holds
( [x,y] in E1 /\ E2 iff [x,y] in E3 )
proof
let x, y be set ; ::_thesis: ( [x,y] in E1 /\ E2 iff [x,y] in E3 )
(field E1) /\ (field E2) = A /\ (field E2) by EQREL_1:9
.= A /\ A by EQREL_1:9
.= A ;
then A14: field (E1 /\ E2) c= A by RELAT_1:19;
hereby  ::_thesis: ( [x,y] in E3 implies [x,y] in E1 /\ E2 )
assume A15: [x,y] in E1 /\ E2 ; ::_thesis: [x,y] in E3
then  ( x in field (E1 /\ E2) & y in field (E1 /\ E2) ) by RELAT_1:15;
then reconsider x9 = x, y9 = y as Element of A by A14;
 [x9,y9] in E3 by A8, A15;
hence  [x,y] in E3 ; ::_thesis: verum
end;
assume A16: [x,y] in E3 ; ::_thesis: [x,y] in E1 /\ E2
 field E3 = A by EQREL_1:9;
then reconsider x9 = x, y9 = y as Element of A by A16, RELAT_1:15;
 [x9,y9] in E1 /\ E2 by A8, A16;
hence  [x,y] in E1 /\ E2 ; ::_thesis: verum
end;
 dom (alpha d) = the carrier of L by FUNCT_2:def_1;
then  inf ((alpha d) .: {a,b}) = inf {((alpha d) . a),((alpha d) . b)} by FUNCT_1:60
.= ((alpha d) . a) "/\" ((alpha d) . b) by YELLOW_0:40
.= E1 /\ E2 by A6, A4, Th8
.= (alpha d) . (a "/\" b) by A2, A13, RELAT_1:def_2
.= (alpha d) . (inf {a,b}) by YELLOW_0:40 ;
hence  alpha d preserves_inf_of {a,b} by A1, WAYBEL_0:def_30; ::_thesis: verum
end;

theorem Th15: :: LATTICE5:15
for A being non empty set
for L being lower-bounded LATTICE
for d being distance_function of A,L st d is onto holds
alpha d is one-to-one
proof
let A be non empty set ; ::_thesis: for L being lower-bounded LATTICE
for d being distance_function of A,L st d is onto holds
alpha d is one-to-one
let L be lower-bounded LATTICE; ::_thesis: for d being distance_function of A,L st d is onto holds
alpha d is one-to-one
let d be distance_function of A,L; ::_thesis: ( d is onto implies alpha d is one-to-one )
set f = alpha d;
assume  d is onto ; ::_thesis: alpha d is one-to-one
then A1: rng d = the carrier of L by FUNCT_2:def_3;
 for a, b being Element of L st (alpha d) . a = (alpha d) . b holds
a = b
proof
let a, b be Element of L; ::_thesis: ( (alpha d) . a = (alpha d) . b implies a = b )
assume A2: (alpha d) . a = (alpha d) . b ; ::_thesis: a = b
consider z1 being set  such that
A3: z1 in [:A,A:] and
A4: d . z1 = a by A1, FUNCT_2:11;
consider x1, y1 being set  such that
A5: ( x1 in A & y1 in A ) and
A6: z1 = [x1,y1] by A3, ZFMISC_1:def_2;
reconsider x1 = x1, y1 = y1 as Element of A by A5;
consider z2 being set  such that
A7: z2 in [:A,A:] and
A8: d . z2 = b by A1, FUNCT_2:11;
consider x2, y2 being set  such that
A9: ( x2 in A & y2 in A ) and
A10: z2 = [x2,y2] by A7, ZFMISC_1:def_2;
reconsider x2 = x2, y2 = y2 as Element of A by A9;
consider E1 being Equivalence_Relation of A such that
A11: E1 = (alpha d) . a and
A12: for x, y being Element of A holds
( [x,y] in E1 iff d . (x,y) <= a ) by Def8;
consider E2 being Equivalence_Relation of A such that
A13: E2 = (alpha d) . b and
A14: for x, y being Element of A holds
( [x,y] in E2 iff d . (x,y) <= b ) by Def8;
A15: d . (x2,y2) = b by A8, A10;
then  [x2,y2] in E2 by A14;
then A16: b <= a by A2, A15, A11, A12, A13;
A17: d . (x1,y1) = a by A4, A6;
then  [x1,y1] in E1 by A12;
then  a <= b by A2, A17, A11, A13, A14;
hence  a = b by A16, ORDERS_2:2; ::_thesis: verum
end;
hence  alpha d is one-to-one by WAYBEL_1:def_1; ::_thesis: verum
end;

begin

definition
let A be set ;
func new_set A ->  set  equals :: LATTICE5:def 9
A \/ {{A},{{A}},{{{A}}}};
correctness
coherence
A \/ {{A},{{A}},{{{A}}}} is set ;
;
end;

:: deftheorem  defines new_set LATTICE5:def_9_:_
 for A being set holds new_set A = A \/ {{A},{{A}},{{{A}}}};

registration
let A be set ;
cluster new_set A ->  non empty ;
coherence
 not new_set A is empty ;
end;

definition
let A be non empty set ;
let L be lower-bounded LATTICE;
let d be BiFunction of A,L;
let q be Element of [:A,A, the carrier of L, the carrier of L:];
func new_bi_fun (d,q) ->  BiFunction of (new_set A),L means :Def10: :: LATTICE5:def 10
( ( for u, v being Element of A holds it . (u,v) = d . (u,v) ) & it . ({A},{A}) = Bottom L & it . ({{A}},{{A}}) = Bottom L & it . ({{{A}}},{{{A}}}) = Bottom L & it . ({{A}},{{{A}}}) = q `3_4 & it . ({{{A}}},{{A}}) = q `3_4 & it . ({A},{{A}}) = q `4_4 & it . ({{A}},{A}) = q `4_4 & it . ({A},{{{A}}}) = (q `3_4) "\/" (q `4_4) & it . ({{{A}}},{A}) = (q `3_4) "\/" (q `4_4) & ( for u being Element of A holds
( it . (u,{A}) = (d . (u,(q `1_4))) "\/" (q `3_4) & it . ({A},u) = (d . (u,(q `1_4))) "\/" (q `3_4) & it . (u,{{A}}) = ((d . (u,(q `1_4))) "\/" (q `3_4)) "\/" (q `4_4) & it . ({{A}},u) = ((d . (u,(q `1_4))) "\/" (q `3_4)) "\/" (q `4_4) & it . (u,{{{A}}}) = (d . (u,(q `2_4))) "\/" (q `4_4) & it . ({{{A}}},u) = (d . (u,(q `2_4))) "\/" (q `4_4) ) ) );
existence
 ex b1 being BiFunction of (new_set A),L st
( ( for u, v being Element of A holds b1 . (u,v) = d . (u,v) ) & b1 . ({A},{A}) = Bottom L & b1 . ({{A}},{{A}}) = Bottom L & b1 . ({{{A}}},{{{A}}}) = Bottom L & b1 . ({{A}},{{{A}}}) = q `3_4 & b1 . ({{{A}}},{{A}}) = q `3_4 & b1 . ({A},{{A}}) = q `4_4 & b1 . ({{A}},{A}) = q `4_4 & b1 . ({A},{{{A}}}) = (q `3_4) "\/" (q `4_4) & b1 . ({{{A}}},{A}) = (q `3_4) "\/" (q `4_4) & ( for u being Element of A holds
( b1 . (u,{A}) = (d . (u,(q `1_4))) "\/" (q `3_4) & b1 . ({A},u) = (d . (u,(q `1_4))) "\/" (q `3_4) & b1 . (u,{{A}}) = ((d . (u,(q `1_4))) "\/" (q `3_4)) "\/" (q `4_4) & b1 . ({{A}},u) = ((d . (u,(q `1_4))) "\/" (q `3_4)) "\/" (q `4_4) & b1 . (u,{{{A}}}) = (d . (u,(q `2_4))) "\/" (q `4_4) & b1 . ({{{A}}},u) = (d . (u,(q `2_4))) "\/" (q `4_4) ) ) )
proof
reconsider a = q `3_4 , b = q `4_4 as Element of L ;
set x = q `1_4 ;
set y = q `2_4 ;
defpred S1[ Element of new_set A, Element of new_set A, Element of L] means ( ( $1 in A & $2 in A implies $3 = d . ($1,$2) ) & ( ( ( $1 = {{A}} & $2 = {{{A}}} ) or ( $2 = {{A}} & $1 = {{{A}}} ) ) implies $3 = a ) & ( ( ( $1 = {A} & $2 = {{A}} ) or ( $2 = {A} & $1 = {{A}} ) ) implies $3 = b ) & ( ( ( $1 = {A} & $2 = {{{A}}} ) or ( $2 = {A} & $1 = {{{A}}} ) ) implies $3 = a "\/" b ) & ( ( $1 = {A} or $1 = {{A}} or $1 = {{{A}}} ) & $2 = $1 implies $3 = Bottom L ) & ( $1 in A & $2 = {A} implies ex p9 being Element of A st
( p9 = $1 & $3 = (d . (p9,(q `1_4))) "\/" a ) ) & ( $1 in A & $2 = {{A}} implies ex p9 being Element of A st
( p9 = $1 & $3 = ((d . (p9,(q `1_4))) "\/" a) "\/" b ) ) & ( $1 in A & $2 = {{{A}}} implies ex p9 being Element of A st
( p9 = $1 & $3 = (d . (p9,(q `2_4))) "\/" b ) ) & ( $2 in A & $1 = {A} implies ex q9 being Element of A st
( q9 = $2 & $3 = (d . (q9,(q `1_4))) "\/" a ) ) & ( $2 in A & $1 = {{A}} implies ex q9 being Element of A st
( q9 = $2 & $3 = ((d . (q9,(q `1_4))) "\/" a) "\/" b ) ) & ( $2 in A & $1 = {{{A}}} implies ex q9 being Element of A st
( q9 = $2 & $3 = (d . (q9,(q `2_4))) "\/" b ) ) );
 {{A}} in {{A},{{A}},{{{A}}}} by ENUMSET1:def_1;
then A1: {{A}} in new_set A by XBOOLE_0:def_3;
A2: for p, q being Element of new_set A ex r being Element of L st S1[p,q,r]
proof
let p, q be Element of new_set A; ::_thesis: ex r being Element of L st S1[p,q,r]
A3: ( p in A or p in {{A},{{A}},{{{A}}}} ) by XBOOLE_0:def_3;
A4: ( q in A or q in {{A},{{A}},{{{A}}}} ) by XBOOLE_0:def_3;
A5: ( ( ( p = {A} or p = {{A}} or p = {{{A}}} ) & p = q ) iff ( ( p = {A} & q = {A} ) or ( p = {{A}} & q = {{A}} ) or ( p = {{{A}}} & q = {{{A}}} ) ) ) ;
A6: not {A} in A by TARSKI:def_1;
A7: {{A}} <> {{{A}}}
proof
assume  {{A}} = {{{A}}} ; ::_thesis: contradiction
then  {{A}} in {{A}} by TARSKI:def_1;
hence  contradiction ; ::_thesis: verum
end;
A8: not {{{A}}} in A
proof
A9: {{A}} in {{{A}}} by TARSKI:def_1;
A10: ( A in {A} & {A} in {{A}} ) by TARSKI:def_1;
assume  {{{A}}} in A ; ::_thesis: contradiction
hence  contradiction by A10, A9, XREGULAR:8; ::_thesis: verum
end;
A11: {A} <> {{{A}}}
proof
assume  {A} = {{{A}}} ; ::_thesis: contradiction
then  {{A}} in {A} by TARSKI:def_1;
hence  contradiction by TARSKI:def_1; ::_thesis: verum
end;
A12: not {{A}} in A
proof
A13: ( A in {A} & {A} in {{A}} ) by TARSKI:def_1;
assume  {{A}} in A ; ::_thesis: contradiction
hence  contradiction by A13, XREGULAR:7; ::_thesis: verum
end;
percases ( ( p in A & q in A ) or ( p = {{A}} & q = {{{A}}} ) or ( q = {{A}} & p = {{{A}}} ) or ( p = {A} & q = {{A}} ) or ( q = {A} & p = {{A}} ) or ( p = {A} & q = {{{A}}} ) or ( q = {A} & p = {{{A}}} ) or ( ( p = {A} or p = {{A}} or p = {{{A}}} ) & q = p ) or ( p in A & q = {A} ) or ( p in A & q = {{A}} ) or ( p in A & q = {{{A}}} ) or ( q in A & p = {A} ) or ( q in A & p = {{A}} ) or ( q in A & p = {{{A}}} ) ) by A3, A4, A5, ENUMSET1:def_1;
suppose ( p in A & q in A ) ; ::_thesis: ex r being Element of L st S1[p,q,r]
then reconsider p9 = p, q9 = q as Element of A ;
take  d . (p9,q9) ; ::_thesis: S1[p,q,d . (p9,q9)]
thus  S1[p,q,d . (p9,q9)] by A6, A12, A8; ::_thesis: verum
end;
supposeA14: ( ( p = {{A}} & q = {{{A}}} ) or ( q = {{A}} & p = {{{A}}} ) ) ; ::_thesis: ex r being Element of L st S1[p,q,r]
take  a ; ::_thesis: S1[p,q,a]
thus  S1[p,q,a] by A7, A11, A12, A8, A14; ::_thesis: verum
end;
supposeA15: ( ( p = {A} & q = {{A}} ) or ( q = {A} & p = {{A}} ) ) ; ::_thesis: ex r being Element of L st S1[p,q,r]
take  b ; ::_thesis: S1[p,q,b]
thus  S1[p,q,b] by A7, A11, A12, A15, TARSKI:def_1; ::_thesis: verum
end;
supposeA16: ( ( p = {A} & q = {{{A}}} ) or ( q = {A} & p = {{{A}}} ) ) ; ::_thesis: ex r being Element of L st S1[p,q,r]
take  a "\/" b ; ::_thesis: S1[p,q,a "\/" b]
thus  S1[p,q,a "\/" b] by A7, A11, A8, A16, TARSKI:def_1; ::_thesis: verum
end;
supposeA17: ( ( p = {A} or p = {{A}} or p = {{{A}}} ) & q = p ) ; ::_thesis: ex r being Element of L st S1[p,q,r]
take  Bottom L ; ::_thesis: S1[p,q, Bottom L]
thus  S1[p,q, Bottom L] by A7, A11, A12, A8, A17, TARSKI:def_1; ::_thesis: verum
end;
supposeA18: ( p in A & q = {A} ) ; ::_thesis: ex r being Element of L st S1[p,q,r]
then reconsider p9 = p as Element of A ;
take  (d . (p9,(q `1_4))) "\/" a ; ::_thesis: S1[p,q,(d . (p9,(q `1_4))) "\/" a]
thus  S1[p,q,(d . (p9,(q `1_4))) "\/" a] by A11, A12, A8, A18, TARSKI:def_1; ::_thesis: verum
end;
supposeA19: ( p in A & q = {{A}} ) ; ::_thesis: ex r being Element of L st S1[p,q,r]
then reconsider p9 = p as Element of A ;
take  ((d . (p9,(q `1_4))) "\/" a) "\/" b ; ::_thesis: S1[p,q,((d . (p9,(q `1_4))) "\/" a) "\/" b]
thus  S1[p,q,((d . (p9,(q `1_4))) "\/" a) "\/" b] by A7, A12, A8, A19, TARSKI:def_1; ::_thesis: verum
end;
supposeA20: ( p in A & q = {{{A}}} ) ; ::_thesis: ex r being Element of L st S1[p,q,r]
then reconsider p9 = p as Element of A ;
take  (d . (p9,(q `2_4))) "\/" b ; ::_thesis: S1[p,q,(d . (p9,(q `2_4))) "\/" b]
thus  S1[p,q,(d . (p9,(q `2_4))) "\/" b] by A7, A11, A12, A8, A20, TARSKI:def_1; ::_thesis: verum
end;
supposeA21: ( q in A & p = {A} ) ; ::_thesis: ex r being Element of L st S1[p,q,r]
then reconsider q9 = q as Element of A ;
take  (d . (q9,(q `1_4))) "\/" a ; ::_thesis: S1[p,q,(d . (q9,(q `1_4))) "\/" a]
thus  S1[p,q,(d . (q9,(q `1_4))) "\/" a] by A11, A12, A8, A21, TARSKI:def_1; ::_thesis: verum
end;
supposeA22: ( q in A & p = {{A}} ) ; ::_thesis: ex r being Element of L st S1[p,q,r]
then reconsider q9 = q as Element of A ;
take  ((d . (q9,(q `1_4))) "\/" a) "\/" b ; ::_thesis: S1[p,q,((d . (q9,(q `1_4))) "\/" a) "\/" b]
thus  S1[p,q,((d . (q9,(q `1_4))) "\/" a) "\/" b] by A7, A12, A8, A22, TARSKI:def_1; ::_thesis: verum
end;
supposeA23: ( q in A & p = {{{A}}} ) ; ::_thesis: ex r being Element of L st S1[p,q,r]
then reconsider q9 = q as Element of A ;
take  (d . (q9,(q `2_4))) "\/" b ; ::_thesis: S1[p,q,(d . (q9,(q `2_4))) "\/" b]
thus  S1[p,q,(d . (q9,(q `2_4))) "\/" b] by A7, A11, A12, A8, A23, TARSKI:def_1; ::_thesis: verum
end;
end;
end;
consider f being Function of [:(new_set A),(new_set A):], the carrier of L such that
A24: for p, q being Element of new_set A holds S1[p,q,f . (p,q)] from BINOP_1:sch_3(A2);
 {{{A}}} in {{A},{{A}},{{{A}}}} by ENUMSET1:def_1;
then A25: {{{A}}} in new_set A by XBOOLE_0:def_3;
reconsider f = f as BiFunction of (new_set A),L ;
 {A} in {{A},{{A}},{{{A}}}} by ENUMSET1:def_1;
then A26: {A} in new_set A by XBOOLE_0:def_3;
A27: for u being Element of A holds
( f . ({A},u) = (d . (u,(q `1_4))) "\/" a & f . ({{A}},u) = ((d . (u,(q `1_4))) "\/" a) "\/" b & f . ({{{A}}},u) = (d . (u,(q `2_4))) "\/" b )
proof
let u be Element of A; ::_thesis: ( f . ({A},u) = (d . (u,(q `1_4))) "\/" a & f . ({{A}},u) = ((d . (u,(q `1_4))) "\/" a) "\/" b & f . ({{{A}}},u) = (d . (u,(q `2_4))) "\/" b )
reconsider u9 = u as Element of new_set A by XBOOLE_0:def_3;
 ex u1 being Element of A st
( u1 = u9 & f . ({A},u9) = (d . (u1,(q `1_4))) "\/" a ) by A26, A24;
hence  f . ({A},u) = (d . (u,(q `1_4))) "\/" a ; ::_thesis: ( f . ({{A}},u) = ((d . (u,(q `1_4))) "\/" a) "\/" b & f . ({{{A}}},u) = (d . (u,(q `2_4))) "\/" b )
 ex u2 being Element of A st
( u2 = u9 & f . ({{A}},u9) = ((d . (u2,(q `1_4))) "\/" a) "\/" b ) by A1, A24;
hence  f . ({{A}},u) = ((d . (u,(q `1_4))) "\/" a) "\/" b ; ::_thesis: f . ({{{A}}},u) = (d . (u,(q `2_4))) "\/" b
 ex u3 being Element of A st
( u3 = u9 & f . ({{{A}}},u9) = (d . (u3,(q `2_4))) "\/" b ) by A25, A24;
hence  f . ({{{A}}},u) = (d . (u,(q `2_4))) "\/" b ; ::_thesis: verum
end;
take  f ; ::_thesis: ( ( for u, v being Element of A holds f . (u,v) = d . (u,v) ) & f . ({A},{A}) = Bottom L & f . ({{A}},{{A}}) = Bottom L & f . ({{{A}}},{{{A}}}) = Bottom L & f . ({{A}},{{{A}}}) = q `3_4 & f . ({{{A}}},{{A}}) = q `3_4 & f . ({A},{{A}}) = q `4_4 & f . ({{A}},{A}) = q `4_4 & f . ({A},{{{A}}}) = (q `3_4) "\/" (q `4_4) & f . ({{{A}}},{A}) = (q `3_4) "\/" (q `4_4) & ( for u being Element of A holds
( f . (u,{A}) = (d . (u,(q `1_4))) "\/" (q `3_4) & f . ({A},u) = (d . (u,(q `1_4))) "\/" (q `3_4) & f . (u,{{A}}) = ((d . (u,(q `1_4))) "\/" (q `3_4)) "\/" (q `4_4) & f . ({{A}},u) = ((d . (u,(q `1_4))) "\/" (q `3_4)) "\/" (q `4_4) & f . (u,{{{A}}}) = (d . (u,(q `2_4))) "\/" (q `4_4) & f . ({{{A}}},u) = (d . (u,(q `2_4))) "\/" (q `4_4) ) ) )
A28: for u, v being Element of A holds f . (u,v) = d . (u,v)
proof
let u, v be Element of A; ::_thesis: f . (u,v) = d . (u,v)
reconsider u9 = u, v9 = v as Element of new_set A by XBOOLE_0:def_3;
thus f . (u,v) = f . (u9,v9)
.= d . (u,v) by A24 ; ::_thesis: verum
end;
 for u being Element of A holds
( f . (u,{A}) = (d . (u,(q `1_4))) "\/" a & f . (u,{{A}}) = ((d . (u,(q `1_4))) "\/" a) "\/" b & f . (u,{{{A}}}) = (d . (u,(q `2_4))) "\/" b )
proof
let u be Element of A; ::_thesis: ( f . (u,{A}) = (d . (u,(q `1_4))) "\/" a & f . (u,{{A}}) = ((d . (u,(q `1_4))) "\/" a) "\/" b & f . (u,{{{A}}}) = (d . (u,(q `2_4))) "\/" b )
reconsider u9 = u as Element of new_set A by XBOOLE_0:def_3;
 ex u1 being Element of A st
( u1 = u9 & f . (u9,{A}) = (d . (u1,(q `1_4))) "\/" a ) by A26, A24;
hence  f . (u,{A}) = (d . (u,(q `1_4))) "\/" a ; ::_thesis: ( f . (u,{{A}}) = ((d . (u,(q `1_4))) "\/" a) "\/" b & f . (u,{{{A}}}) = (d . (u,(q `2_4))) "\/" b )
 ex u2 being Element of A st
( u2 = u9 & f . (u9,{{A}}) = ((d . (u2,(q `1_4))) "\/" a) "\/" b ) by A1, A24;
hence  f . (u,{{A}}) = ((d . (u,(q `1_4))) "\/" a) "\/" b ; ::_thesis: f . (u,{{{A}}}) = (d . (u,(q `2_4))) "\/" b
 ex u3 being Element of A st
( u3 = u9 & f . (u9,{{{A}}}) = (d . (u3,(q `2_4))) "\/" b ) by A25, A24;
hence  f . (u,{{{A}}}) = (d . (u,(q `2_4))) "\/" b ; ::_thesis: verum
end;
hence  ( ( for u, v being Element of A holds f . (u,v) = d . (u,v) ) & f . ({A},{A}) = Bottom L & f . ({{A}},{{A}}) = Bottom L & f . ({{{A}}},{{{A}}}) = Bottom L & f . ({{A}},{{{A}}}) = q `3_4 & f . ({{{A}}},{{A}}) = q `3_4 & f . ({A},{{A}}) = q `4_4 & f . ({{A}},{A}) = q `4_4 & f . ({A},{{{A}}}) = (q `3_4) "\/" (q `4_4) & f . ({{{A}}},{A}) = (q `3_4) "\/" (q `4_4) & ( for u being Element of A holds
( f . (u,{A}) = (d . (u,(q `1_4))) "\/" (q `3_4) & f . ({A},u) = (d . (u,(q `1_4))) "\/" (q `3_4) & f . (u,{{A}}) = ((d . (u,(q `1_4))) "\/" (q `3_4)) "\/" (q `4_4) & f . ({{A}},u) = ((d . (u,(q `1_4))) "\/" (q `3_4)) "\/" (q `4_4) & f . (u,{{{A}}}) = (d . (u,(q `2_4))) "\/" (q `4_4) & f . ({{{A}}},u) = (d . (u,(q `2_4))) "\/" (q `4_4) ) ) ) by A26, A1, A25, A24, A28, A27; ::_thesis: verum
end;
uniqueness
 for b1, b2 being BiFunction of (new_set A),L st ( for u, v being Element of A holds b1 . (u,v) = d . (u,v) ) & b1 . ({A},{A}) = Bottom L & b1 . ({{A}},{{A}}) = Bottom L & b1 . ({{{A}}},{{{A}}}) = Bottom L & b1 . ({{A}},{{{A}}}) = q `3_4 & b1 . ({{{A}}},{{A}}) = q `3_4 & b1 . ({A},{{A}}) = q `4_4 & b1 . ({{A}},{A}) = q `4_4 & b1 . ({A},{{{A}}}) = (q `3_4) "\/" (q `4_4) & b1 . ({{{A}}},{A}) = (q `3_4) "\/" (q `4_4) & ( for u being Element of A holds
( b1 . (u,{A}) = (d . (u,(q `1_4))) "\/" (q `3_4) & b1 . ({A},u) = (d . (u,(q `1_4))) "\/" (q `3_4) & b1 . (u,{{A}}) = ((d . (u,(q `1_4))) "\/" (q `3_4)) "\/" (q `4_4) & b1 . ({{A}},u) = ((d . (u,(q `1_4))) "\/" (q `3_4)) "\/" (q `4_4) & b1 . (u,{{{A}}}) = (d . (u,(q `2_4))) "\/" (q `4_4) & b1 . ({{{A}}},u) = (d . (u,(q `2_4))) "\/" (q `4_4) ) ) & ( for u, v being Element of A holds b2 . (u,v) = d . (u,v) ) & b2 . ({A},{A}) = Bottom L & b2 . ({{A}},{{A}}) = Bottom L & b2 . ({{{A}}},{{{A}}}) = Bottom L & b2 . ({{A}},{{{A}}}) = q `3_4 & b2 . ({{{A}}},{{A}}) = q `3_4 & b2 . ({A},{{A}}) = q `4_4 & b2 . ({{A}},{A}) = q `4_4 & b2 . ({A},{{{A}}}) = (q `3_4) "\/" (q `4_4) & b2 . ({{{A}}},{A}) = (q `3_4) "\/" (q `4_4) & ( for u being Element of A holds
( b2 . (u,{A}) = (d . (u,(q `1_4))) "\/" (q `3_4) & b2 . ({A},u) = (d . (u,(q `1_4))) "\/" (q `3_4) & b2 . (u,{{A}}) = ((d . (u,(q `1_4))) "\/" (q `3_4)) "\/" (q `4_4) & b2 . ({{A}},u) = ((d . (u,(q `1_4))) "\/" (q `3_4)) "\/" (q `4_4) & b2 . (u,{{{A}}}) = (d . (u,(q `2_4))) "\/" (q `4_4) & b2 . ({{{A}}},u) = (d . (u,(q `2_4))) "\/" (q `4_4) ) ) holds
b1 = b2
proof
set x = q `1_4 ;
set y = q `2_4 ;
set a = q `3_4 ;
set b = q `4_4 ;
let f1, f2 be BiFunction of (new_set A),L; ::_thesis: ( ( for u, v being Element of A holds f1 . (u,v) = d . (u,v) ) & f1 . ({A},{A}) = Bottom L & f1 . ({{A}},{{A}}) = Bottom L & f1 . ({{{A}}},{{{A}}}) = Bottom L & f1 . ({{A}},{{{A}}}) = q `3_4 & f1 . ({{{A}}},{{A}}) = q `3_4 & f1 . ({A},{{A}}) = q `4_4 & f1 . ({{A}},{A}) = q `4_4 & f1 . ({A},{{{A}}}) = (q `3_4) "\/" (q `4_4) & f1 . ({{{A}}},{A}) = (q `3_4) "\/" (q `4_4) & ( for u being Element of A holds
( f1 . (u,{A}) = (d . (u,(q `1_4))) "\/" (q `3_4) & f1 . ({A},u) = (d . (u,(q `1_4))) "\/" (q `3_4) & f1 . (u,{{A}}) = ((d . (u,(q `1_4))) "\/" (q `3_4)) "\/" (q `4_4) & f1 . ({{A}},u) = ((d . (u,(q `1_4))) "\/" (q `3_4)) "\/" (q `4_4) & f1 . (u,{{{A}}}) = (d . (u,(q `2_4))) "\/" (q `4_4) & f1 . ({{{A}}},u) = (d . (u,(q `2_4))) "\/" (q `4_4) ) ) & ( for u, v being Element of A holds f2 . (u,v) = d . (u,v) ) & f2 . ({A},{A}) = Bottom L & f2 . ({{A}},{{A}}) = Bottom L & f2 . ({{{A}}},{{{A}}}) = Bottom L & f2 . ({{A}},{{{A}}}) = q `3_4 & f2 . ({{{A}}},{{A}}) = q `3_4 & f2 . ({A},{{A}}) = q `4_4 & f2 . ({{A}},{A}) = q `4_4 & f2 . ({A},{{{A}}}) = (q `3_4) "\/" (q `4_4) & f2 . ({{{A}}},{A}) = (q `3_4) "\/" (q `4_4) & ( for u being Element of A holds
( f2 . (u,{A}) = (d . (u,(q `1_4))) "\/" (q `3_4) & f2 . ({A},u) = (d . (u,(q `1_4))) "\/" (q `3_4) & f2 . (u,{{A}}) = ((d . (u,(q `1_4))) "\/" (q `3_4)) "\/" (q `4_4) & f2 . ({{A}},u) = ((d . (u,(q `1_4))) "\/" (q `3_4)) "\/" (q `4_4) & f2 . (u,{{{A}}}) = (d . (u,(q `2_4))) "\/" (q `4_4) & f2 . ({{{A}}},u) = (d . (u,(q `2_4))) "\/" (q `4_4) ) ) implies f1 = f2 )
assume that
A29: for u, v being Element of A holds f1 . (u,v) = d . (u,v) and
A30: f1 . ({A},{A}) = Bottom L and
A31: f1 . ({{A}},{{A}}) = Bottom L and
A32: f1 . ({{{A}}},{{{A}}}) = Bottom L and
A33: f1 . ({{A}},{{{A}}}) = q `3_4 and
A34: f1 . ({{{A}}},{{A}}) = q `3_4 and
A35: f1 . ({A},{{A}}) = q `4_4 and
A36: f1 . ({{A}},{A}) = q `4_4 and
A37: f1 . ({A},{{{A}}}) = (q `3_4) "\/" (q `4_4) and
A38: f1 . ({{{A}}},{A}) = (q `3_4) "\/" (q `4_4) and
A39: for u being Element of A holds
( f1 . (u,{A}) = (d . (u,(q `1_4))) "\/" (q `3_4) & f1 . ({A},u) = (d . (u,(q `1_4))) "\/" (q `3_4) & f1 . (u,{{A}}) = ((d . (u,(q `1_4))) "\/" (q `3_4)) "\/" (q `4_4) & f1 . ({{A}},u) = ((d . (u,(q `1_4))) "\/" (q `3_4)) "\/" (q `4_4) & f1 . (u,{{{A}}}) = (d . (u,(q `2_4))) "\/" (q `4_4) & f1 . ({{{A}}},u) = (d . (u,(q `2_4))) "\/" (q `4_4) ) and
A40: for u, v being Element of A holds f2 . (u,v) = d . (u,v) and
A41: f2 . ({A},{A}) = Bottom L and
A42: f2 . ({{A}},{{A}}) = Bottom L and
A43: f2 . ({{{A}}},{{{A}}}) = Bottom L and
A44: f2 . ({{A}},{{{A}}}) = q `3_4 and
A45: f2 . ({{{A}}},{{A}}) = q `3_4 and
A46: f2 . ({A},{{A}}) = q `4_4 and
A47: f2 . ({{A}},{A}) = q `4_4 and
A48: f2 . ({A},{{{A}}}) = (q `3_4) "\/" (q `4_4) and
A49: f2 . ({{{A}}},{A}) = (q `3_4) "\/" (q `4_4) and
A50: for u being Element of A holds
( f2 . (u,{A}) = (d . (u,(q `1_4))) "\/" (q `3_4) & f2 . ({A},u) = (d . (u,(q `1_4))) "\/" (q `3_4) & f2 . (u,{{A}}) = ((d . (u,(q `1_4))) "\/" (q `3_4)) "\/" (q `4_4) & f2 . ({{A}},u) = ((d . (u,(q `1_4))) "\/" (q `3_4)) "\/" (q `4_4) & f2 . (u,{{{A}}}) = (d . (u,(q `2_4))) "\/" (q `4_4) & f2 . ({{{A}}},u) = (d . (u,(q `2_4))) "\/" (q `4_4) ) ; ::_thesis: f1 = f2
now__::_thesis:_for_p,_q_being_Element_of_new_set_A_holds_f1_._(p,q)_=_f2_._(p,q)
let p, q be Element of new_set A; ::_thesis: f1 . (b1,b2) = f2 . (b1,b2)
A51: ( p in A or p in {{A},{{A}},{{{A}}}} ) by XBOOLE_0:def_3;
A52: ( q in A or q in {{A},{{A}},{{{A}}}} ) by XBOOLE_0:def_3;
percases ( ( p in A & q in A ) or ( p in A & q = {A} ) or ( p in A & q = {{A}} ) or ( p in A & q = {{{A}}} ) or ( p = {A} & q in A ) or ( p = {A} & q = {A} ) or ( p = {A} & q = {{A}} ) or ( p = {A} & q = {{{A}}} ) or ( p = {{A}} & q in A ) or ( p = {{A}} & q = {A} ) or ( p = {{A}} & q = {{A}} ) or ( p = {{A}} & q = {{{A}}} ) or ( p = {{{A}}} & q in A ) or ( p = {{{A}}} & q = {A} ) or ( p = {{{A}}} & q = {{A}} ) or ( p = {{{A}}} & q = {{{A}}} ) ) by A51, A52, ENUMSET1:def_1;
supposeA53: ( p in A & q in A ) ; ::_thesis: f1 . (b1,b2) = f2 . (b1,b2)
hence f1 . (p,q) = d . (p,q) by A29
.= f2 . (p,q) by A40, A53 ;
::_thesis: verum
end;
supposeA54: ( p in A & q = {A} ) ; ::_thesis: f1 . (b1,b2) = f2 . (b1,b2)
then reconsider p9 = p as Element of A ;
thus f1 . (p,q) = (d . (p9,(q `1_4))) "\/" (q `3_4) by A39, A54
.= f2 . (p,q) by A50, A54 ; ::_thesis: verum
end;
supposeA55: ( p in A & q = {{A}} ) ; ::_thesis: f1 . (b1,b2) = f2 . (b1,b2)
then reconsider p9 = p as Element of A ;
thus f1 . (p,q) = ((d . (p9,(q `1_4))) "\/" (q `3_4)) "\/" (q `4_4) by A39, A55
.= f2 . (p,q) by A50, A55 ; ::_thesis: verum
end;
supposeA56: ( p in A & q = {{{A}}} ) ; ::_thesis: f1 . (b1,b2) = f2 . (b1,b2)
then reconsider p9 = p as Element of A ;
thus f1 . (p,q) = (d . (p9,(q `2_4))) "\/" (q `4_4) by A39, A56
.= f2 . (p,q) by A50, A56 ; ::_thesis: verum
end;
supposeA57: ( p = {A} & q in A ) ; ::_thesis: f1 . (b1,b2) = f2 . (b1,b2)
then reconsider q9 = q as Element of A ;
thus f1 . (p,q) = (d . (q9,(q `1_4))) "\/" (q `3_4) by A39, A57
.= f2 . (p,q) by A50, A57 ; ::_thesis: verum
end;
suppose ( p = {A} & q = {A} ) ; ::_thesis: f1 . (b1,b2) = f2 . (b1,b2)
hence  f1 . (p,q) = f2 . (p,q) by A30, A41; ::_thesis: verum
end;
suppose ( p = {A} & q = {{A}} ) ; ::_thesis: f1 . (b1,b2) = f2 . (b1,b2)
hence  f1 . (p,q) = f2 . (p,q) by A35, A46; ::_thesis: verum
end;
suppose ( p = {A} & q = {{{A}}} ) ; ::_thesis: f1 . (b1,b2) = f2 . (b1,b2)
hence  f1 . (p,q) = f2 . (p,q) by A37, A48; ::_thesis: verum
end;
supposeA58: ( p = {{A}} & q in A ) ; ::_thesis: f1 . (b1,b2) = f2 . (b1,b2)
then reconsider q9 = q as Element of A ;
thus f1 . (p,q) = ((d . (q9,(q `1_4))) "\/" (q `3_4)) "\/" (q `4_4) by A39, A58
.= f2 . (p,q) by A50, A58 ; ::_thesis: verum
end;
suppose ( p = {{A}} & q = {A} ) ; ::_thesis: f1 . (b1,b2) = f2 . (b1,b2)
hence  f1 . (p,q) = f2 . (p,q) by A36, A47; ::_thesis: verum
end;
suppose ( p = {{A}} & q = {{A}} ) ; ::_thesis: f1 . (b1,b2) = f2 . (b1,b2)
hence  f1 . (p,q) = f2 . (p,q) by A31, A42; ::_thesis: verum
end;
suppose ( p = {{A}} & q = {{{A}}} ) ; ::_thesis: f1 . (b1,b2) = f2 . (b1,b2)
hence  f1 . (p,q) = f2 . (p,q) by A33, A44; ::_thesis: verum
end;
supposeA59: ( p = {{{A}}} & q in A ) ; ::_thesis: f1 . (b1,b2) = f2 . (b1,b2)
then reconsider q9 = q as Element of A ;
thus f1 . (p,q) = (d . (q9,(q `2_4))) "\/" (q `4_4) by A39, A59
.= f2 . (p,q) by A50, A59 ; ::_thesis: verum
end;
suppose ( p = {{{A}}} & q = {A} ) ; ::_thesis: f1 . (b1,b2) = f2 . (b1,b2)
hence  f1 . (p,q) = f2 . (p,q) by A38, A49; ::_thesis: verum
end;
suppose ( p = {{{A}}} & q = {{A}} ) ; ::_thesis: f1 . (b1,b2) = f2 . (b1,b2)
hence  f1 . (p,q) = f2 . (p,q) by A34, A45; ::_thesis: verum
end;
suppose ( p = {{{A}}} & q = {{{A}}} ) ; ::_thesis: f1 . (b1,b2) = f2 . (b1,b2)
hence  f1 . (p,q) = f2 . (p,q) by A32, A43; ::_thesis: verum
end;
end;
end;
hence  f1 = f2 by BINOP_1:2; ::_thesis: verum
end;
end;

:: deftheorem Def10 defines new_bi_fun LATTICE5:def_10_:_
 for A being non empty set
for L being lower-bounded LATTICE
for d being BiFunction of A,L
for q being Element of [:A,A, the carrier of L, the carrier of L:]
for b5 being BiFunction of (new_set A),L holds
( b5 = new_bi_fun (d,q) iff ( ( for u, v being Element of A holds b5 . (u,v) = d . (u,v) ) & b5 . ({A},{A}) = Bottom L & b5 . ({{A}},{{A}}) = Bottom L & b5 . ({{{A}}},{{{A}}}) = Bottom L & b5 . ({{A}},{{{A}}}) = q `3_4 & b5 . ({{{A}}},{{A}}) = q `3_4 & b5 . ({A},{{A}}) = q `4_4 & b5 . ({{A}},{A}) = q `4_4 & b5 . ({A},{{{A}}}) = (q `3_4) "\/" (q `4_4) & b5 . ({{{A}}},{A}) = (q `3_4) "\/" (q `4_4) & ( for u being Element of A holds
( b5 . (u,{A}) = (d . (u,(q `1_4))) "\/" (q `3_4) & b5 . ({A},u) = (d . (u,(q `1_4))) "\/" (q `3_4) & b5 . (u,{{A}}) = ((d . (u,(q `1_4))) "\/" (q `3_4)) "\/" (q `4_4) & b5 . ({{A}},u) = ((d . (u,(q `1_4))) "\/" (q `3_4)) "\/" (q `4_4) & b5 . (u,{{{A}}}) = (d . (u,(q `2_4))) "\/" (q `4_4) & b5 . ({{{A}}},u) = (d . (u,(q `2_4))) "\/" (q `4_4) ) ) ) );

theorem Th16: :: LATTICE5:16
for A being non empty set
for L being lower-bounded LATTICE
for d being BiFunction of A,L st d is zeroed holds
for q being Element of [:A,A, the carrier of L, the carrier of L:] holds new_bi_fun (d,q) is zeroed
proof
let A be non empty set ; ::_thesis: for L being lower-bounded LATTICE
for d being BiFunction of A,L st d is zeroed holds
for q being Element of [:A,A, the carrier of L, the carrier of L:] holds new_bi_fun (d,q) is zeroed
let L be lower-bounded LATTICE; ::_thesis: for d being BiFunction of A,L st d is zeroed holds
for q being Element of [:A,A, the carrier of L, the carrier of L:] holds new_bi_fun (d,q) is zeroed
let d be BiFunction of A,L; ::_thesis: ( d is zeroed implies for q being Element of [:A,A, the carrier of L, the carrier of L:] holds new_bi_fun (d,q) is zeroed )
assume A1: d is zeroed ; ::_thesis: for q being Element of [:A,A, the carrier of L, the carrier of L:] holds new_bi_fun (d,q) is zeroed
let q be Element of [:A,A, the carrier of L, the carrier of L:]; ::_thesis: new_bi_fun (d,q) is zeroed
set f = new_bi_fun (d,q);
 for u being Element of new_set A holds (new_bi_fun (d,q)) . (u,u) = Bottom L
proof
let u be Element of new_set A; ::_thesis: (new_bi_fun (d,q)) . (u,u) = Bottom L
A2: ( u in A or u in {{A},{{A}},{{{A}}}} ) by XBOOLE_0:def_3;
percases ( u in A or u = {A} or u = {{A}} or u = {{{A}}} ) by A2, ENUMSET1:def_1;
suppose u in A ; ::_thesis: (new_bi_fun (d,q)) . (u,u) = Bottom L
then reconsider u9 = u as Element of A ;
thus (new_bi_fun (d,q)) . (u,u) = d . (u9,u9) by Def10
.= Bottom L by A1, Def6 ; ::_thesis: verum
end;
suppose ( u = {A} or u = {{A}} or u = {{{A}}} ) ; ::_thesis: (new_bi_fun (d,q)) . (u,u) = Bottom L
hence  (new_bi_fun (d,q)) . (u,u) = Bottom L by Def10; ::_thesis: verum
end;
end;
end;
hence  new_bi_fun (d,q) is zeroed by Def6; ::_thesis: verum
end;

theorem Th17: :: LATTICE5:17
for A being non empty set
for L being lower-bounded LATTICE
for d being BiFunction of A,L st d is symmetric holds
for q being Element of [:A,A, the carrier of L, the carrier of L:] holds new_bi_fun (d,q) is symmetric
proof
let A be non empty set ; ::_thesis: for L being lower-bounded LATTICE
for d being BiFunction of A,L st d is symmetric holds
for q being Element of [:A,A, the carrier of L, the carrier of L:] holds new_bi_fun (d,q) is symmetric
let L be lower-bounded LATTICE; ::_thesis: for d being BiFunction of A,L st d is symmetric holds
for q being Element of [:A,A, the carrier of L, the carrier of L:] holds new_bi_fun (d,q) is symmetric
let d be BiFunction of A,L; ::_thesis: ( d is symmetric implies for q being Element of [:A,A, the carrier of L, the carrier of L:] holds new_bi_fun (d,q) is symmetric )
assume A1: d is symmetric ; ::_thesis: for q being Element of [:A,A, the carrier of L, the carrier of L:] holds new_bi_fun (d,q) is symmetric
let q be Element of [:A,A, the carrier of L, the carrier of L:]; ::_thesis: new_bi_fun (d,q) is symmetric
set f = new_bi_fun (d,q);
set x = q `1_4 ;
set y = q `2_4 ;
set a = q `3_4 ;
set b = q `4_4 ;
let p, q be Element of new_set A; :: according to LATTICE5:def_5 ::_thesis: (new_bi_fun (d,q)) . (p,q) = (new_bi_fun (d,q)) . (q,p)
A2: ( p in A or p in {{A},{{A}},{{{A}}}} ) by XBOOLE_0:def_3;
A3: ( q in A or q in {{A},{{A}},{{{A}}}} ) by XBOOLE_0:def_3;
percases ( ( p in A & q in A ) or ( p in A & q = {A} ) or ( p in A & q = {{A}} ) or ( p in A & q = {{{A}}} ) or ( p = {A} & q in A ) or ( p = {A} & q = {A} ) or ( p = {A} & q = {{A}} ) or ( p = {A} & q = {{{A}}} ) or ( p = {{A}} & q in A ) or ( p = {{A}} & q = {A} ) or ( p = {{A}} & q = {{A}} ) or ( p = {{A}} & q = {{{A}}} ) or ( p = {{{A}}} & q in A ) or ( p = {{{A}}} & q = {A} ) or ( p = {{{A}}} & q = {{A}} ) or ( p = {{{A}}} & q = {{{A}}} ) ) by A2, A3, ENUMSET1:def_1;
suppose ( p in A & q in A ) ; ::_thesis: (new_bi_fun (d,q)) . (p,q) = (new_bi_fun (d,q)) . (q,p)
then reconsider p9 = p, q9 = q as Element of A ;
thus (new_bi_fun (d,q)) . (p,q) = d . (p9,q9) by Def10
.= d . (q9,p9) by A1, Def5
.= (new_bi_fun (d,q)) . (q,p) by Def10 ; ::_thesis: verum
end;
supposeA4: ( p in A & q = {A} ) ; ::_thesis: (new_bi_fun (d,q)) . (p,q) = (new_bi_fun (d,q)) . (q,p)
then reconsider p9 = p as Element of A ;
thus (new_bi_fun (d,q)) . (p,q) = (d . (p9,(q `1_4))) "\/" (q `3_4) by A4, Def10
.= (new_bi_fun (d,q)) . (q,p) by A4, Def10 ; ::_thesis: verum
end;
supposeA5: ( p in A & q = {{A}} ) ; ::_thesis: (new_bi_fun (d,q)) . (p,q) = (new_bi_fun (d,q)) . (q,p)
then reconsider p9 = p as Element of A ;
thus (new_bi_fun (d,q)) . (p,q) = ((d . (p9,(q `1_4))) "\/" (q `3_4)) "\/" (q `4_4) by A5, Def10
.= (new_bi_fun (d,q)) . (q,p) by A5, Def10 ; ::_thesis: verum
end;
supposeA6: ( p in A & q = {{{A}}} ) ; ::_thesis: (new_bi_fun (d,q)) . (p,q) = (new_bi_fun (d,q)) . (q,p)
then reconsider p9 = p as Element of A ;
thus (new_bi_fun (d,q)) . (p,q) = (d . (p9,(q `2_4))) "\/" (q `4_4) by A6, Def10
.= (new_bi_fun (d,q)) . (q,p) by A6, Def10 ; ::_thesis: verum
end;
supposeA7: ( p = {A} & q in A ) ; ::_thesis: (new_bi_fun (d,q)) . (p,q) = (new_bi_fun (d,q)) . (q,p)
then reconsider q9 = q as Element of A ;
thus (new_bi_fun (d,q)) . (p,q) = (d . (q9,(q `1_4))) "\/" (q `3_4) by A7, Def10
.= (new_bi_fun (d,q)) . (q,p) by A7, Def10 ; ::_thesis: verum
end;
suppose ( p = {A} & q = {A} ) ; ::_thesis: (new_bi_fun (d,q)) . (p,q) = (new_bi_fun (d,q)) . (q,p)
hence  (new_bi_fun (d,q)) . (p,q) = (new_bi_fun (d,q)) . (q,p) ; ::_thesis: verum
end;
supposeA8: ( p = {A} & q = {{A}} ) ; ::_thesis: (new_bi_fun (d,q)) . (p,q) = (new_bi_fun (d,q)) . (q,p)
hence (new_bi_fun (d,q)) . (p,q) = q `4_4 by Def10
.= (new_bi_fun (d,q)) . (q,p) by A8, Def10 ;
::_thesis: verum
end;
supposeA9: ( p = {A} & q = {{{A}}} ) ; ::_thesis: (new_bi_fun (d,q)) . (p,q) = (new_bi_fun (d,q)) . (q,p)
hence (new_bi_fun (d,q)) . (p,q) = (q `3_4) "\/" (q `4_4) by Def10
.= (new_bi_fun (d,q)) . (q,p) by A9, Def10 ;
::_thesis: verum
end;
supposeA10: ( p = {{A}} & q in A ) ; ::_thesis: (new_bi_fun (d,q)) . (p,q) = (new_bi_fun (d,q)) . (q,p)
then reconsider q9 = q as Element of A ;
thus (new_bi_fun (d,q)) . (p,q) = ((d . (q9,(q `1_4))) "\/" (q `3_4)) "\/" (q `4_4) by A10, Def10
.= (new_bi_fun (d,q)) . (q,p) by A10, Def10 ; ::_thesis: verum
end;
supposeA11: ( p = {{A}} & q = {A} ) ; ::_thesis: (new_bi_fun (d,q)) . (p,q) = (new_bi_fun (d,q)) . (q,p)
hence (new_bi_fun (d,q)) . (p,q) = q `4_4 by Def10
.= (new_bi_fun (d,q)) . (q,p) by A11, Def10 ;
::_thesis: verum
end;
suppose ( p = {{A}} & q = {{A}} ) ; ::_thesis: (new_bi_fun (d,q)) . (p,q) = (new_bi_fun (d,q)) . (q,p)
hence  (new_bi_fun (d,q)) . (p,q) = (new_bi_fun (d,q)) . (q,p) ; ::_thesis: verum
end;
supposeA12: ( p = {{A}} & q = {{{A}}} ) ; ::_thesis: (new_bi_fun (d,q)) . (p,q) = (new_bi_fun (d,q)) . (q,p)
hence (new_bi_fun (d,q)) . (p,q) = q `3_4 by Def10
.= (new_bi_fun (d,q)) . (q,p) by A12, Def10 ;
::_thesis: verum
end;
supposeA13: ( p = {{{A}}} & q in A ) ; ::_thesis: (new_bi_fun (d,q)) . (p,q) = (new_bi_fun (d,q)) . (q,p)
then reconsider q9 = q as Element of A ;
thus (new_bi_fun (d,q)) . (p,q) = (d . (q9,(q `2_4))) "\/" (q `4_4) by A13, Def10
.= (new_bi_fun (d,q)) . (q,p) by A13, Def10 ; ::_thesis: verum
end;
supposeA14: ( p = {{{A}}} & q = {A} ) ; ::_thesis: (new_bi_fun (d,q)) . (p,q) = (new_bi_fun (d,q)) . (q,p)
hence (new_bi_fun (d,q)) . (p,q) = (q `3_4) "\/" (q `4_4) by Def10
.= (new_bi_fun (d,q)) . (q,p) by A14, Def10 ;
::_thesis: verum
end;
supposeA15: ( p = {{{A}}} & q = {{A}} ) ; ::_thesis: (new_bi_fun (d,q)) . (p,q) = (new_bi_fun (d,q)) . (q,p)
hence (new_bi_fun (d,q)) . (p,q) = q `3_4 by Def10
.= (new_bi_fun (d,q)) . (q,p) by A15, Def10 ;
::_thesis: verum
end;
suppose ( p = {{{A}}} & q = {{{A}}} ) ; ::_thesis: (new_bi_fun (d,q)) . (p,q) = (new_bi_fun (d,q)) . (q,p)
hence  (new_bi_fun (d,q)) . (p,q) = (new_bi_fun (d,q)) . (q,p) ; ::_thesis: verum
end;
end;
end;

theorem Th18: :: LATTICE5:18
for A being non empty set
for L being lower-bounded LATTICE
for d being BiFunction of A,L st d is symmetric & d is u.t.i. holds
for q being Element of [:A,A, the carrier of L, the carrier of L:] st d . ((q `1_4),(q `2_4)) <= (q `3_4) "\/" (q `4_4) holds
new_bi_fun (d,q) is u.t.i.
proof
let A be non empty set ; ::_thesis: for L being lower-bounded LATTICE
for d being BiFunction of A,L st d is symmetric & d is u.t.i. holds
for q being Element of [:A,A, the carrier of L, the carrier of L:] st d . ((q `1_4),(q `2_4)) <= (q `3_4) "\/" (q `4_4) holds
new_bi_fun (d,q) is u.t.i.
let L be lower-bounded LATTICE; ::_thesis: for d being BiFunction of A,L st d is symmetric & d is u.t.i. holds
for q being Element of [:A,A, the carrier of L, the carrier of L:] st d . ((q `1_4),(q `2_4)) <= (q `3_4) "\/" (q `4_4) holds
new_bi_fun (d,q) is u.t.i.
let d be BiFunction of A,L; ::_thesis: ( d is symmetric & d is u.t.i. implies for q being Element of [:A,A, the carrier of L, the carrier of L:] st d . ((q `1_4),(q `2_4)) <= (q `3_4) "\/" (q `4_4) holds
new_bi_fun (d,q) is u.t.i. )
assume that
A1: d is symmetric and
A2: d is u.t.i. ; ::_thesis: for q being Element of [:A,A, the carrier of L, the carrier of L:] st d . ((q `1_4),(q `2_4)) <= (q `3_4) "\/" (q `4_4) holds
new_bi_fun (d,q) is u.t.i.
reconsider B = {{A},{{A}},{{{A}}}} as non empty set ;
let q be Element of [:A,A, the carrier of L, the carrier of L:]; ::_thesis: ( d . ((q `1_4),(q `2_4)) <= (q `3_4) "\/" (q `4_4) implies new_bi_fun (d,q) is u.t.i. )
set x = q `1_4 ;
set y = q `2_4 ;
set f = new_bi_fun (d,q);
reconsider a = q `3_4 , b = q `4_4 as Element of L ;
A3: for p, q, u being Element of new_set A st p in A & q in B & u in A holds
(new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u))
proof
let p, q, u be Element of new_set A; ::_thesis: ( p in A & q in B & u in A implies (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u)) )
assume A4: ( p in A & q in B & u in A ) ; ::_thesis: (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u))
percases ( ( p in A & u in A & q = {A} ) or ( p in A & u in A & q = {{A}} ) or ( p in A & u in A & q = {{{A}}} ) ) by A4, ENUMSET1:def_1;
supposeA5: ( p in A & u in A & q = {A} ) ; ::_thesis: (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u))
then reconsider p9 = p, u9 = u as Element of A ;
 d . (p9,u9) <= (d . (p9,(q `1_4))) "\/" (d . ((q `1_4),u9)) by A2, Def7;
then A6: ( (d . (p9,(q `1_4))) "\/" (d . (u9,(q `1_4))) <= ((d . (p9,(q `1_4))) "\/" (d . (u9,(q `1_4)))) "\/" a & d . (p9,u9) <= (d . (p9,(q `1_4))) "\/" (d . (u9,(q `1_4))) ) by A1, Def5, YELLOW_0:22;
((d . (p9,(q `1_4))) "\/" (d . (u9,(q `1_4)))) "\/" a = (d . (p9,(q `1_4))) "\/" ((d . (u9,(q `1_4))) "\/" a) by LATTICE3:14
.= (d . (p9,(q `1_4))) "\/" ((d . (u9,(q `1_4))) "\/" (a "\/" a)) by YELLOW_5:1
.= (d . (p9,(q `1_4))) "\/" (((d . (u9,(q `1_4))) "\/" a) "\/" a) by LATTICE3:14
.= ((d . (p9,(q `1_4))) "\/" a) "\/" ((d . (u9,(q `1_4))) "\/" a) by LATTICE3:14 ;
then A7: d . (p9,u9) <= ((d . (p9,(q `1_4))) "\/" a) "\/" ((d . (u9,(q `1_4))) "\/" a) by A6, ORDERS_2:3;
 ( (new_bi_fun (d,q)) . (p,q) = (d . (p9,(q `1_4))) "\/" a & (new_bi_fun (d,q)) . (q,u) = (d . (u9,(q `1_4))) "\/" a ) by A5, Def10;
hence  (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u)) by A7, Def10; ::_thesis: verum
end;
supposeA8: ( p in A & u in A & q = {{A}} ) ; ::_thesis: (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u))
then reconsider p9 = p, u9 = u as Element of A ;
 d . (p9,u9) <= (d . (p9,(q `1_4))) "\/" (d . ((q `1_4),u9)) by A2, Def7;
then A9: ( (d . (p9,(q `1_4))) "\/" (d . (u9,(q `1_4))) <= ((d . (p9,(q `1_4))) "\/" (d . (u9,(q `1_4)))) "\/" (a "\/" b) & d . (p9,u9) <= (d . (p9,(q `1_4))) "\/" (d . (u9,(q `1_4))) ) by A1, Def5, YELLOW_0:22;
((d . (p9,(q `1_4))) "\/" (d . (u9,(q `1_4)))) "\/" (a "\/" b) = (d . (p9,(q `1_4))) "\/" ((d . (u9,(q `1_4))) "\/" (a "\/" b)) by LATTICE3:14
.= (d . (p9,(q `1_4))) "\/" ((d . (u9,(q `1_4))) "\/" ((a "\/" b) "\/" (a "\/" b))) by YELLOW_5:1
.= (d . (p9,(q `1_4))) "\/" (((d . (u9,(q `1_4))) "\/" (a "\/" b)) "\/" (a "\/" b)) by LATTICE3:14
.= ((d . (p9,(q `1_4))) "\/" (a "\/" b)) "\/" ((d . (u9,(q `1_4))) "\/" (a "\/" b)) by LATTICE3:14
.= (((d . (p9,(q `1_4))) "\/" a) "\/" b) "\/" ((d . (u9,(q `1_4))) "\/" (a "\/" b)) by LATTICE3:14
.= (((d . (p9,(q `1_4))) "\/" a) "\/" b) "\/" (((d . (u9,(q `1_4))) "\/" a) "\/" b) by LATTICE3:14 ;
then A10: d . (p9,u9) <= (((d . (p9,(q `1_4))) "\/" a) "\/" b) "\/" (((d . (u9,(q `1_4))) "\/" a) "\/" b) by A9, ORDERS_2:3;
 ( (new_bi_fun (d,q)) . (p,q) = ((d . (p9,(q `1_4))) "\/" a) "\/" b & (new_bi_fun (d,q)) . (q,u) = ((d . (u9,(q `1_4))) "\/" a) "\/" b ) by A8, Def10;
hence  (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u)) by A10, Def10; ::_thesis: verum
end;
supposeA11: ( p in A & u in A & q = {{{A}}} ) ; ::_thesis: (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u))
then reconsider p9 = p, u9 = u as Element of A ;
 d . (p9,u9) <= (d . (p9,(q `2_4))) "\/" (d . ((q `2_4),u9)) by A2, Def7;
then A12: ( (d . (p9,(q `2_4))) "\/" (d . (u9,(q `2_4))) <= ((d . (p9,(q `2_4))) "\/" (d . (u9,(q `2_4)))) "\/" b & d . (p9,u9) <= (d . (p9,(q `2_4))) "\/" (d . (u9,(q `2_4))) ) by A1, Def5, YELLOW_0:22;
((d . (p9,(q `2_4))) "\/" (d . (u9,(q `2_4)))) "\/" b = (d . (p9,(q `2_4))) "\/" ((d . (u9,(q `2_4))) "\/" b) by LATTICE3:14
.= (d . (p9,(q `2_4))) "\/" ((d . (u9,(q `2_4))) "\/" (b "\/" b)) by YELLOW_5:1
.= (d . (p9,(q `2_4))) "\/" (((d . (u9,(q `2_4))) "\/" b) "\/" b) by LATTICE3:14
.= ((d . (p9,(q `2_4))) "\/" b) "\/" ((d . (u9,(q `2_4))) "\/" b) by LATTICE3:14 ;
then A13: d . (p9,u9) <= ((d . (p9,(q `2_4))) "\/" b) "\/" ((d . (u9,(q `2_4))) "\/" b) by A12, ORDERS_2:3;
 ( (new_bi_fun (d,q)) . (p,q) = (d . (p9,(q `2_4))) "\/" b & (new_bi_fun (d,q)) . (q,u) = (d . (u9,(q `2_4))) "\/" b ) by A11, Def10;
hence  (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u)) by A13, Def10; ::_thesis: verum
end;
end;
end;
assume A14: d . ((q `1_4),(q `2_4)) <= (q `3_4) "\/" (q `4_4) ; ::_thesis: new_bi_fun (d,q) is u.t.i.
A15: for p, q, u being Element of new_set A st p in B & q in B & u in A holds
(new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u))
proof
let p, q, u be Element of new_set A; ::_thesis: ( p in B & q in B & u in A implies (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u)) )
assume that
A16: ( p in B & q in B ) and
A17: u in A ; ::_thesis: (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u))
reconsider u9 = u as Element of A by A17;
percases ( ( u in A & q = {A} & p = {A} ) or ( u in A & q = {A} & p = {{A}} ) or ( u in A & q = {A} & p = {{{A}}} ) or ( u in A & q = {{A}} & p = {A} ) or ( u in A & q = {{A}} & p = {{A}} ) or ( u in A & q = {{A}} & p = {{{A}}} ) or ( u in A & q = {{{A}}} & p = {A} ) or ( u in A & q = {{{A}}} & p = {{A}} ) or ( u in A & q = {{{A}}} & p = {{{A}}} ) ) by A16, A17, ENUMSET1:def_1;
supposeA18: ( u in A & q = {A} & p = {A} ) ; ::_thesis: (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u))
then ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u)) = (Bottom L) "\/" ((new_bi_fun (d,q)) . (q,u)) by Def10
.= (new_bi_fun (d,q)) . (p,u) by A18, WAYBEL_1:3 ;
hence  (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u)) ; ::_thesis: verum
end;
supposeA19: ( u in A & q = {A} & p = {{A}} ) ; ::_thesis: (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u))
then ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u)) = b "\/" ((new_bi_fun (d,q)) . (q,u)) by Def10
.= ((d . (u9,(q `1_4))) "\/" a) "\/" b by A19, Def10 ;
hence  (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u)) by A19, Def10; ::_thesis: verum
end;
supposeA20: ( u in A & q = {A} & p = {{{A}}} ) ; ::_thesis: (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u))
b "\/" (a "\/" b) = (b "\/" b) "\/" a by LATTICE3:14
.= b "\/" a by YELLOW_5:1
.= b "\/" (a "\/" a) by YELLOW_5:1
.= a "\/" (a "\/" b) by LATTICE3:14 ;
then A21: ((d . (u9,(q `1_4))) "\/" b) "\/" (a "\/" b) = (d . (u9,(q `1_4))) "\/" (a "\/" (a "\/" b)) by LATTICE3:14
.= (a "\/" b) "\/" ((d . (u9,(q `1_4))) "\/" a) by LATTICE3:14
.= ((new_bi_fun (d,q)) . (p,q)) "\/" ((d . (u9,(q `1_4))) "\/" a) by A20, Def10 ;
 d . (u9,(q `2_4)) <= (d . (u9,(q `1_4))) "\/" (d . ((q `1_4),(q `2_4))) by A2, Def7;
then A22: (d . (u9,(q `2_4))) "\/" b <= ((d . (u9,(q `1_4))) "\/" (d . ((q `1_4),(q `2_4)))) "\/" b by WAYBEL_1:2;
 (d . (u9,(q `1_4))) "\/" b <= (d . (u9,(q `1_4))) "\/" b ;
then A23: ( ((d . (u9,(q `1_4))) "\/" (d . ((q `1_4),(q `2_4)))) "\/" b = ((d . (u9,(q `1_4))) "\/" b) "\/" (d . ((q `1_4),(q `2_4))) & ((d . (u9,(q `1_4))) "\/" b) "\/" (d . ((q `1_4),(q `2_4))) <= ((d . (u9,(q `1_4))) "\/" b) "\/" (a "\/" b) ) by A14, LATTICE3:14, YELLOW_3:3;
 (new_bi_fun (d,q)) . (p,u) = (d . (u9,(q `2_4))) "\/" b by A20, Def10;
then  (new_bi_fun (d,q)) . (p,u) <= ((d . (u9,(q `1_4))) "\/" b) "\/" (a "\/" b) by A22, A23, ORDERS_2:3;
hence  (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u)) by A20, A21, Def10; ::_thesis: verum
end;
supposeA24: ( u in A & q = {{A}} & p = {A} ) ; ::_thesis: (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u))
then ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u)) = b "\/" ((new_bi_fun (d,q)) . (q,u)) by Def10
.= b "\/" (((d . (u9,(q `1_4))) "\/" a) "\/" b) by A24, Def10
.= b "\/" (b "\/" ((new_bi_fun (d,q)) . (p,u))) by A24, Def10
.= (b "\/" b) "\/" ((new_bi_fun (d,q)) . (p,u)) by LATTICE3:14
.= b "\/" ((new_bi_fun (d,q)) . (p,u)) by YELLOW_5:1 ;
hence  (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u)) by YELLOW_0:22; ::_thesis: verum
end;
supposeA25: ( u in A & q = {{A}} & p = {{A}} ) ; ::_thesis: (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u))
then ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u)) = (Bottom L) "\/" ((new_bi_fun (d,q)) . (q,u)) by Def10
.= (new_bi_fun (d,q)) . (p,u) by A25, WAYBEL_1:3 ;
hence  (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u)) ; ::_thesis: verum
end;
supposeA26: ( u in A & q = {{A}} & p = {{{A}}} ) ; ::_thesis: (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u))
b "\/" (a "\/" b) = (b "\/" b) "\/" a by LATTICE3:14
.= b "\/" a by YELLOW_5:1
.= b "\/" (a "\/" a) by YELLOW_5:1
.= (a "\/" b) "\/" a by LATTICE3:14 ;
then A27: ((d . (u9,(q `1_4))) "\/" b) "\/" (a "\/" b) = (d . (u9,(q `1_4))) "\/" ((a "\/" b) "\/" a) by LATTICE3:14
.= ((d . (u9,(q `1_4))) "\/" (a "\/" b)) "\/" a by LATTICE3:14
.= (((d . (u9,(q `1_4))) "\/" a) "\/" b) "\/" a by LATTICE3:14
.= ((new_bi_fun (d,q)) . (p,q)) "\/" (((d . (u9,(q `1_4))) "\/" a) "\/" b) by A26, Def10 ;
 (d . (u9,(q `1_4))) "\/" b <= (d . (u9,(q `1_4))) "\/" b ;
then A28: ( ((d . (u9,(q `1_4))) "\/" (d . ((q `1_4),(q `2_4)))) "\/" b = ((d . (u9,(q `1_4))) "\/" b) "\/" (d . ((q `1_4),(q `2_4))) & ((d . (u9,(q `1_4))) "\/" b) "\/" (d . ((q `1_4),(q `2_4))) <= ((d . (u9,(q `1_4))) "\/" b) "\/" (a "\/" b) ) by A14, LATTICE3:14, YELLOW_3:3;
 d . (u9,(q `2_4)) <= (d . (u9,(q `1_4))) "\/" (d . ((q `1_4),(q `2_4))) by A2, Def7;
then A29: (d . (u9,(q `2_4))) "\/" b <= ((d . (u9,(q `1_4))) "\/" (d . ((q `1_4),(q `2_4)))) "\/" b by WAYBEL_1:2;
 (new_bi_fun (d,q)) . (p,u) = (d . (u9,(q `2_4))) "\/" b by A26, Def10;
then  (new_bi_fun (d,q)) . (p,u) <= ((d . (u9,(q `1_4))) "\/" b) "\/" (a "\/" b) by A29, A28, ORDERS_2:3;
hence  (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u)) by A26, A27, Def10; ::_thesis: verum
end;
supposeA30: ( u in A & q = {{{A}}} & p = {A} ) ; ::_thesis: (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u))
A31: a "\/" (a "\/" b) = (a "\/" a) "\/" b by LATTICE3:14
.= a "\/" b by YELLOW_5:1
.= a "\/" (b "\/" b) by YELLOW_5:1
.= b "\/" (a "\/" b) by LATTICE3:14 ;
A32: (a "\/" (d . (u9,(q `2_4)))) "\/" (a "\/" b) = (d . (u9,(q `2_4))) "\/" (a "\/" (a "\/" b)) by LATTICE3:14
.= ((d . (u9,(q `2_4))) "\/" b) "\/" (a "\/" b) by A31, LATTICE3:14
.= ((new_bi_fun (d,q)) . (p,q)) "\/" ((d . (u9,(q `2_4))) "\/" b) by A30, Def10 ;
 a "\/" (d . (u9,(q `2_4))) <= a "\/" (d . (u9,(q `2_4))) ;
then A33: (a "\/" (d . (u9,(q `2_4)))) "\/" (d . ((q `1_4),(q `2_4))) <= (a "\/" (d . (u9,(q `2_4)))) "\/" (a "\/" b) by A14, YELLOW_3:3;
 d . (u9,(q `1_4)) <= (d . (u9,(q `2_4))) "\/" (d . ((q `2_4),(q `1_4))) by A2, Def7;
then A34: (d . (u9,(q `1_4))) "\/" a <= ((d . (u9,(q `2_4))) "\/" (d . ((q `2_4),(q `1_4)))) "\/" a by WAYBEL_1:2;
A35: ((d . (u9,(q `2_4))) "\/" (d . ((q `2_4),(q `1_4)))) "\/" a = (d . ((q `2_4),(q `1_4))) "\/" ((d . (u9,(q `2_4))) "\/" a) by LATTICE3:14
.= (a "\/" (d . (u9,(q `2_4)))) "\/" (d . ((q `1_4),(q `2_4))) by A1, Def5 ;
 (new_bi_fun (d,q)) . (p,u) = (d . (u9,(q `1_4))) "\/" a by A30, Def10;
then  (new_bi_fun (d,q)) . (p,u) <= (a "\/" (d . (u9,(q `2_4)))) "\/" (a "\/" b) by A34, A35, A33, ORDERS_2:3;
hence  (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u)) by A30, A32, Def10; ::_thesis: verum
end;
supposeA36: ( u in A & q = {{{A}}} & p = {{A}} ) ; ::_thesis: (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u))
then A37: (new_bi_fun (d,q)) . (p,u) = ((d . (u9,(q `1_4))) "\/" a) "\/" b by Def10
.= (d . (u9,(q `1_4))) "\/" (a "\/" b) by LATTICE3:14 ;
 (a "\/" b) "\/" (d . (u9,(q `2_4))) <= (a "\/" b) "\/" (d . (u9,(q `2_4))) ;
then A38: ((a "\/" b) "\/" (d . (u9,(q `2_4)))) "\/" (d . ((q `1_4),(q `2_4))) <= ((a "\/" b) "\/" (d . (u9,(q `2_4)))) "\/" (a "\/" b) by A14, YELLOW_3:3;
 d . (u9,(q `1_4)) <= (d . (u9,(q `2_4))) "\/" (d . ((q `2_4),(q `1_4))) by A2, Def7;
then A39: (d . (u9,(q `1_4))) "\/" (a "\/" b) <= ((d . (u9,(q `2_4))) "\/" (d . ((q `2_4),(q `1_4)))) "\/" (a "\/" b) by WAYBEL_1:2;
A40: ((d . (u9,(q `2_4))) "\/" (d . ((q `2_4),(q `1_4)))) "\/" (a "\/" b) = ((a "\/" b) "\/" (d . (u9,(q `2_4)))) "\/" (d . ((q `2_4),(q `1_4))) by LATTICE3:14
.= ((a "\/" b) "\/" (d . (u9,(q `2_4)))) "\/" (d . ((q `1_4),(q `2_4))) by A1, Def5 ;
((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u)) = a "\/" ((new_bi_fun (d,q)) . (q,u)) by A36, Def10
.= a "\/" (b "\/" (d . (u9,(q `2_4)))) by A36, Def10
.= (a "\/" b) "\/" (d . (u9,(q `2_4))) by LATTICE3:14
.= ((a "\/" b) "\/" (a "\/" b)) "\/" (d . (u9,(q `2_4))) by YELLOW_5:1
.= (a "\/" b) "\/" ((d . (u9,(q `2_4))) "\/" (a "\/" b)) by LATTICE3:14 ;
hence  (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u)) by A37, A39, A40, A38, ORDERS_2:3; ::_thesis: verum
end;
supposeA41: ( u in A & q = {{{A}}} & p = {{{A}}} ) ; ::_thesis: (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u))
then ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u)) = (Bottom L) "\/" ((new_bi_fun (d,q)) . (q,u)) by Def10
.= (new_bi_fun (d,q)) . (p,u) by A41, WAYBEL_1:3 ;
hence  (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u)) ; ::_thesis: verum
end;
end;
end;
A42: for p, q, u being Element of new_set A st p in B & q in A & u in A holds
(new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u))
proof
let p, q, u be Element of new_set A; ::_thesis: ( p in B & q in A & u in A implies (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u)) )
assume that
A43: p in B and
A44: ( q in A & u in A ) ; ::_thesis: (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u))
reconsider q9 = q, u9 = u as Element of A by A44;
percases ( ( p = {A} & q in A & u in A ) or ( p = {{A}} & q in A & u in A ) or ( p = {{{A}}} & q in A & u in A ) ) by A43, A44, ENUMSET1:def_1;
supposeA45: ( p = {A} & q in A & u in A ) ; ::_thesis: (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u))
 d . (u9,(q `1_4)) <= (d . (u9,q9)) "\/" (d . (q9,(q `1_4))) by A2, Def7;
then  d . (u9,(q `1_4)) <= (d . (q9,u9)) "\/" (d . (q9,(q `1_4))) by A1, Def5;
then  (d . (u9,(q `1_4))) "\/" a <= ((d . (q9,(q `1_4))) "\/" (d . (q9,u9))) "\/" a by WAYBEL_1:2;
then A46: (d . (u9,(q `1_4))) "\/" a <= ((d . (q9,(q `1_4))) "\/" a) "\/" (d . (q9,u9)) by LATTICE3:14;
A47: (new_bi_fun (d,q)) . (q,u) = d . (q9,u9) by Def10;
 (new_bi_fun (d,q)) . (p,q) = (d . (q9,(q `1_4))) "\/" a by A45, Def10;
hence  (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u)) by A45, A47, A46, Def10; ::_thesis: verum
end;
supposeA48: ( p = {{A}} & q in A & u in A ) ; ::_thesis: (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u))
 d . (u9,(q `1_4)) <= (d . (u9,q9)) "\/" (d . (q9,(q `1_4))) by A2, Def7;
then  d . (u9,(q `1_4)) <= (d . (q9,u9)) "\/" (d . (q9,(q `1_4))) by A1, Def5;
then  (d . (u9,(q `1_4))) "\/" (a "\/" b) <= ((d . (q9,(q `1_4))) "\/" (d . (q9,u9))) "\/" (a "\/" b) by WAYBEL_1:2;
then  ((d . (u9,(q `1_4))) "\/" a) "\/" b <= ((d . (q9,(q `1_4))) "\/" (d . (q9,u9))) "\/" (a "\/" b) by LATTICE3:14;
then  ((d . (u9,(q `1_4))) "\/" a) "\/" b <= ((d . (q9,(q `1_4))) "\/" (a "\/" b)) "\/" (d . (q9,u9)) by LATTICE3:14;
then A49: ((d . (u9,(q `1_4))) "\/" a) "\/" b <= (((d . (q9,(q `1_4))) "\/" a) "\/" b) "\/" (d . (q9,u9)) by LATTICE3:14;
A50: (new_bi_fun (d,q)) . (q,u) = d . (q9,u9) by Def10;
 (new_bi_fun (d,q)) . (p,q) = ((d . (q9,(q `1_4))) "\/" a) "\/" b by A48, Def10;
hence  (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u)) by A48, A50, A49, Def10; ::_thesis: verum
end;
supposeA51: ( p = {{{A}}} & q in A & u in A ) ; ::_thesis: (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u))
 d . (u9,(q `2_4)) <= (d . (u9,q9)) "\/" (d . (q9,(q `2_4))) by A2, Def7;
then  d . (u9,(q `2_4)) <= (d . (q9,u9)) "\/" (d . (q9,(q `2_4))) by A1, Def5;
then  (d . (u9,(q `2_4))) "\/" b <= ((d . (q9,(q `2_4))) "\/" (d . (q9,u9))) "\/" b by WAYBEL_1:2;
then A52: (d . (u9,(q `2_4))) "\/" b <= ((d . (q9,(q `2_4))) "\/" b) "\/" (d . (q9,u9)) by LATTICE3:14;
A53: (new_bi_fun (d,q)) . (q,u) = d . (q9,u9) by Def10;
 (new_bi_fun (d,q)) . (p,q) = (d . (q9,(q `2_4))) "\/" b by A51, Def10;
hence  (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u)) by A51, A53, A52, Def10; ::_thesis: verum
end;
end;
end;
A54: for p, q, u being Element of new_set A st p in A & q in A & u in B holds
(new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u))
proof
let p, q, u be Element of new_set A; ::_thesis: ( p in A & q in A & u in B implies (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u)) )
assume A55: ( p in A & q in A & u in B ) ; ::_thesis: (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u))
percases ( ( p in A & q in A & u = {A} ) or ( p in A & q in A & u = {{A}} ) or ( p in A & q in A & u = {{{A}}} ) ) by A55, ENUMSET1:def_1;
supposeA56: ( p in A & q in A & u = {A} ) ; ::_thesis: (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u))
then reconsider p9 = p, q9 = q as Element of A ;
A57: (new_bi_fun (d,q)) . (p,q) = d . (p9,q9) by Def10;
 d . (p9,(q `1_4)) <= (d . (p9,q9)) "\/" (d . (q9,(q `1_4))) by A2, Def7;
then A58: (d . (p9,(q `1_4))) "\/" a <= ((d . (p9,q9)) "\/" (d . (q9,(q `1_4)))) "\/" a by WAYBEL_1:2;
 ( (new_bi_fun (d,q)) . (p,u) = (d . (p9,(q `1_4))) "\/" a & (new_bi_fun (d,q)) . (q,u) = (d . (q9,(q `1_4))) "\/" a ) by A56, Def10;
hence  (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u)) by A57, A58, LATTICE3:14; ::_thesis: verum
end;
supposeA59: ( p in A & q in A & u = {{A}} ) ; ::_thesis: (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u))
then reconsider p9 = p, q9 = q as Element of A ;
A60: (new_bi_fun (d,q)) . (p,q) = d . (p9,q9) by Def10;
 d . (p9,(q `1_4)) <= (d . (p9,q9)) "\/" (d . (q9,(q `1_4))) by A2, Def7;
then  (d . (p9,(q `1_4))) "\/" a <= ((d . (p9,q9)) "\/" (d . (q9,(q `1_4)))) "\/" a by WAYBEL_1:2;
then  ((d . (p9,(q `1_4))) "\/" a) "\/" b <= (((d . (p9,q9)) "\/" (d . (q9,(q `1_4)))) "\/" a) "\/" b by WAYBEL_1:2;
then A61: ((d . (p9,(q `1_4))) "\/" a) "\/" b <= ((d . (p9,q9)) "\/" ((d . (q9,(q `1_4))) "\/" a)) "\/" b by LATTICE3:14;
 ( (new_bi_fun (d,q)) . (p,u) = ((d . (p9,(q `1_4))) "\/" a) "\/" b & (new_bi_fun (d,q)) . (q,u) = ((d . (q9,(q `1_4))) "\/" a) "\/" b ) by A59, Def10;
hence  (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u)) by A60, A61, LATTICE3:14; ::_thesis: verum
end;
supposeA62: ( p in A & q in A & u = {{{A}}} ) ; ::_thesis: (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u))
then reconsider p9 = p, q9 = q as Element of A ;
A63: (new_bi_fun (d,q)) . (p,q) = d . (p9,q9) by Def10;
 d . (p9,(q `2_4)) <= (d . (p9,q9)) "\/" (d . (q9,(q `2_4))) by A2, Def7;
then A64: (d . (p9,(q `2_4))) "\/" b <= ((d . (p9,q9)) "\/" (d . (q9,(q `2_4)))) "\/" b by WAYBEL_1:2;
 ( (new_bi_fun (d,q)) . (p,u) = (d . (p9,(q `2_4))) "\/" b & (new_bi_fun (d,q)) . (q,u) = (d . (q9,(q `2_4))) "\/" b ) by A62, Def10;
hence  (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u)) by A63, A64, LATTICE3:14; ::_thesis: verum
end;
end;
end;
A65: for p, q, u being Element of new_set A st p in B & q in B & u in B holds
(new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u))
proof
let p, q, u be Element of new_set A; ::_thesis: ( p in B & q in B & u in B implies (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u)) )
assume A66: ( p in B & q in B & u in B ) ; ::_thesis: (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u))
percases ( ( p = {A} & q = {A} & u = {A} ) or ( p = {A} & q = {A} & u = {{A}} ) or ( p = {A} & q = {A} & u = {{{A}}} ) or ( p = {A} & q = {{A}} & u = {A} ) or ( p = {A} & q = {{A}} & u = {{A}} ) or ( p = {A} & q = {{A}} & u = {{{A}}} ) or ( p = {A} & q = {{{A}}} & u = {A} ) or ( p = {A} & q = {{{A}}} & u = {{A}} ) or ( p = {A} & q = {{{A}}} & u = {{{A}}} ) or ( p = {{A}} & q = {A} & u = {A} ) or ( p = {{A}} & q = {A} & u = {{A}} ) or ( p = {{A}} & q = {A} & u = {{{A}}} ) or ( p = {{A}} & q = {{A}} & u = {A} ) or ( p = {{A}} & q = {{A}} & u = {{A}} ) or ( p = {{A}} & q = {{A}} & u = {{{A}}} ) or ( p = {{A}} & q = {{{A}}} & u = {A} ) or ( p = {{A}} & q = {{{A}}} & u = {{A}} ) or ( p = {{A}} & q = {{{A}}} & u = {{{A}}} ) or ( p = {{{A}}} & q = {A} & u = {A} ) or ( p = {{{A}}} & q = {A} & u = {{A}} ) or ( p = {{{A}}} & q = {A} & u = {{{A}}} ) or ( p = {{{A}}} & q = {{A}} & u = {A} ) or ( p = {{{A}}} & q = {{A}} & u = {{A}} ) or ( p = {{{A}}} & q = {{A}} & u = {{{A}}} ) or ( p = {{{A}}} & q = {{{A}}} & u = {A} ) or ( p = {{{A}}} & q = {{{A}}} & u = {{A}} ) or ( p = {{{A}}} & q = {{{A}}} & u = {{{A}}} ) ) by A66, ENUMSET1:def_1;
supposeA67: ( p = {A} & q = {A} & u = {A} ) ; ::_thesis: (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u))
 Bottom L <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u)) by YELLOW_0:44;
hence  (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u)) by A67, Def10; ::_thesis: verum
end;
supposeA68: ( p = {A} & q = {A} & u = {{A}} ) ; ::_thesis: (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u))
then ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u)) = (Bottom L) "\/" ((new_bi_fun (d,q)) . (p,u)) by Def10
.= (Bottom L) "\/" b by A68, Def10
.= b by WAYBEL_1:3 ;
hence  (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u)) by A68, Def10; ::_thesis: verum
end;
supposeA69: ( p = {A} & q = {A} & u = {{{A}}} ) ; ::_thesis: (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u))
then ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u)) = (Bottom L) "\/" ((new_bi_fun (d,q)) . (p,u)) by Def10
.= (Bottom L) "\/" (a "\/" b) by A69, Def10
.= a "\/" b by WAYBEL_1:3 ;
hence  (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u)) by A69, Def10; ::_thesis: verum
end;
supposeA70: ( p = {A} & q = {{A}} & u = {A} ) ; ::_thesis: (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u))
 Bottom L <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u)) by YELLOW_0:44;
hence  (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u)) by A70, Def10; ::_thesis: verum
end;
supposeA71: ( p = {A} & q = {{A}} & u = {{A}} ) ; ::_thesis: (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u))
then ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u)) = b "\/" ((new_bi_fun (d,q)) . (q,u)) by Def10
.= (Bottom L) "\/" b by A71, Def10
.= b by WAYBEL_1:3 ;
hence  (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u)) by A71, Def10; ::_thesis: verum
end;
supposeA72: ( p = {A} & q = {{A}} & u = {{{A}}} ) ; ::_thesis: (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u))
then ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u)) = b "\/" ((new_bi_fun (d,q)) . (q,u)) by Def10
.= a "\/" b by A72, Def10 ;
hence  (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u)) by A72, Def10; ::_thesis: verum
end;
supposeA73: ( p = {A} & q = {{{A}}} & u = {A} ) ; ::_thesis: (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u))
 Bottom L <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u)) by YELLOW_0:44;
hence  (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u)) by A73, Def10; ::_thesis: verum
end;
supposeA74: ( p = {A} & q = {{{A}}} & u = {{A}} ) ; ::_thesis: (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u))
then A75: (new_bi_fun (d,q)) . (p,u) = b by Def10;
((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u)) = (a "\/" b) "\/" ((new_bi_fun (d,q)) . (q,u)) by A74, Def10
.= (b "\/" a) "\/" a by A74, Def10
.= b "\/" (a "\/" a) by LATTICE3:14
.= b "\/" a by YELLOW_5:1 ;
hence  (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u)) by A75, YELLOW_0:22; ::_thesis: verum
end;
supposeA76: ( p = {A} & q = {{{A}}} & u = {{{A}}} ) ; ::_thesis: (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u))
then ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u)) = (a "\/" b) "\/" ((new_bi_fun (d,q)) . (q,u)) by Def10
.= (Bottom L) "\/" (a "\/" b) by A76, Def10
.= a "\/" b by WAYBEL_1:3
.= (new_bi_fun (d,q)) . (p,q) by A76, Def10 ;
hence  (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u)) by A76; ::_thesis: verum
end;
supposeA77: ( p = {{A}} & q = {A} & u = {A} ) ; ::_thesis: (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u))
then ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u)) = b "\/" ((new_bi_fun (d,q)) . (q,u)) by Def10
.= (Bottom L) "\/" b by A77, Def10
.= b by WAYBEL_1:3
.= (new_bi_fun (d,q)) . (p,q) by A77, Def10 ;
hence  (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u)) by A77; ::_thesis: verum
end;
supposeA78: ( p = {{A}} & q = {A} & u = {{A}} ) ; ::_thesis: (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u))
 Bottom L <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u)) by YELLOW_0:44;
hence  (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u)) by A78, Def10; ::_thesis: verum
end;
supposeA79: ( p = {{A}} & q = {A} & u = {{{A}}} ) ; ::_thesis: (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u))
then A80: (new_bi_fun (d,q)) . (p,u) = a by Def10;
((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u)) = b "\/" ((new_bi_fun (d,q)) . (q,u)) by A79, Def10
.= b "\/" (b "\/" a) by A79, Def10
.= (b "\/" b) "\/" a by LATTICE3:14
.= b "\/" a by YELLOW_5:1 ;
hence  (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u)) by A80, YELLOW_0:22; ::_thesis: verum
end;
supposeA81: ( p = {{A}} & q = {{A}} & u = {A} ) ; ::_thesis: (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u))
then ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u)) = (Bottom L) "\/" ((new_bi_fun (d,q)) . (p,u)) by Def10
.= (Bottom L) "\/" b by A81, Def10
.= b by WAYBEL_1:3 ;
hence  (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u)) by A81, Def10; ::_thesis: verum
end;
supposeA82: ( p = {{A}} & q = {{A}} & u = {{A}} ) ; ::_thesis: (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u))
 Bottom L <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u)) by YELLOW_0:44;
hence  (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u)) by A82, Def10; ::_thesis: verum
end;
supposeA83: ( p = {{A}} & q = {{A}} & u = {{{A}}} ) ; ::_thesis: (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u))
then ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u)) = (Bottom L) "\/" ((new_bi_fun (d,q)) . (p,u)) by Def10
.= (Bottom L) "\/" a by A83, Def10
.= a by WAYBEL_1:3 ;
hence  (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u)) by A83, Def10; ::_thesis: verum
end;
supposeA84: ( p = {{A}} & q = {{{A}}} & u = {A} ) ; ::_thesis: (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u))
then A85: (new_bi_fun (d,q)) . (p,u) = b by Def10;
((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u)) = a "\/" ((new_bi_fun (d,q)) . (q,u)) by A84, Def10
.= a "\/" (a "\/" b) by A84, Def10
.= (a "\/" a) "\/" b by LATTICE3:14
.= a "\/" b by YELLOW_5:1 ;
hence  (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u)) by A85, YELLOW_0:22; ::_thesis: verum
end;
supposeA86: ( p = {{A}} & q = {{{A}}} & u = {{A}} ) ; ::_thesis: (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u))
 Bottom L <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u)) by YELLOW_0:44;
hence  (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u)) by A86, Def10; ::_thesis: verum
end;
supposeA87: ( p = {{A}} & q = {{{A}}} & u = {{{A}}} ) ; ::_thesis: (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u))
then ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u)) = a "\/" ((new_bi_fun (d,q)) . (q,u)) by Def10
.= (Bottom L) "\/" a by A87, Def10
.= a by WAYBEL_1:3
.= (new_bi_fun (d,q)) . (p,q) by A87, Def10 ;
hence  (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u)) by A87; ::_thesis: verum
end;
supposeA88: ( p = {{{A}}} & q = {A} & u = {A} ) ; ::_thesis: (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u))
then ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u)) = (a "\/" b) "\/" ((new_bi_fun (d,q)) . (q,u)) by Def10
.= (Bottom L) "\/" (a "\/" b) by A88, Def10
.= a "\/" b by WAYBEL_1:3
.= (new_bi_fun (d,q)) . (p,q) by A88, Def10 ;
hence  (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u)) by A88; ::_thesis: verum
end;
supposeA89: ( p = {{{A}}} & q = {A} & u = {{A}} ) ; ::_thesis: (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u))
then A90: (new_bi_fun (d,q)) . (p,u) = a by Def10;
((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u)) = (a "\/" b) "\/" ((new_bi_fun (d,q)) . (q,u)) by A89, Def10
.= (a "\/" b) "\/" b by A89, Def10
.= a "\/" (b "\/" b) by LATTICE3:14
.= a "\/" b by YELLOW_5:1 ;
hence  (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u)) by A90, YELLOW_0:22; ::_thesis: verum
end;
supposeA91: ( p = {{{A}}} & q = {A} & u = {{{A}}} ) ; ::_thesis: (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u))
 Bottom L <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u)) by YELLOW_0:44;
hence  (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u)) by A91, Def10; ::_thesis: verum
end;
supposeA92: ( p = {{{A}}} & q = {{A}} & u = {A} ) ; ::_thesis: (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u))
then ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u)) = a "\/" ((new_bi_fun (d,q)) . (q,u)) by Def10
.= a "\/" b by A92, Def10 ;
hence  (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u)) by A92, Def10; ::_thesis: verum
end;
supposeA93: ( p = {{{A}}} & q = {{A}} & u = {{A}} ) ; ::_thesis: (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u))
then ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u)) = a "\/" ((new_bi_fun (d,q)) . (q,u)) by Def10
.= (Bottom L) "\/" a by A93, Def10
.= a by WAYBEL_1:3
.= (new_bi_fun (d,q)) . (p,q) by A93, Def10 ;
hence  (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u)) by A93; ::_thesis: verum
end;
supposeA94: ( p = {{{A}}} & q = {{A}} & u = {{{A}}} ) ; ::_thesis: (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u))
 Bottom L <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u)) by YELLOW_0:44;
hence  (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u)) by A94, Def10; ::_thesis: verum
end;
supposeA95: ( p = {{{A}}} & q = {{{A}}} & u = {A} ) ; ::_thesis: (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u))
then ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u)) = (Bottom L) "\/" ((new_bi_fun (d,q)) . (p,u)) by Def10
.= (Bottom L) "\/" (a "\/" b) by A95, Def10
.= a "\/" b by WAYBEL_1:3 ;
hence  (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u)) by A95, Def10; ::_thesis: verum
end;
supposeA96: ( p = {{{A}}} & q = {{{A}}} & u = {{A}} ) ; ::_thesis: (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u))
then ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u)) = (Bottom L) "\/" ((new_bi_fun (d,q)) . (p,u)) by Def10
.= (Bottom L) "\/" a by A96, Def10
.= a by WAYBEL_1:3 ;
hence  (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u)) by A96, Def10; ::_thesis: verum
end;
supposeA97: ( p = {{{A}}} & q = {{{A}}} & u = {{{A}}} ) ; ::_thesis: (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u))
 Bottom L <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u)) by YELLOW_0:44;
hence  (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u)) by A97, Def10; ::_thesis: verum
end;
end;
end;
A98: for p, q, u being Element of new_set A st p in B & q in A & u in B holds
(new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u))
proof
let p, q, u be Element of new_set A; ::_thesis: ( p in B & q in A & u in B implies (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u)) )
assume that
A99: p in B and
A100: q in A and
A101: u in B ; ::_thesis: (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u))
reconsider q9 = q as Element of A by A100;
percases ( ( q in A & p = {A} & u = {A} ) or ( q in A & p = {A} & u = {{A}} ) or ( q in A & p = {A} & u = {{{A}}} ) or ( q in A & p = {{A}} & u = {A} ) or ( q in A & p = {{A}} & u = {{A}} ) or ( q in A & p = {{A}} & u = {{{A}}} ) or ( q in A & p = {{{A}}} & u = {A} ) or ( q in A & p = {{{A}}} & u = {{A}} ) or ( q in A & p = {{{A}}} & u = {{{A}}} ) ) by A99, A100, A101, ENUMSET1:def_1;
suppose ( q in A & p = {A} & u = {A} ) ; ::_thesis: (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u))
then  (new_bi_fun (d,q)) . (p,u) = Bottom L by Def10;
hence  (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u)) by YELLOW_0:44; ::_thesis: verum
end;
supposeA102: ( q in A & p = {A} & u = {{A}} ) ; ::_thesis: (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u))
then A103: ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u)) = ((new_bi_fun (d,q)) . (p,q)) "\/" (((d . (q9,(q `1_4))) "\/" a) "\/" b) by Def10
.= ((new_bi_fun (d,q)) . (p,q)) "\/" ((d . (q9,(q `1_4))) "\/" (a "\/" b)) by LATTICE3:14
.= (((new_bi_fun (d,q)) . (p,q)) "\/" (d . (q9,(q `1_4)))) "\/" (a "\/" b) by LATTICE3:14
.= ((((new_bi_fun (d,q)) . (p,q)) "\/" (d . (q9,(q `1_4)))) "\/" a) "\/" b by LATTICE3:14 ;
 (new_bi_fun (d,q)) . (p,u) = b by A102, Def10;
hence  (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u)) by A103, YELLOW_0:22; ::_thesis: verum
end;
supposeA104: ( q in A & p = {A} & u = {{{A}}} ) ; ::_thesis: (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u))
then A105: (new_bi_fun (d,q)) . (p,u) = a "\/" b by Def10;
((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u)) = ((d . (q9,(q `1_4))) "\/" a) "\/" ((new_bi_fun (d,q)) . (q,u)) by A104, Def10
.= ((d . (q9,(q `1_4))) "\/" a) "\/" ((d . (q9,(q `2_4))) "\/" b) by A104, Def10
.= (d . (q9,(q `1_4))) "\/" (a "\/" ((d . (q9,(q `2_4))) "\/" b)) by LATTICE3:14
.= (d . (q9,(q `1_4))) "\/" ((d . (q9,(q `2_4))) "\/" (a "\/" b)) by LATTICE3:14
.= ((d . (q9,(q `1_4))) "\/" (d . (q9,(q `2_4)))) "\/" (a "\/" b) by LATTICE3:14 ;
hence  (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u)) by A105, YELLOW_0:22; ::_thesis: verum
end;
supposeA106: ( q in A & p = {{A}} & u = {A} ) ; ::_thesis: (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u))
then A107: ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u)) = (((d . (q9,(q `1_4))) "\/" a) "\/" b) "\/" ((new_bi_fun (d,q)) . (q,u)) by Def10
.= ((new_bi_fun (d,q)) . (q,u)) "\/" ((d . (q9,(q `1_4))) "\/" (a "\/" b)) by LATTICE3:14
.= (((new_bi_fun (d,q)) . (q,u)) "\/" (d . (q9,(q `1_4)))) "\/" (a "\/" b) by LATTICE3:14
.= ((((new_bi_fun (d,q)) . (q,u)) "\/" (d . (q9,(q `1_4)))) "\/" a) "\/" b by LATTICE3:14 ;
 (new_bi_fun (d,q)) . (p,u) = b by A106, Def10;
hence  (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u)) by A107, YELLOW_0:22; ::_thesis: verum
end;
suppose ( q in A & p = {{A}} & u = {{A}} ) ; ::_thesis: (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u))
then  (new_bi_fun (d,q)) . (p,u) = Bottom L by Def10;
hence  (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u)) by YELLOW_0:44; ::_thesis: verum
end;
supposeA108: ( q in A & p = {{A}} & u = {{{A}}} ) ; ::_thesis: (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u))
then A109: ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u)) = (((d . (q9,(q `1_4))) "\/" a) "\/" b) "\/" ((new_bi_fun (d,q)) . (q,u)) by Def10
.= (a "\/" ((d . (q9,(q `1_4))) "\/" b)) "\/" ((new_bi_fun (d,q)) . (q,u)) by LATTICE3:14
.= a "\/" (((d . (q9,(q `1_4))) "\/" b) "\/" ((new_bi_fun (d,q)) . (q,u))) by LATTICE3:14 ;
 (new_bi_fun (d,q)) . (p,u) = a by A108, Def10;
hence  (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u)) by A109, YELLOW_0:22; ::_thesis: verum
end;
supposeA110: ( q in A & p = {{{A}}} & u = {A} ) ; ::_thesis: (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u))
then A111: (new_bi_fun (d,q)) . (p,u) = a "\/" b by Def10;
((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u)) = ((d . (q9,(q `2_4))) "\/" b) "\/" ((new_bi_fun (d,q)) . (q,u)) by A110, Def10
.= ((d . (q9,(q `2_4))) "\/" b) "\/" ((d . (q9,(q `1_4))) "\/" a) by A110, Def10
.= (d . (q9,(q `2_4))) "\/" (b "\/" ((d . (q9,(q `1_4))) "\/" a)) by LATTICE3:14
.= (d . (q9,(q `2_4))) "\/" ((d . (q9,(q `1_4))) "\/" (b "\/" a)) by LATTICE3:14
.= ((d . (q9,(q `2_4))) "\/" (d . (q9,(q `1_4)))) "\/" (a "\/" b) by LATTICE3:14 ;
hence  (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u)) by A111, YELLOW_0:22; ::_thesis: verum
end;
supposeA112: ( q in A & p = {{{A}}} & u = {{A}} ) ; ::_thesis: (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u))
then A113: ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u)) = ((new_bi_fun (d,q)) . (p,q)) "\/" (((d . (q9,(q `1_4))) "\/" a) "\/" b) by Def10
.= ((new_bi_fun (d,q)) . (p,q)) "\/" ((d . (q9,(q `1_4))) "\/" (a "\/" b)) by LATTICE3:14
.= (((new_bi_fun (d,q)) . (p,q)) "\/" (d . (q9,(q `1_4)))) "\/" (a "\/" b) by LATTICE3:14
.= ((((new_bi_fun (d,q)) . (p,q)) "\/" (d . (q9,(q `1_4)))) "\/" b) "\/" a by LATTICE3:14 ;
 (new_bi_fun (d,q)) . (p,u) = a by A112, Def10;
hence  (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u)) by A113, YELLOW_0:22; ::_thesis: verum
end;
suppose ( q in A & p = {{{A}}} & u = {{{A}}} ) ; ::_thesis: (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u))
then  (new_bi_fun (d,q)) . (p,u) = Bottom L by Def10;
hence  (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u)) by YELLOW_0:44; ::_thesis: verum
end;
end;
end;
A114: for p, q, u being Element of new_set A st p in A & q in B & u in B holds
(new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u))
proof
let p, q, u be Element of new_set A; ::_thesis: ( p in A & q in B & u in B implies (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u)) )
assume that
A115: p in A and
A116: ( q in B & u in B ) ; ::_thesis: (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u))
reconsider p9 = p as Element of A by A115;
percases ( ( p in A & q = {A} & u = {A} ) or ( p in A & q = {A} & u = {{A}} ) or ( p in A & q = {A} & u = {{{A}}} ) or ( p in A & q = {{A}} & u = {A} ) or ( p in A & q = {{A}} & u = {{A}} ) or ( p in A & q = {{A}} & u = {{{A}}} ) or ( p in A & q = {{{A}}} & u = {A} ) or ( p in A & q = {{{A}}} & u = {{A}} ) or ( p in A & q = {{{A}}} & u = {{{A}}} ) ) by A115, A116, ENUMSET1:def_1;
supposeA117: ( p in A & q = {A} & u = {A} ) ; ::_thesis: (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u))
then ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u)) = (Bottom L) "\/" ((new_bi_fun (d,q)) . (p,q)) by Def10
.= (new_bi_fun (d,q)) . (p,q) by WAYBEL_1:3 ;
hence  (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u)) by A117; ::_thesis: verum
end;
supposeA118: ( p in A & q = {A} & u = {{A}} ) ; ::_thesis: (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u))
then (new_bi_fun (d,q)) . (p,u) = ((d . (p9,(q `1_4))) "\/" a) "\/" b by Def10
.= ((new_bi_fun (d,q)) . (p,q)) "\/" b by A118, Def10 ;
hence  (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u)) by A118, Def10; ::_thesis: verum
end;
supposeA119: ( p in A & q = {A} & u = {{{A}}} ) ; ::_thesis: (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u))
 d . (p9,(q `2_4)) <= (d . (p9,(q `1_4))) "\/" (d . ((q `1_4),(q `2_4))) by A2, Def7;
then  (d . (p9,(q `2_4))) "\/" b <= ((d . (p9,(q `1_4))) "\/" (d . ((q `1_4),(q `2_4)))) "\/" b by WAYBEL_1:2;
then A120: (new_bi_fun (d,q)) . (p,u) <= ((d . (p9,(q `1_4))) "\/" (d . ((q `1_4),(q `2_4)))) "\/" b by A119, Def10;
 (d . (p9,(q `1_4))) "\/" b <= (d . (p9,(q `1_4))) "\/" b ;
then  ( ((d . (p9,(q `1_4))) "\/" (d . ((q `1_4),(q `2_4)))) "\/" b = ((d . (p9,(q `1_4))) "\/" b) "\/" (d . ((q `1_4),(q `2_4))) & ((d . (p9,(q `1_4))) "\/" b) "\/" (d . ((q `1_4),(q `2_4))) <= ((d . (p9,(q `1_4))) "\/" b) "\/" (a "\/" b) ) by A14, LATTICE3:14, YELLOW_3:3;
then A121: (new_bi_fun (d,q)) . (p,u) <= ((d . (p9,(q `1_4))) "\/" b) "\/" (a "\/" b) by A120, ORDERS_2:3;
A122: ((d . (p9,(q `1_4))) "\/" b) "\/" (a "\/" b) = (d . (p9,(q `1_4))) "\/" ((b "\/" a) "\/" b) by LATTICE3:14
.= (d . (p9,(q `1_4))) "\/" (a "\/" (b "\/" b)) by LATTICE3:14
.= (d . (p9,(q `1_4))) "\/" (a "\/" b) by YELLOW_5:1
.= (d . (p9,(q `1_4))) "\/" ((a "\/" a) "\/" b) by YELLOW_5:1
.= (d . (p9,(q `1_4))) "\/" (a "\/" (a "\/" b)) by LATTICE3:14
.= ((d . (p9,(q `1_4))) "\/" a) "\/" (a "\/" b) by LATTICE3:14 ;
 (new_bi_fun (d,q)) . (p,q) = (d . (p9,(q `1_4))) "\/" a by A119, Def10;
hence  (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u)) by A119, A121, A122, Def10; ::_thesis: verum
end;
supposeA123: ( p in A & q = {{A}} & u = {A} ) ; ::_thesis: (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u))
then (new_bi_fun (d,q)) . (p,q) = ((d . (p9,(q `1_4))) "\/" a) "\/" b by Def10
.= ((new_bi_fun (d,q)) . (p,u)) "\/" b by A123, Def10 ;
then ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u)) = (((new_bi_fun (d,q)) . (p,u)) "\/" b) "\/" b by A123, Def10
.= ((new_bi_fun (d,q)) . (p,u)) "\/" (b "\/" b) by LATTICE3:14
.= ((new_bi_fun (d,q)) . (p,u)) "\/" b by YELLOW_5:1 ;
hence  (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u)) by YELLOW_0:22; ::_thesis: verum
end;
supposeA124: ( p in A & q = {{A}} & u = {{A}} ) ; ::_thesis: (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u))
then ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u)) = (Bottom L) "\/" ((new_bi_fun (d,q)) . (p,q)) by Def10
.= (new_bi_fun (d,q)) . (p,q) by WAYBEL_1:3 ;
hence  (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u)) by A124; ::_thesis: verum
end;
supposeA125: ( p in A & q = {{A}} & u = {{{A}}} ) ; ::_thesis: (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u))
 d . (p9,(q `2_4)) <= (d . (p9,(q `1_4))) "\/" (d . ((q `1_4),(q `2_4))) by A2, Def7;
then  (d . (p9,(q `2_4))) "\/" b <= ((d . (p9,(q `1_4))) "\/" (d . ((q `1_4),(q `2_4)))) "\/" b by WAYBEL_1:2;
then A126: (new_bi_fun (d,q)) . (p,u) <= ((d . (p9,(q `1_4))) "\/" (d . ((q `1_4),(q `2_4)))) "\/" b by A125, Def10;
 (d . (p9,(q `1_4))) "\/" b <= (d . (p9,(q `1_4))) "\/" b ;
then  ( ((d . (p9,(q `1_4))) "\/" (d . ((q `1_4),(q `2_4)))) "\/" b = ((d . (p9,(q `1_4))) "\/" b) "\/" (d . ((q `1_4),(q `2_4))) & ((d . (p9,(q `1_4))) "\/" b) "\/" (d . ((q `1_4),(q `2_4))) <= ((d . (p9,(q `1_4))) "\/" b) "\/" (a "\/" b) ) by A14, LATTICE3:14, YELLOW_3:3;
then A127: (new_bi_fun (d,q)) . (p,u) <= ((d . (p9,(q `1_4))) "\/" b) "\/" (a "\/" b) by A126, ORDERS_2:3;
A128: ((d . (p9,(q `1_4))) "\/" b) "\/" (a "\/" b) = (d . (p9,(q `1_4))) "\/" ((b "\/" a) "\/" b) by LATTICE3:14
.= (d . (p9,(q `1_4))) "\/" (a "\/" (b "\/" b)) by LATTICE3:14
.= (d . (p9,(q `1_4))) "\/" (a "\/" b) by YELLOW_5:1
.= (d . (p9,(q `1_4))) "\/" ((a "\/" a) "\/" b) by YELLOW_5:1
.= (d . (p9,(q `1_4))) "\/" (a "\/" (a "\/" b)) by LATTICE3:14
.= ((d . (p9,(q `1_4))) "\/" (a "\/" b)) "\/" a by LATTICE3:14
.= (((d . (p9,(q `1_4))) "\/" a) "\/" b) "\/" a by LATTICE3:14 ;
 (new_bi_fun (d,q)) . (p,q) = ((d . (p9,(q `1_4))) "\/" a) "\/" b by A125, Def10;
hence  (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u)) by A125, A127, A128, Def10; ::_thesis: verum
end;
supposeA129: ( p in A & q = {{{A}}} & u = {A} ) ; ::_thesis: (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u))
 d . (p9,(q `1_4)) <= (d . (p9,(q `2_4))) "\/" (d . ((q `2_4),(q `1_4))) by A2, Def7;
then  (d . (p9,(q `1_4))) "\/" a <= ((d . (p9,(q `2_4))) "\/" (d . ((q `2_4),(q `1_4)))) "\/" a by WAYBEL_1:2;
then A130: (new_bi_fun (d,q)) . (p,u) <= ((d . (p9,(q `2_4))) "\/" (d . ((q `2_4),(q `1_4)))) "\/" a by A129, Def10;
 ( d . ((q `2_4),(q `1_4)) <= a "\/" b & (d . (p9,(q `2_4))) "\/" a <= (d . (p9,(q `2_4))) "\/" a ) by A1, A14, Def5;
then  ( ((d . (p9,(q `2_4))) "\/" (d . ((q `2_4),(q `1_4)))) "\/" a = ((d . (p9,(q `2_4))) "\/" a) "\/" (d . ((q `2_4),(q `1_4))) & ((d . (p9,(q `2_4))) "\/" a) "\/" (d . ((q `2_4),(q `1_4))) <= ((d . (p9,(q `2_4))) "\/" a) "\/" (a "\/" b) ) by LATTICE3:14, YELLOW_3:3;
then A131: (new_bi_fun (d,q)) . (p,u) <= ((d . (p9,(q `2_4))) "\/" a) "\/" (a "\/" b) by A130, ORDERS_2:3;
A132: ((d . (p9,(q `2_4))) "\/" a) "\/" (a "\/" b) = (((d . (p9,(q `2_4))) "\/" a) "\/" a) "\/" b by LATTICE3:14
.= ((d . (p9,(q `2_4))) "\/" (a "\/" a)) "\/" b by LATTICE3:14
.= ((d . (p9,(q `2_4))) "\/" a) "\/" b by YELLOW_5:1
.= (d . (p9,(q `2_4))) "\/" (a "\/" b) by LATTICE3:14
.= (d . (p9,(q `2_4))) "\/" (a "\/" (b "\/" b)) by YELLOW_5:1
.= (d . (p9,(q `2_4))) "\/" ((a "\/" b) "\/" b) by LATTICE3:14
.= ((d . (p9,(q `2_4))) "\/" b) "\/" (a "\/" b) by LATTICE3:14 ;
 (new_bi_fun (d,q)) . (p,q) = (d . (p9,(q `2_4))) "\/" b by A129, Def10;
hence  (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u)) by A129, A131, A132, Def10; ::_thesis: verum
end;
supposeA133: ( p in A & q = {{{A}}} & u = {{A}} ) ; ::_thesis: (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u))
 d . (p9,(q `1_4)) <= (d . (p9,(q `2_4))) "\/" (d . ((q `2_4),(q `1_4))) by A2, Def7;
then A134: (d . (p9,(q `1_4))) "\/" (a "\/" b) <= ((d . (p9,(q `2_4))) "\/" (d . ((q `2_4),(q `1_4)))) "\/" (a "\/" b) by WAYBEL_1:2;
 (new_bi_fun (d,q)) . (p,u) = ((d . (p9,(q `1_4))) "\/" a) "\/" b by A133, Def10;
then A135: (new_bi_fun (d,q)) . (p,u) <= ((d . (p9,(q `2_4))) "\/" (d . ((q `2_4),(q `1_4)))) "\/" (a "\/" b) by A134, LATTICE3:14;
A136: ((d . (p9,(q `2_4))) "\/" a) "\/" (a "\/" b) = (((d . (p9,(q `2_4))) "\/" a) "\/" a) "\/" b by LATTICE3:14
.= ((d . (p9,(q `2_4))) "\/" (a "\/" a)) "\/" b by LATTICE3:14
.= ((d . (p9,(q `2_4))) "\/" a) "\/" b by YELLOW_5:1
.= ((d . (p9,(q `2_4))) "\/" b) "\/" a by LATTICE3:14 ;
A137: (new_bi_fun (d,q)) . (p,q) = (d . (p9,(q `2_4))) "\/" b by A133, Def10;
A138: (d . (p9,(q `2_4))) "\/" (a "\/" b) <= (d . (p9,(q `2_4))) "\/" (a "\/" b) ;
 ( d . ((q `2_4),(q `1_4)) <= a "\/" b & ((d . (p9,(q `2_4))) "\/" (d . ((q `2_4),(q `1_4)))) "\/" (a "\/" b) = ((d . (p9,(q `2_4))) "\/" (a "\/" b)) "\/" (d . ((q `2_4),(q `1_4))) ) by A1, A14, Def5, LATTICE3:14;
then A139: ((d . (p9,(q `2_4))) "\/" (d . ((q `2_4),(q `1_4)))) "\/" (a "\/" b) <= ((d . (p9,(q `2_4))) "\/" (a "\/" b)) "\/" (a "\/" b) by A138, YELLOW_3:3;
((d . (p9,(q `2_4))) "\/" a) "\/" (a "\/" b) = ((d . (p9,(q `2_4))) "\/" a) "\/" ((a "\/" b) "\/" (a "\/" b)) by YELLOW_5:1
.= (d . (p9,(q `2_4))) "\/" (a "\/" ((a "\/" b) "\/" (a "\/" b))) by LATTICE3:14
.= (d . (p9,(q `2_4))) "\/" ((a "\/" (a "\/" b)) "\/" (a "\/" b)) by LATTICE3:14
.= (d . (p9,(q `2_4))) "\/" (((a "\/" a) "\/" b) "\/" (a "\/" b)) by LATTICE3:14
.= (d . (p9,(q `2_4))) "\/" ((a "\/" b) "\/" (a "\/" b)) by YELLOW_5:1
.= ((d . (p9,(q `2_4))) "\/" (a "\/" b)) "\/" (a "\/" b) by LATTICE3:14 ;
then  (new_bi_fun (d,q)) . (p,u) <= ((d . (p9,(q `2_4))) "\/" a) "\/" (a "\/" b) by A139, A135, ORDERS_2:3;
hence  (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u)) by A133, A137, A136, Def10; ::_thesis: verum
end;
supposeA140: ( p in A & q = {{{A}}} & u = {{{A}}} ) ; ::_thesis: (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u))
then ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u)) = (Bottom L) "\/" ((new_bi_fun (d,q)) . (p,q)) by Def10
.= (new_bi_fun (d,q)) . (p,q) by WAYBEL_1:3 ;
hence  (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u)) by A140; ::_thesis: verum
end;
end;
end;
A141: for p, q, u being Element of new_set A st p in A & q in A & u in A holds
(new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u))
proof
let p, q, u be Element of new_set A; ::_thesis: ( p in A & q in A & u in A implies (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u)) )
assume  ( p in A & q in A & u in A ) ; ::_thesis: (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u))
then reconsider p9 = p, q9 = q, u9 = u as Element of A ;
A142: (new_bi_fun (d,q)) . (q,u) = d . (q9,u9) by Def10;
 ( (new_bi_fun (d,q)) . (p,u) = d . (p9,u9) & (new_bi_fun (d,q)) . (p,q) = d . (p9,q9) ) by Def10;
hence  (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u)) by A2, A142, Def7; ::_thesis: verum
end;
 for p, q, u being Element of new_set A holds (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u))
proof
let p, q, u be Element of new_set A; ::_thesis: (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u))
percases ( ( p in A & q in A & u in A ) or ( p in A & q in A & u in B ) or ( p in A & q in B & u in A ) or ( p in A & q in B & u in B ) or ( p in B & q in A & u in A ) or ( p in B & q in A & u in B ) or ( p in B & q in B & u in A ) or ( p in B & q in B & u in B ) ) by XBOOLE_0:def_3;
suppose ( p in A & q in A & u in A ) ; ::_thesis: (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u))
hence  (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u)) by A141; ::_thesis: verum
end;
suppose ( p in A & q in A & u in B ) ; ::_thesis: (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u))
hence  (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u)) by A54; ::_thesis: verum
end;
suppose ( p in A & q in B & u in A ) ; ::_thesis: (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u))
hence  (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u)) by A3; ::_thesis: verum
end;
suppose ( p in A & q in B & u in B ) ; ::_thesis: (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u))
hence  (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u)) by A114; ::_thesis: verum
end;
suppose ( p in B & q in A & u in A ) ; ::_thesis: (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u))
hence  (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u)) by A42; ::_thesis: verum
end;
suppose ( p in B & q in A & u in B ) ; ::_thesis: (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u))
hence  (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u)) by A98; ::_thesis: verum
end;
suppose ( p in B & q in B & u in A ) ; ::_thesis: (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u))
hence  (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u)) by A15; ::_thesis: verum
end;
suppose ( p in B & q in B & u in B ) ; ::_thesis: (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u))
hence  (new_bi_fun (d,q)) . (p,u) <= ((new_bi_fun (d,q)) . (p,q)) "\/" ((new_bi_fun (d,q)) . (q,u)) by A65; ::_thesis: verum
end;
end;
end;
hence  new_bi_fun (d,q) is u.t.i. by Def7; ::_thesis: verum
end;

theorem Th19: :: LATTICE5:19
for A being non empty set
for L being lower-bounded LATTICE
for d being BiFunction of A,L
for q being Element of [:A,A, the carrier of L, the carrier of L:] holds d c= new_bi_fun (d,q)
proof
let A be non empty set ; ::_thesis: for L being lower-bounded LATTICE
for d being BiFunction of A,L
for q being Element of [:A,A, the carrier of L, the carrier of L:] holds d c= new_bi_fun (d,q)
let L be lower-bounded LATTICE; ::_thesis: for d being BiFunction of A,L
for q being Element of [:A,A, the carrier of L, the carrier of L:] holds d c= new_bi_fun (d,q)
let d be BiFunction of A,L; ::_thesis: for q being Element of [:A,A, the carrier of L, the carrier of L:] holds d c= new_bi_fun (d,q)
let q be Element of [:A,A, the carrier of L, the carrier of L:]; ::_thesis: d c= new_bi_fun (d,q)
set g = new_bi_fun (d,q);
A1: A c= new_set A by XBOOLE_1:7;
A2: for z being set st z in dom d holds
d . z = (new_bi_fun (d,q)) . z
proof
let z be set ; ::_thesis: ( z in dom d implies d . z = (new_bi_fun (d,q)) . z )
assume A3: z in dom d ; ::_thesis: d . z = (new_bi_fun (d,q)) . z
then consider x, y being set  such that
A4: [x,y] = z by RELAT_1:def_1;
reconsider x9 = x, y9 = y as Element of A by A3, A4, ZFMISC_1:87;
d . [x,y] = d . (x9,y9)
.= (new_bi_fun (d,q)) . (x9,y9) by Def10
.= (new_bi_fun (d,q)) . [x,y] ;
hence  d . z = (new_bi_fun (d,q)) . z by A4; ::_thesis: verum
end;
 ( dom d = [:A,A:] & dom (new_bi_fun (d,q)) = [:(new_set A),(new_set A):] ) by FUNCT_2:def_1;
then  dom d c= dom (new_bi_fun (d,q)) by A1, ZFMISC_1:96;
hence  d c= new_bi_fun (d,q) by A2, GRFUNC_1:2; ::_thesis: verum
end;

definition
let A be non empty set ;
let L be lower-bounded LATTICE;
let d be BiFunction of A,L;
func DistEsti d ->  Cardinal means :Def11: :: LATTICE5:def 11
it, {  [x,y,a,b] where x, y is Element of A, a, b is Element of L : d . (x,y) <= a "\/" b  }  are_equipotent ;
existence
 ex b1 being Cardinal st b1, {  [x,y,a,b] where x, y is Element of A, a, b is Element of L : d . (x,y) <= a "\/" b  }  are_equipotent
proof
set D =  {  [x,y,a,b] where x, y is Element of A, a, b is Element of L : d . (x,y) <= a "\/" b  }  ;
take  card  {  [x,y,a,b] where x, y is Element of A, a, b is Element of L : d . (x,y) <= a "\/" b  }  ; ::_thesis: card  {  [x,y,a,b] where x, y is Element of A, a, b is Element of L : d . (x,y) <= a "\/" b  }  , {  [x,y,a,b] where x, y is Element of A, a, b is Element of L : d . (x,y) <= a "\/" b  }  are_equipotent
thus  card  {  [x,y,a,b] where x, y is Element of A, a, b is Element of L : d . (x,y) <= a "\/" b  }  , {  [x,y,a,b] where x, y is Element of A, a, b is Element of L : d . (x,y) <= a "\/" b  }  are_equipotent by CARD_1:def_2; ::_thesis: verum
end;
uniqueness
 for b1, b2 being Cardinal st b1, {  [x,y,a,b] where x, y is Element of A, a, b is Element of L : d . (x,y) <= a "\/" b  }  are_equipotent & b2, {  [x,y,a,b] where x, y is Element of A, a, b is Element of L : d . (x,y) <= a "\/" b  }  are_equipotent holds
b1 = b2
proof
set D =  {  [x,y,a,b] where x, y is Element of A, a, b is Element of L : d . (x,y) <= a "\/" b  }  ;
let c1, c2 be Cardinal; ::_thesis: ( c1, {  [x,y,a,b] where x, y is Element of A, a, b is Element of L : d . (x,y) <= a "\/" b  }  are_equipotent & c2, {  [x,y,a,b] where x, y is Element of A, a, b is Element of L : d . (x,y) <= a "\/" b  }  are_equipotent implies c1 = c2 )
assume  ( c1, {  [x,y,a,b] where x, y is Element of A, a, b is Element of L : d . (x,y) <= a "\/" b  }  are_equipotent & c2, {  [x,y,a,b] where x, y is Element of A, a, b is Element of L : d . (x,y) <= a "\/" b  }  are_equipotent ) ; ::_thesis: c1 = c2
then  c1,c2 are_equipotent by WELLORD2:15;
hence  c1 = c2 by CARD_1:2; ::_thesis: verum
end;
end;

:: deftheorem Def11 defines DistEsti LATTICE5:def_11_:_
 for A being non empty set
for L being lower-bounded LATTICE
for d being BiFunction of A,L
for b4 being Cardinal holds
( b4 = DistEsti d iff b4, {  [x,y,a,b] where x, y is Element of A, a, b is Element of L : d . (x,y) <= a "\/" b  }  are_equipotent );

theorem Th20: :: LATTICE5:20
for A being non empty set
for L being lower-bounded LATTICE
for d being distance_function of A,L holds DistEsti d <> {}
proof
let A be non empty set ; ::_thesis: for L being lower-bounded LATTICE
for d being distance_function of A,L holds DistEsti d <> {}
let L be lower-bounded LATTICE; ::_thesis: for d being distance_function of A,L holds DistEsti d <> {}
let d be distance_function of A,L; ::_thesis: DistEsti d <> {}
set X =  {  [x,y,a,b] where x, y is Element of A, a, b is Element of L : d . (x,y) <= a "\/" b  }  ;
set x9 = the Element of A;
consider z being set  such that
A1: z = [ the Element of A, the Element of A,(Bottom L),(Bottom L)] ;
A2: DistEsti d, {  [x,y,a,b] where x, y is Element of A, a, b is Element of L : d . (x,y) <= a "\/" b  }  are_equipotent by Def11;
d . ( the Element of A, the Element of A) = Bottom L by Def6
.= (Bottom L) "\/" (Bottom L) by YELLOW_5:1 ;
then  z in  {  [x,y,a,b] where x, y is Element of A, a, b is Element of L : d . (x,y) <= a "\/" b  }  by A1;
hence  DistEsti d <> {} by A2, CARD_1:26; ::_thesis: verum
end;


definition
let A be non empty set ;
let O be Ordinal;
func ConsecutiveSet (A,O) ->  set  means :Def12: :: LATTICE5:def 12
 ex L0 being T-Sequence st
( it = last L0 & dom L0 = succ O & L0 . {} = A & ( for C being Ordinal st succ C in succ O holds
L0 . (succ C) = new_set (L0 . C) ) & ( for C being Ordinal st C in succ O & C <> {} & C is limit_ordinal holds
L0 . C = union (rng (L0 | C)) ) );
correctness
existence
ex b1 being set ex L0 being T-Sequence st
( b1 = last L0 & dom L0 = succ O & L0 . {} = A & ( for C being Ordinal st succ C in succ O holds
L0 . (succ C) = new_set (L0 . C) ) & ( for C being Ordinal st C in succ O & C <> {} & C is limit_ordinal holds
L0 . C = union (rng (L0 | C)) ) );
uniqueness
for b1, b2 being set st ex L0 being T-Sequence st
( b1 = last L0 & dom L0 = succ O & L0 . {} = A & ( for C being Ordinal st succ C in succ O holds
L0 . (succ C) = new_set (L0 . C) ) & ( for C being Ordinal st C in succ O & C <> {} & C is limit_ordinal holds
L0 . C = union (rng (L0 | C)) ) ) & ex L0 being T-Sequence st
( b2 = last L0 & dom L0 = succ O & L0 . {} = A & ( for C being Ordinal st succ C in succ O holds
L0 . (succ C) = new_set (L0 . C) ) & ( for C being Ordinal st C in succ O & C <> {} & C is limit_ordinal holds
L0 . C = union (rng (L0 | C)) ) ) holds
b1 = b2;
proof
deffunc H1( Ordinal, T-Sequence) ->  set = union (rng $2);
deffunc H2( Ordinal, set ) ->  set = new_set $2;
thus  ( ex x being set ex L1 being T-Sequence st
( x = last L1 & dom L1 = succ O & L1 . {} = A & ( for C being Ordinal st succ C in succ O holds
L1 . (succ C) = H2(C,L1 . C) ) & ( for C being Ordinal st C in succ O & C <> {} & C is limit_ordinal holds
L1 . C = H1(C,L1 | C) ) ) & ( for x1, x2 being set st ex L1 being T-Sequence st
( x1 = last L1 & dom L1 = succ O & L1 . {} = A & ( for C being Ordinal st succ C in succ O holds
L1 . (succ C) = H2(C,L1 . C) ) & ( for C being Ordinal st C in succ O & C <> {} & C is limit_ordinal holds
L1 . C = H1(C,L1 | C) ) ) & ex L1 being T-Sequence st
( x2 = last L1 & dom L1 = succ O & L1 . {} = A & ( for C being Ordinal st succ C in succ O holds
L1 . (succ C) = H2(C,L1 . C) ) & ( for C being Ordinal st C in succ O & C <> {} & C is limit_ordinal holds
L1 . C = H1(C,L1 | C) ) ) holds
x1 = x2 ) ) from ORDINAL2:sch_7(); ::_thesis: verum
end;
end;

:: deftheorem Def12 defines ConsecutiveSet LATTICE5:def_12_:_
 for A being non empty set
for O being Ordinal
for b3 being set holds
( b3 = ConsecutiveSet (A,O) iff ex L0 being T-Sequence st
( b3 = last L0 & dom L0 = succ O & L0 . {} = A & ( for C being Ordinal st succ C in succ O holds
L0 . (succ C) = new_set (L0 . C) ) & ( for C being Ordinal st C in succ O & C <> {} & C is limit_ordinal holds
L0 . C = union (rng (L0 | C)) ) ) );

theorem Th21: :: LATTICE5:21
for A being non empty set holds ConsecutiveSet (A,{}) = A
proof
let A be non empty set ; ::_thesis: ConsecutiveSet (A,{}) = A
deffunc H1( Ordinal, T-Sequence) ->  set = union (rng $2);
deffunc H2( Ordinal, set ) ->  set = new_set $2;
deffunc H3( Ordinal) ->  set = ConsecutiveSet (A,$1);
A1: for O being Ordinal
for x being set holds
( x = H3(O) iff ex L0 being T-Sequence st
( x = last L0 & dom L0 = succ O & L0 . {} = A & ( for C being Ordinal st succ C in succ O holds
L0 . (succ C) = H2(C,L0 . C) ) & ( for C being Ordinal st C in succ O & C <> {} & C is limit_ordinal holds
L0 . C = H1(C,L0 | C) ) ) ) by Def12;
thus  H3( {} ) = A from ORDINAL2:sch_8(A1); ::_thesis: verum
end;

theorem Th22: :: LATTICE5:22
for A being non empty set
for O being Ordinal holds ConsecutiveSet (A,(succ O)) = new_set (ConsecutiveSet (A,O))
proof
let A be non empty set ; ::_thesis: for O being Ordinal holds ConsecutiveSet (A,(succ O)) = new_set (ConsecutiveSet (A,O))
let O be Ordinal; ::_thesis: ConsecutiveSet (A,(succ O)) = new_set (ConsecutiveSet (A,O))
deffunc H1( Ordinal, T-Sequence) ->  set = union (rng $2);
deffunc H2( Ordinal, set ) ->  set = new_set $2;
deffunc H3( Ordinal) ->  set = ConsecutiveSet (A,$1);
A1: for O being Ordinal
for It being set holds
( It = H3(O) iff ex L0 being T-Sequence st
( It = last L0 & dom L0 = succ O & L0 . {} = A & ( for C being Ordinal st succ C in succ O holds
L0 . (succ C) = H2(C,L0 . C) ) & ( for C being Ordinal st C in succ O & C <> {} & C is limit_ordinal holds
L0 . C = H1(C,L0 | C) ) ) ) by Def12;
 for O being Ordinal holds H3( succ O) = H2(O,H3(O)) from ORDINAL2:sch_9(A1);
hence  ConsecutiveSet (A,(succ O)) = new_set (ConsecutiveSet (A,O)) ; ::_thesis: verum
end;

theorem Th23: :: LATTICE5:23
for A being non empty set
for T being T-Sequence
for O being Ordinal st O <> {} & O is limit_ordinal & dom T = O & ( for O1 being Ordinal st O1 in O holds
T . O1 = ConsecutiveSet (A,O1) ) holds
ConsecutiveSet (A,O) = union (rng T)
proof
let A be non empty set ; ::_thesis: for T being T-Sequence
for O being Ordinal st O <> {} & O is limit_ordinal & dom T = O & ( for O1 being Ordinal st O1 in O holds
T . O1 = ConsecutiveSet (A,O1) ) holds
ConsecutiveSet (A,O) = union (rng T)
let T be T-Sequence; ::_thesis: for O being Ordinal st O <> {} & O is limit_ordinal & dom T = O & ( for O1 being Ordinal st O1 in O holds
T . O1 = ConsecutiveSet (A,O1) ) holds
ConsecutiveSet (A,O) = union (rng T)
let O be Ordinal; ::_thesis: ( O <> {} & O is limit_ordinal & dom T = O & ( for O1 being Ordinal st O1 in O holds
T . O1 = ConsecutiveSet (A,O1) ) implies ConsecutiveSet (A,O) = union (rng T) )
deffunc H1( Ordinal, T-Sequence) ->  set = union (rng $2);
deffunc H2( Ordinal, set ) ->  set = new_set $2;
deffunc H3( Ordinal) ->  set = ConsecutiveSet (A,$1);
assume that
A1: ( O <> {} & O is limit_ordinal ) and
A2: dom T = O and
A3: for O1 being Ordinal st O1 in O holds
T . O1 = H3(O1) ; ::_thesis: ConsecutiveSet (A,O) = union (rng T)
A4: for O being Ordinal
for x being set holds
( x = H3(O) iff ex L0 being T-Sequence st
( x = last L0 & dom L0 = succ O & L0 . {} = A & ( for C being Ordinal st succ C in succ O holds
L0 . (succ C) = H2(C,L0 . C) ) & ( for C being Ordinal st C in succ O & C <> {} & C is limit_ordinal holds
L0 . C = H1(C,L0 | C) ) ) ) by Def12;
thus  H3(O) = H1(O,T) from ORDINAL2:sch_10(A4, A1, A2, A3); ::_thesis: verum
end;

registration
let A be non empty set ;
let O be Ordinal;
cluster ConsecutiveSet (A,O) ->  non empty ;
coherence
 not ConsecutiveSet (A,O) is empty
proof
defpred S1[ Ordinal] means  not ConsecutiveSet (A,A) is empty ;
A1: for O being Ordinal st S1[O] holds
S1[ succ O]
proof
let O1 be Ordinal; ::_thesis: ( S1[O1] implies S1[ succ O1] )
assume  not ConsecutiveSet (A,O1) is empty ; ::_thesis: S1[ succ O1]
 ConsecutiveSet (A,(succ O1)) = new_set (ConsecutiveSet (A,O1)) by Th22;
hence  S1[ succ O1] ; ::_thesis: verum
end;
A2: for O being Ordinal st O <> {} & O is limit_ordinal & ( for B being Ordinal st B in O holds
S1[B] ) holds
S1[O]
proof
deffunc H1( Ordinal) ->  set = ConsecutiveSet (A,A);
let O1 be Ordinal; ::_thesis: ( O1 <> {} & O1 is limit_ordinal & ( for B being Ordinal st B in O1 holds
S1[B] ) implies S1[O1] )
assume that
A3: O1 <> {} and
A4: O1 is limit_ordinal and
 for O2 being Ordinal st O2 in O1 holds
not ConsecutiveSet (A,O2) is empty ; ::_thesis: S1[O1]
A5: {} in O1 by A3, ORDINAL3:8;
consider Ls being T-Sequence such that
A6: ( dom Ls = O1 & ( for O2 being Ordinal st O2 in O1 holds
Ls . O2 = H1(O2) ) ) from ORDINAL2:sch_2();
Ls . {} = ConsecutiveSet (A,{}) by A3, A6, ORDINAL3:8
.= A by Th21 ;
then A7: A in rng Ls by A6, A5, FUNCT_1:def_3;
 ConsecutiveSet (A,O1) = union (rng Ls) by A3, A4, A6, Th23;
then  A c= ConsecutiveSet (A,O1) by A7, ZFMISC_1:74;
hence  S1[O1] ; ::_thesis: verum
end;
A8: S1[ {} ] by Th21;
 for O being Ordinal holds S1[O] from ORDINAL2:sch_1(A8, A1, A2);
hence  not ConsecutiveSet (A,O) is empty ; ::_thesis: verum
end;
end;

theorem Th24: :: LATTICE5:24
for A being non empty set
for O being Ordinal holds A c= ConsecutiveSet (A,O)
proof
let A be non empty set ; ::_thesis: for O being Ordinal holds A c= ConsecutiveSet (A,O)
let O be Ordinal; ::_thesis: A c= ConsecutiveSet (A,O)
defpred S1[ Ordinal] means A c= ConsecutiveSet (A,$1);
A1: for O1 being Ordinal st S1[O1] holds
S1[ succ O1]
proof
let O1 be Ordinal; ::_thesis: ( S1[O1] implies S1[ succ O1] )
 ConsecutiveSet (A,(succ O1)) = new_set (ConsecutiveSet (A,O1)) by Th22;
then A2: ConsecutiveSet (A,O1) c= ConsecutiveSet (A,(succ O1)) by XBOOLE_1:7;
assume  A c= ConsecutiveSet (A,O1) ; ::_thesis: S1[ succ O1]
hence  S1[ succ O1] by A2, XBOOLE_1:1; ::_thesis: verum
end;
A3: for O2 being Ordinal st O2 <> {} & O2 is limit_ordinal & ( for O1 being Ordinal st O1 in O2 holds
S1[O1] ) holds
S1[O2]
proof
deffunc H1( Ordinal) ->  set = ConsecutiveSet (A,$1);
let O2 be Ordinal; ::_thesis: ( O2 <> {} & O2 is limit_ordinal & ( for O1 being Ordinal st O1 in O2 holds
S1[O1] ) implies S1[O2] )
assume that
A4: O2 <> {} and
A5: O2 is limit_ordinal and
 for O1 being Ordinal st O1 in O2 holds
A c= ConsecutiveSet (A,O1) ; ::_thesis: S1[O2]
A6: {} in O2 by A4, ORDINAL3:8;
consider Ls being T-Sequence such that
A7: ( dom Ls = O2 & ( for O1 being Ordinal st O1 in O2 holds
Ls . O1 = H1(O1) ) ) from ORDINAL2:sch_2();
Ls . {} = ConsecutiveSet (A,{}) by A4, A7, ORDINAL3:8
.= A by Th21 ;
then A8: A in rng Ls by A7, A6, FUNCT_1:def_3;
 ConsecutiveSet (A,O2) = union (rng Ls) by A4, A5, A7, Th23;
hence  S1[O2] by A8, ZFMISC_1:74; ::_thesis: verum
end;
A9: S1[ {} ] by Th21;
 for O being Ordinal holds S1[O] from ORDINAL2:sch_1(A9, A1, A3);
hence  A c= ConsecutiveSet (A,O) ; ::_thesis: verum
end;

definition
let A be non empty set ;
let L be lower-bounded LATTICE;
let d be BiFunction of A,L;
mode QuadrSeq of d ->  T-Sequence of [:A,A, the carrier of L, the carrier of L:] means :Def13: :: LATTICE5:def 13
( dom it is Cardinal & it is one-to-one & rng it =  {  [x,y,a,b] where x, y is Element of A, a, b is Element of L : d . (x,y) <= a "\/" b  }  );
existence
 ex b1 being T-Sequence of [:A,A, the carrier of L, the carrier of L:] st
( dom b1 is Cardinal & b1 is one-to-one & rng b1 =  {  [x,y,a,b] where x, y is Element of A, a, b is Element of L : d . (x,y) <= a "\/" b  }  )
proof
set X =  {  [x,y,a,b] where x, y is Element of A, a, b is Element of L : d . (x,y) <= a "\/" b  }  ;
 card  {  [x,y,a,b] where x, y is Element of A, a, b is Element of L : d . (x,y) <= a "\/" b  }  , {  [x,y,a,b] where x, y is Element of A, a, b is Element of L : d . (x,y) <= a "\/" b  }  are_equipotent by CARD_1:def_2;
then consider f being Function such that
A1: f is one-to-one and
A2: dom f = card  {  [x,y,a,b] where x, y is Element of A, a, b is Element of L : d . (x,y) <= a "\/" b  }  and
A3: rng f =  {  [x,y,a,b] where x, y is Element of A, a, b is Element of L : d . (x,y) <= a "\/" b  }  by WELLORD2:def_4;
reconsider f = f as T-Sequence by A2, ORDINAL1:def_7;
 rng f c= [:A,A, the carrier of L, the carrier of L:]
proof
let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in rng f or z in [:A,A, the carrier of L, the carrier of L:] )
assume  z in rng f ; ::_thesis: z in [:A,A, the carrier of L, the carrier of L:]
then  ex x, y being Element of A ex a, b being Element of L st
( z = [x,y,a,b] & d . (x,y) <= a "\/" b ) by A3;
hence  z in [:A,A, the carrier of L, the carrier of L:] ; ::_thesis: verum
end;
then reconsider f = f as T-Sequence of [:A,A, the carrier of L, the carrier of L:] by RELAT_1:def_19;
take  f ; ::_thesis: ( dom f is Cardinal & f is one-to-one & rng f =  {  [x,y,a,b] where x, y is Element of A, a, b is Element of L : d . (x,y) <= a "\/" b  }  )
thus  dom f is Cardinal by A2; ::_thesis: ( f is one-to-one & rng f =  {  [x,y,a,b] where x, y is Element of A, a, b is Element of L : d . (x,y) <= a "\/" b  }  )
thus  f is one-to-one by A1; ::_thesis: rng f =  {  [x,y,a,b] where x, y is Element of A, a, b is Element of L : d . (x,y) <= a "\/" b  }
thus  rng f =  {  [x,y,a,b] where x, y is Element of A, a, b is Element of L : d . (x,y) <= a "\/" b  }  by A3; ::_thesis: verum
end;
end;

:: deftheorem Def13 defines QuadrSeq LATTICE5:def_13_:_
 for A being non empty set
for L being lower-bounded LATTICE
for d being BiFunction of A,L
for b4 being T-Sequence of [:A,A, the carrier of L, the carrier of L:] holds
( b4 is QuadrSeq of d iff ( dom b4 is Cardinal & b4 is one-to-one & rng b4 =  {  [x,y,a,b] where x, y is Element of A, a, b is Element of L : d . (x,y) <= a "\/" b  }  ) );

definition
let A be non empty set ;
let L be lower-bounded LATTICE;
let d be BiFunction of A,L;
let q be QuadrSeq of d;
let O be Ordinal;
assume A1: O in dom q ;
func Quadr (q,O) ->  Element of [:(ConsecutiveSet (A,O)),(ConsecutiveSet (A,O)), the carrier of L, the carrier of L:] equals :Def14: :: LATTICE5:def 14
q . O;
correctness
coherence
q . O is Element of [:(ConsecutiveSet (A,O)),(ConsecutiveSet (A,O)), the carrier of L, the carrier of L:];
proof
 q . O in rng q by A1, FUNCT_1:def_3;
then  q . O in  {  [x,y,a,b] where x, y is Element of A, a, b is Element of L : d . (x,y) <= a "\/" b  }  by Def13;
then consider x, y being Element of A, a, b being Element of L such that
A2: q . O = [x,y,a,b] and
 d . (x,y) <= a "\/" b ;
reconsider a = a, b = b as Element of L ;
A3: ( x in A & y in A ) ;
 A c= ConsecutiveSet (A,O) by Th24;
then reconsider x = x, y = y as Element of ConsecutiveSet (A,O) by A3;
reconsider z = [x,y,a,b] as Element of [:(ConsecutiveSet (A,O)),(ConsecutiveSet (A,O)), the carrier of L, the carrier of L:] ;
 z = q . O by A2;
hence  q . O is Element of [:(ConsecutiveSet (A,O)),(ConsecutiveSet (A,O)), the carrier of L, the carrier of L:] ; ::_thesis: verum
end;
end;

:: deftheorem Def14 defines Quadr LATTICE5:def_14_:_
 for A being non empty set
for L being lower-bounded LATTICE
for d being BiFunction of A,L
for q being QuadrSeq of d
for O being Ordinal st O in dom q holds
Quadr (q,O) = q . O;

theorem Th25: :: LATTICE5:25
for A being non empty set
for L being lower-bounded LATTICE
for O being Ordinal
for d being BiFunction of A,L
for q being QuadrSeq of d holds
( O in DistEsti d iff O in dom q )
proof
let A be non empty set ; ::_thesis: for L being lower-bounded LATTICE
for O being Ordinal
for d being BiFunction of A,L
for q being QuadrSeq of d holds
( O in DistEsti d iff O in dom q )
let L be lower-bounded LATTICE; ::_thesis: for O being Ordinal
for d being BiFunction of A,L
for q being QuadrSeq of d holds
( O in DistEsti d iff O in dom q )
let O be Ordinal; ::_thesis: for d being BiFunction of A,L
for q being QuadrSeq of d holds
( O in DistEsti d iff O in dom q )
let d be BiFunction of A,L; ::_thesis: for q being QuadrSeq of d holds
( O in DistEsti d iff O in dom q )
let q be QuadrSeq of d; ::_thesis: ( O in DistEsti d iff O in dom q )
reconsider N = dom q as Cardinal by Def13;
reconsider M = DistEsti d as Cardinal ;
 q is one-to-one by Def13;
then A1: dom q, rng q are_equipotent by WELLORD2:def_4;
 DistEsti d, {  [x,y,a,b] where x, y is Element of A, a, b is Element of L : d . (x,y) <= a "\/" b  }  are_equipotent by Def11;
then  DistEsti d, rng q are_equipotent by Def13;
then  DistEsti d, dom q are_equipotent by A1, WELLORD2:15;
then A2: M = N by CARD_1:2;
hence  ( O in DistEsti d implies O in dom q ) ; ::_thesis: ( O in dom q implies O in DistEsti d )
assume  O in dom q ; ::_thesis: O in DistEsti d
hence  O in DistEsti d by A2; ::_thesis: verum
end;

definition
let A be non empty set ;
let L be lower-bounded LATTICE;
let z be set ;
assume A1: z is BiFunction of A,L ;
func BiFun (z,A,L) ->  BiFunction of A,L equals :Def15: :: LATTICE5:def 15
z;
coherence
 z is BiFunction of A,L by A1;
end;

:: deftheorem Def15 defines BiFun LATTICE5:def_15_:_
 for A being non empty set
for L being lower-bounded LATTICE
for z being set st z is BiFunction of A,L holds
BiFun (z,A,L) = z;

definition
let A be non empty set ;
let L be lower-bounded LATTICE;
let d be BiFunction of A,L;
let q be QuadrSeq of d;
let O be Ordinal;
func ConsecutiveDelta (q,O) ->  set  means :Def16: :: LATTICE5:def 16
 ex L0 being T-Sequence st
( it = last L0 & dom L0 = succ O & L0 . {} = d & ( for C being Ordinal st succ C in succ O holds
L0 . (succ C) = new_bi_fun ((BiFun ((L0 . C),(ConsecutiveSet (A,C)),L)),(Quadr (q,C))) ) & ( for C being Ordinal st C in succ O & C <> {} & C is limit_ordinal holds
L0 . C = union (rng (L0 | C)) ) );
correctness
existence
ex b1 being set ex L0 being T-Sequence st
( b1 = last L0 & dom L0 = succ O & L0 . {} = d & ( for C being Ordinal st succ C in succ O holds
L0 . (succ C) = new_bi_fun ((BiFun ((L0 . C),(ConsecutiveSet (A,C)),L)),(Quadr (q,C))) ) & ( for C being Ordinal st C in succ O & C <> {} & C is limit_ordinal holds
L0 . C = union (rng (L0 | C)) ) );
uniqueness
for b1, b2 being set st ex L0 being T-Sequence st
( b1 = last L0 & dom L0 = succ O & L0 . {} = d & ( for C being Ordinal st succ C in succ O holds
L0 . (succ C) = new_bi_fun ((BiFun ((L0 . C),(ConsecutiveSet (A,C)),L)),(Quadr (q,C))) ) & ( for C being Ordinal st C in succ O & C <> {} & C is limit_ordinal holds
L0 . C = union (rng (L0 | C)) ) ) & ex L0 being T-Sequence st
( b2 = last L0 & dom L0 = succ O & L0 . {} = d & ( for C being Ordinal st succ C in succ O holds
L0 . (succ C) = new_bi_fun ((BiFun ((L0 . C),(ConsecutiveSet (A,C)),L)),(Quadr (q,C))) ) & ( for C being Ordinal st C in succ O & C <> {} & C is limit_ordinal holds
L0 . C = union (rng (L0 | C)) ) ) holds
b1 = b2;
proof
deffunc H1( Ordinal, T-Sequence) ->  set = union (rng $2);
deffunc H2( Ordinal, set ) ->  BiFunction of (new_set (ConsecutiveSet (A,$1))),L = new_bi_fun ((BiFun ($2,(ConsecutiveSet (A,$1)),L)),(Quadr (q,$1)));
thus  ( ex x being set ex L1 being T-Sequence st
( x = last L1 & dom L1 = succ O & L1 . {} = d & ( for C being Ordinal st succ C in succ O holds
L1 . (succ C) = H2(C,L1 . C) ) & ( for C being Ordinal st C in succ O & C <> {} & C is limit_ordinal holds
L1 . C = H1(C,L1 | C) ) ) & ( for x1, x2 being set st ex L1 being T-Sequence st
( x1 = last L1 & dom L1 = succ O & L1 . {} = d & ( for C being Ordinal st succ C in succ O holds
L1 . (succ C) = H2(C,L1 . C) ) & ( for C being Ordinal st C in succ O & C <> {} & C is limit_ordinal holds
L1 . C = H1(C,L1 | C) ) ) & ex L1 being T-Sequence st
( x2 = last L1 & dom L1 = succ O & L1 . {} = d & ( for C being Ordinal st succ C in succ O holds
L1 . (succ C) = H2(C,L1 . C) ) & ( for C being Ordinal st C in succ O & C <> {} & C is limit_ordinal holds
L1 . C = H1(C,L1 | C) ) ) holds
x1 = x2 ) ) from ORDINAL2:sch_7(); ::_thesis: verum
end;
end;

:: deftheorem Def16 defines ConsecutiveDelta LATTICE5:def_16_:_
 for A being non empty set
for L being lower-bounded LATTICE
for d being BiFunction of A,L
for q being QuadrSeq of d
for O being Ordinal
for b6 being set holds
( b6 = ConsecutiveDelta (q,O) iff ex L0 being T-Sequence st
( b6 = last L0 & dom L0 = succ O & L0 . {} = d & ( for C being Ordinal st succ C in succ O holds
L0 . (succ C) = new_bi_fun ((BiFun ((L0 . C),(ConsecutiveSet (A,C)),L)),(Quadr (q,C))) ) & ( for C being Ordinal st C in succ O & C <> {} & C is limit_ordinal holds
L0 . C = union (rng (L0 | C)) ) ) );

theorem Th26: :: LATTICE5:26
for A being non empty set
for L being lower-bounded LATTICE
for d being BiFunction of A,L
for q being QuadrSeq of d holds ConsecutiveDelta (q,{}) = d
proof
let A be non empty set ; ::_thesis: for L being lower-bounded LATTICE
for d being BiFunction of A,L
for q being QuadrSeq of d holds ConsecutiveDelta (q,{}) = d
let L be lower-bounded LATTICE; ::_thesis: for d being BiFunction of A,L
for q being QuadrSeq of d holds ConsecutiveDelta (q,{}) = d
deffunc H1( Ordinal, T-Sequence) ->  set = union (rng $2);
let d be BiFunction of A,L; ::_thesis: for q being QuadrSeq of d holds ConsecutiveDelta (q,{}) = d
let q be QuadrSeq of d; ::_thesis: ConsecutiveDelta (q,{}) = d
deffunc H2( Ordinal, set ) ->  BiFunction of (new_set (ConsecutiveSet (A,$1))),L = new_bi_fun ((BiFun ($2,(ConsecutiveSet (A,$1)),L)),(Quadr (q,$1)));
deffunc H3( Ordinal) ->  set = ConsecutiveDelta (q,$1);
A1: for O being Ordinal
for It being set holds
( It = H3(O) iff ex L0 being T-Sequence st
( It = last L0 & dom L0 = succ O & L0 . {} = d & ( for C being Ordinal st succ C in succ O holds
L0 . (succ C) = H2(C,L0 . C) ) & ( for C being Ordinal st C in succ O & C <> {} & C is limit_ordinal holds
L0 . C = H1(C,L0 | C) ) ) ) by Def16;
thus  H3( {} ) = d from ORDINAL2:sch_8(A1); ::_thesis: verum
end;

theorem Th27: :: LATTICE5:27
for A being non empty set
for L being lower-bounded LATTICE
for O being Ordinal
for d being BiFunction of A,L
for q being QuadrSeq of d holds ConsecutiveDelta (q,(succ O)) = new_bi_fun ((BiFun ((ConsecutiveDelta (q,O)),(ConsecutiveSet (A,O)),L)),(Quadr (q,O)))
proof
let A be non empty set ; ::_thesis: for L being lower-bounded LATTICE
for O being Ordinal
for d being BiFunction of A,L
for q being QuadrSeq of d holds ConsecutiveDelta (q,(succ O)) = new_bi_fun ((BiFun ((ConsecutiveDelta (q,O)),(ConsecutiveSet (A,O)),L)),(Quadr (q,O)))
let L be lower-bounded LATTICE; ::_thesis: for O being Ordinal
for d being BiFunction of A,L
for q being QuadrSeq of d holds ConsecutiveDelta (q,(succ O)) = new_bi_fun ((BiFun ((ConsecutiveDelta (q,O)),(ConsecutiveSet (A,O)),L)),(Quadr (q,O)))
let O be Ordinal; ::_thesis: for d being BiFunction of A,L
for q being QuadrSeq of d holds ConsecutiveDelta (q,(succ O)) = new_bi_fun ((BiFun ((ConsecutiveDelta (q,O)),(ConsecutiveSet (A,O)),L)),(Quadr (q,O)))
deffunc H1( Ordinal, T-Sequence) ->  set = union (rng $2);
let d be BiFunction of A,L; ::_thesis: for q being QuadrSeq of d holds ConsecutiveDelta (q,(succ O)) = new_bi_fun ((BiFun ((ConsecutiveDelta (q,O)),(ConsecutiveSet (A,O)),L)),(Quadr (q,O)))
let q be QuadrSeq of d; ::_thesis: ConsecutiveDelta (q,(succ O)) = new_bi_fun ((BiFun ((ConsecutiveDelta (q,O)),(ConsecutiveSet (A,O)),L)),(Quadr (q,O)))
deffunc H2( Ordinal, set ) ->  BiFunction of (new_set (ConsecutiveSet (A,$1))),L = new_bi_fun ((BiFun ($2,(ConsecutiveSet (A,$1)),L)),(Quadr (q,$1)));
deffunc H3( Ordinal) ->  set = ConsecutiveDelta (q,$1);
A1: for O being Ordinal
for It being set holds
( It = H3(O) iff ex L0 being T-Sequence st
( It = last L0 & dom L0 = succ O & L0 . {} = d & ( for C being Ordinal st succ C in succ O holds
L0 . (succ C) = H2(C,L0 . C) ) & ( for C being Ordinal st C in succ O & C <> {} & C is limit_ordinal holds
L0 . C = H1(C,L0 | C) ) ) ) by Def16;
 for O being Ordinal holds H3( succ O) = H2(O,H3(O)) from ORDINAL2:sch_9(A1);
hence  ConsecutiveDelta (q,(succ O)) = new_bi_fun ((BiFun ((ConsecutiveDelta (q,O)),(ConsecutiveSet (A,O)),L)),(Quadr (q,O))) ; ::_thesis: verum
end;

theorem Th28: :: LATTICE5:28
for A being non empty set
for L being lower-bounded LATTICE
for T being T-Sequence
for O being Ordinal
for d being BiFunction of A,L
for q being QuadrSeq of d st O <> {} & O is limit_ordinal & dom T = O & ( for O1 being Ordinal st O1 in O holds
T . O1 = ConsecutiveDelta (q,O1) ) holds
ConsecutiveDelta (q,O) = union (rng T)
proof
let A be non empty set ; ::_thesis: for L being lower-bounded LATTICE
for T being T-Sequence
for O being Ordinal
for d being BiFunction of A,L
for q being QuadrSeq of d st O <> {} & O is limit_ordinal & dom T = O & ( for O1 being Ordinal st O1 in O holds
T . O1 = ConsecutiveDelta (q,O1) ) holds
ConsecutiveDelta (q,O) = union (rng T)
let L be lower-bounded LATTICE; ::_thesis: for T being T-Sequence
for O being Ordinal
for d being BiFunction of A,L
for q being QuadrSeq of d st O <> {} & O is limit_ordinal & dom T = O & ( for O1 being Ordinal st O1 in O holds
T . O1 = ConsecutiveDelta (q,O1) ) holds
ConsecutiveDelta (q,O) = union (rng T)
let T be T-Sequence; ::_thesis: for O being Ordinal
for d being BiFunction of A,L
for q being QuadrSeq of d st O <> {} & O is limit_ordinal & dom T = O & ( for O1 being Ordinal st O1 in O holds
T . O1 = ConsecutiveDelta (q,O1) ) holds
ConsecutiveDelta (q,O) = union (rng T)
let O be Ordinal; ::_thesis: for d being BiFunction of A,L
for q being QuadrSeq of d st O <> {} & O is limit_ordinal & dom T = O & ( for O1 being Ordinal st O1 in O holds
T . O1 = ConsecutiveDelta (q,O1) ) holds
ConsecutiveDelta (q,O) = union (rng T)
deffunc H1( Ordinal, T-Sequence) ->  set = union (rng $2);
let d be BiFunction of A,L; ::_thesis: for q being QuadrSeq of d st O <> {} & O is limit_ordinal & dom T = O & ( for O1 being Ordinal st O1 in O holds
T . O1 = ConsecutiveDelta (q,O1) ) holds
ConsecutiveDelta (q,O) = union (rng T)
let q be QuadrSeq of d; ::_thesis: ( O <> {} & O is limit_ordinal & dom T = O & ( for O1 being Ordinal st O1 in O holds
T . O1 = ConsecutiveDelta (q,O1) ) implies ConsecutiveDelta (q,O) = union (rng T) )
deffunc H2( Ordinal, set ) ->  BiFunction of (new_set (ConsecutiveSet (A,$1))),L = new_bi_fun ((BiFun ($2,(ConsecutiveSet (A,$1)),L)),(Quadr (q,$1)));
deffunc H3( Ordinal) ->  set = ConsecutiveDelta (q,$1);
assume that
A1: ( O <> {} & O is limit_ordinal ) and
A2: dom T = O and
A3: for O1 being Ordinal st O1 in O holds
T . O1 = H3(O1) ; ::_thesis: ConsecutiveDelta (q,O) = union (rng T)
A4: for O being Ordinal
for It being set holds
( It = H3(O) iff ex L0 being T-Sequence st
( It = last L0 & dom L0 = succ O & L0 . {} = d & ( for C being Ordinal st succ C in succ O holds
L0 . (succ C) = H2(C,L0 . C) ) & ( for C being Ordinal st C in succ O & C <> {} & C is limit_ordinal holds
L0 . C = H1(C,L0 | C) ) ) ) by Def16;
thus  H3(O) = H1(O,T) from ORDINAL2:sch_10(A4, A1, A2, A3); ::_thesis: verum
end;

theorem Th29: :: LATTICE5:29
for A being non empty set
for O1, O2 being Ordinal st O1 c= O2 holds
ConsecutiveSet (A,O1) c= ConsecutiveSet (A,O2)
proof
let A be non empty set ; ::_thesis: for O1, O2 being Ordinal st O1 c= O2 holds
ConsecutiveSet (A,O1) c= ConsecutiveSet (A,O2)
let O1, O2 be Ordinal; ::_thesis: ( O1 c= O2 implies ConsecutiveSet (A,O1) c= ConsecutiveSet (A,O2) )
defpred S1[ Ordinal] means ( O1 c= $1 implies ConsecutiveSet (A,O1) c= ConsecutiveSet (A,$1) );
A1: for O2 being Ordinal st S1[O2] holds
S1[ succ O2]
proof
let O2 be Ordinal; ::_thesis: ( S1[O2] implies S1[ succ O2] )
assume A2: ( O1 c= O2 implies ConsecutiveSet (A,O1) c= ConsecutiveSet (A,O2) ) ; ::_thesis: S1[ succ O2]
assume A3: O1 c= succ O2 ; ::_thesis: ConsecutiveSet (A,O1) c= ConsecutiveSet (A,(succ O2))
percases ( O1 = succ O2 or O1 <> succ O2 ) ;
suppose O1 = succ O2 ; ::_thesis: ConsecutiveSet (A,O1) c= ConsecutiveSet (A,(succ O2))
hence  ConsecutiveSet (A,O1) c= ConsecutiveSet (A,(succ O2)) ; ::_thesis: verum
end;
suppose O1 <> succ O2 ; ::_thesis: ConsecutiveSet (A,O1) c= ConsecutiveSet (A,(succ O2))
then  O1 c< succ O2 by A3, XBOOLE_0:def_8;
then A4: O1 in succ O2 by ORDINAL1:11;
 ConsecutiveSet (A,O2) c= new_set (ConsecutiveSet (A,O2)) by XBOOLE_1:7;
then  ConsecutiveSet (A,O1) c= new_set (ConsecutiveSet (A,O2)) by A2, A4, ORDINAL1:22, XBOOLE_1:1;
hence  ConsecutiveSet (A,O1) c= ConsecutiveSet (A,(succ O2)) by Th22; ::_thesis: verum
end;
end;
end;
A5: for O2 being Ordinal st O2 <> {} & O2 is limit_ordinal & ( for O3 being Ordinal st O3 in O2 holds
S1[O3] ) holds
S1[O2]
proof
deffunc H1( Ordinal) ->  set = ConsecutiveSet (A,$1);
let O2 be Ordinal; ::_thesis: ( O2 <> {} & O2 is limit_ordinal & ( for O3 being Ordinal st O3 in O2 holds
S1[O3] ) implies S1[O2] )
assume that
A6: ( O2 <> {} & O2 is limit_ordinal ) and
 for O3 being Ordinal st O3 in O2 & O1 c= O3 holds
ConsecutiveSet (A,O1) c= ConsecutiveSet (A,O3) ; ::_thesis: S1[O2]
consider L being T-Sequence such that
A7: ( dom L = O2 & ( for O3 being Ordinal st O3 in O2 holds
L . O3 = H1(O3) ) ) from ORDINAL2:sch_2();
A8: ConsecutiveSet (A,O2) = union (rng L) by A6, A7, Th23;
assume A9: O1 c= O2 ; ::_thesis: ConsecutiveSet (A,O1) c= ConsecutiveSet (A,O2)
percases ( O1 = O2 or O1 <> O2 ) ;
suppose O1 = O2 ; ::_thesis: ConsecutiveSet (A,O1) c= ConsecutiveSet (A,O2)
hence  ConsecutiveSet (A,O1) c= ConsecutiveSet (A,O2) ; ::_thesis: verum
end;
suppose O1 <> O2 ; ::_thesis: ConsecutiveSet (A,O1) c= ConsecutiveSet (A,O2)
then A10: O1 c< O2 by A9, XBOOLE_0:def_8;
then  O1 in O2 by ORDINAL1:11;
then A11: L . O1 in rng L by A7, FUNCT_1:def_3;
 L . O1 = ConsecutiveSet (A,O1) by A7, A10, ORDINAL1:11;
hence  ConsecutiveSet (A,O1) c= ConsecutiveSet (A,O2) by A8, A11, ZFMISC_1:74; ::_thesis: verum
end;
end;
end;
A12: S1[ {} ] ;
 for O2 being Ordinal holds S1[O2] from ORDINAL2:sch_1(A12, A1, A5);
hence  ( O1 c= O2 implies ConsecutiveSet (A,O1) c= ConsecutiveSet (A,O2) ) ; ::_thesis: verum
end;

theorem Th30: :: LATTICE5:30
for A being non empty set
for L being lower-bounded LATTICE
for O being Ordinal
for d being BiFunction of A,L
for q being QuadrSeq of d holds ConsecutiveDelta (q,O) is BiFunction of (ConsecutiveSet (A,O)),L
proof
let A be non empty set ; ::_thesis: for L being lower-bounded LATTICE
for O being Ordinal
for d being BiFunction of A,L
for q being QuadrSeq of d holds ConsecutiveDelta (q,O) is BiFunction of (ConsecutiveSet (A,O)),L
let L be lower-bounded LATTICE; ::_thesis: for O being Ordinal
for d being BiFunction of A,L
for q being QuadrSeq of d holds ConsecutiveDelta (q,O) is BiFunction of (ConsecutiveSet (A,O)),L
let O be Ordinal; ::_thesis: for d being BiFunction of A,L
for q being QuadrSeq of d holds ConsecutiveDelta (q,O) is BiFunction of (ConsecutiveSet (A,O)),L
let d be BiFunction of A,L; ::_thesis: for q being QuadrSeq of d holds ConsecutiveDelta (q,O) is BiFunction of (ConsecutiveSet (A,O)),L
let q be QuadrSeq of d; ::_thesis: ConsecutiveDelta (q,O) is BiFunction of (ConsecutiveSet (A,O)),L
defpred S1[ Ordinal] means  ConsecutiveDelta (q,$1) is BiFunction of (ConsecutiveSet (A,$1)),L;
A1: for O1 being Ordinal st S1[O1] holds
S1[ succ O1]
proof
let O1 be Ordinal; ::_thesis: ( S1[O1] implies S1[ succ O1] )
assume  ConsecutiveDelta (q,O1) is BiFunction of (ConsecutiveSet (A,O1)),L ; ::_thesis: S1[ succ O1]
then reconsider CD = ConsecutiveDelta (q,O1) as BiFunction of (ConsecutiveSet (A,O1)),L ;
A2: ConsecutiveSet (A,(succ O1)) = new_set (ConsecutiveSet (A,O1)) by Th22;
ConsecutiveDelta (q,(succ O1)) = new_bi_fun ((BiFun ((ConsecutiveDelta (q,O1)),(ConsecutiveSet (A,O1)),L)),(Quadr (q,O1))) by Th27
.= new_bi_fun (CD,(Quadr (q,O1))) by Def15 ;
hence  S1[ succ O1] by A2; ::_thesis: verum
end;
A3: for O1 being Ordinal st O1 <> {} & O1 is limit_ordinal & ( for O2 being Ordinal st O2 in O1 holds
S1[O2] ) holds
S1[O1]
proof
deffunc H1( Ordinal) ->  set = ConsecutiveDelta (q,$1);
let O1 be Ordinal; ::_thesis: ( O1 <> {} & O1 is limit_ordinal & ( for O2 being Ordinal st O2 in O1 holds
S1[O2] ) implies S1[O1] )
assume that
A4: O1 <> {} and
A5: O1 is limit_ordinal and
A6: for O2 being Ordinal st O2 in O1 holds
ConsecutiveDelta (q,O2) is BiFunction of (ConsecutiveSet (A,O2)),L ; ::_thesis: S1[O1]
consider Ls being T-Sequence such that
A7: ( dom Ls = O1 & ( for O2 being Ordinal st O2 in O1 holds
Ls . O2 = H1(O2) ) ) from ORDINAL2:sch_2();
A8: for O, O2 being Ordinal st O c= O2 & O2 in dom Ls holds
Ls . O c= Ls . O2
proof
let O be Ordinal; ::_thesis: for O2 being Ordinal st O c= O2 & O2 in dom Ls holds
Ls . O c= Ls . O2
defpred S2[ Ordinal] means ( O c= $1 & $1 in dom Ls implies Ls . O c= Ls . $1 );
A9: for O2 being Ordinal st O2 <> {} & O2 is limit_ordinal & ( for O3 being Ordinal st O3 in O2 holds
S2[O3] ) holds
S2[O2]
proof
deffunc H2( Ordinal) ->  set = ConsecutiveDelta (q,$1);
let O2 be Ordinal; ::_thesis: ( O2 <> {} & O2 is limit_ordinal & ( for O3 being Ordinal st O3 in O2 holds
S2[O3] ) implies S2[O2] )
assume that
A10: ( O2 <> {} & O2 is limit_ordinal ) and
 for O3 being Ordinal st O3 in O2 & O c= O3 & O3 in dom Ls holds
Ls . O c= Ls . O3 ; ::_thesis: S2[O2]
assume that
A11: O c= O2 and
A12: O2 in dom Ls ; ::_thesis: Ls . O c= Ls . O2
consider Lt being T-Sequence such that
A13: ( dom Lt = O2 & ( for O3 being Ordinal st O3 in O2 holds
Lt . O3 = H2(O3) ) ) from ORDINAL2:sch_2();
A14: Ls . O2 = ConsecutiveDelta (q,O2) by A7, A12
.= union (rng Lt) by A10, A13, Th28 ;
percases ( O = O2 or O <> O2 ) ;
suppose O = O2 ; ::_thesis: Ls . O c= Ls . O2
hence  Ls . O c= Ls . O2 ; ::_thesis: verum
end;
suppose O <> O2 ; ::_thesis: Ls . O c= Ls . O2
then A15: O c< O2 by A11, XBOOLE_0:def_8;
then A16: O in O2 by ORDINAL1:11;
then Ls . O = ConsecutiveDelta (q,O) by A7, A12, ORDINAL1:10
.= Lt . O by A13, A15, ORDINAL1:11 ;
then  Ls . O in rng Lt by A13, A16, FUNCT_1:def_3;
hence  Ls . O c= Ls . O2 by A14, ZFMISC_1:74; ::_thesis: verum
end;
end;
end;
A17: for O2 being Ordinal st S2[O2] holds
S2[ succ O2]
proof
let O2 be Ordinal; ::_thesis: ( S2[O2] implies S2[ succ O2] )
assume A18: ( O c= O2 & O2 in dom Ls implies Ls . O c= Ls . O2 ) ; ::_thesis: S2[ succ O2]
assume that
A19: O c= succ O2 and
A20: succ O2 in dom Ls ; ::_thesis: Ls . O c= Ls . (succ O2)
percases ( O = succ O2 or O <> succ O2 ) ;
suppose O = succ O2 ; ::_thesis: Ls . O c= Ls . (succ O2)
hence  Ls . O c= Ls . (succ O2) ; ::_thesis: verum
end;
suppose O <> succ O2 ; ::_thesis: Ls . O c= Ls . (succ O2)
then  O c< succ O2 by A19, XBOOLE_0:def_8;
then A21: O in succ O2 by ORDINAL1:11;
A22: O2 in succ O2 by ORDINAL1:6;
then  O2 in dom Ls by A20, ORDINAL1:10;
then reconsider cd2 = ConsecutiveDelta (q,O2) as BiFunction of (ConsecutiveSet (A,O2)),L by A6, A7;
Ls . (succ O2) = ConsecutiveDelta (q,(succ O2)) by A7, A20
.= new_bi_fun ((BiFun ((ConsecutiveDelta (q,O2)),(ConsecutiveSet (A,O2)),L)),(Quadr (q,O2))) by Th27
.= new_bi_fun (cd2,(Quadr (q,O2))) by Def15 ;
then  ConsecutiveDelta (q,O2) c= Ls . (succ O2) by Th19;
then  Ls . O2 c= Ls . (succ O2) by A7, A20, A22, ORDINAL1:10;
hence  Ls . O c= Ls . (succ O2) by A18, A20, A21, A22, ORDINAL1:10, ORDINAL1:22, XBOOLE_1:1; ::_thesis: verum
end;
end;
end;
A23: S2[ {} ] ;
thus  for O2 being Ordinal holds S2[O2] from ORDINAL2:sch_1(A23, A17, A9); ::_thesis: verum
end;
 for x, y being set st x in rng Ls & y in rng Ls holds
x,y are_c=-comparable
proof
let x, y be set ; ::_thesis: ( x in rng Ls & y in rng Ls implies x,y are_c=-comparable )
assume that
A24: x in rng Ls and
A25: y in rng Ls ; ::_thesis: x,y are_c=-comparable
consider o1 being set  such that
A26: o1 in dom Ls and
A27: Ls . o1 = x by A24, FUNCT_1:def_3;
consider o2 being set  such that
A28: o2 in dom Ls and
A29: Ls . o2 = y by A25, FUNCT_1:def_3;
reconsider o19 = o1, o29 = o2 as Ordinal by A26, A28;
 ( o19 c= o29 or o29 c= o19 ) ;
then  ( x c= y or y c= x ) by A8, A26, A27, A28, A29;
hence  x,y are_c=-comparable by XBOOLE_0:def_9; ::_thesis: verum
end;
then A30: rng Ls is c=-linear by ORDINAL1:def_8;
set Y = the carrier of L;
set X = [:(ConsecutiveSet (A,O1)),(ConsecutiveSet (A,O1)):];
set f = union (rng Ls);
 rng Ls c= PFuncs ([:(ConsecutiveSet (A,O1)),(ConsecutiveSet (A,O1)):], the carrier of L)
proof
let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in rng Ls or z in PFuncs ([:(ConsecutiveSet (A,O1)),(ConsecutiveSet (A,O1)):], the carrier of L) )
assume  z in rng Ls ; ::_thesis: z in PFuncs ([:(ConsecutiveSet (A,O1)),(ConsecutiveSet (A,O1)):], the carrier of L)
then consider o being set  such that
A31: o in dom Ls and
A32: z = Ls . o by FUNCT_1:def_3;
reconsider o = o as Ordinal by A31;
 Ls . o = ConsecutiveDelta (q,o) by A7, A31;
then reconsider h = Ls . o as BiFunction of (ConsecutiveSet (A,o)),L by A6, A7, A31;
 o c= O1 by A7, A31, ORDINAL1:def_2;
then  ( dom h = [:(ConsecutiveSet (A,o)),(ConsecutiveSet (A,o)):] & ConsecutiveSet (A,o) c= ConsecutiveSet (A,O1) ) by Th29, FUNCT_2:def_1;
then  ( rng h c= the carrier of L & dom h c= [:(ConsecutiveSet (A,O1)),(ConsecutiveSet (A,O1)):] ) by ZFMISC_1:96;
hence  z in PFuncs ([:(ConsecutiveSet (A,O1)),(ConsecutiveSet (A,O1)):], the carrier of L) by A32, PARTFUN1:def_3; ::_thesis: verum
end;
then  union (rng Ls) in PFuncs ([:(ConsecutiveSet (A,O1)),(ConsecutiveSet (A,O1)):], the carrier of L) by A30, TREES_2:40;
then A33: ex g being Function st
( union (rng Ls) = g & dom g c= [:(ConsecutiveSet (A,O1)),(ConsecutiveSet (A,O1)):] & rng g c= the carrier of L ) by PARTFUN1:def_3;
reconsider o1 = O1 as non empty Ordinal by A4;
set YY =  {  [:(ConsecutiveSet (A,O2)),(ConsecutiveSet (A,O2)):] where O2 is Element of o1 : verum  }  ;
deffunc H2( Ordinal) ->  set = ConsecutiveSet (A,$1);
consider Ts being T-Sequence such that
A34: ( dom Ts = O1 & ( for O2 being Ordinal st O2 in O1 holds
Ts . O2 = H2(O2) ) ) from ORDINAL2:sch_2();
 Ls is Function-yielding
proof
let x be set ; :: according to FUNCOP_1:def_6 ::_thesis: ( not x in proj1 Ls or Ls . x is set )
assume A35: x in dom Ls ; ::_thesis: Ls . x is set
then reconsider o = x as Ordinal ;
 Ls . o = ConsecutiveDelta (q,o) by A7, A35;
hence  Ls . x is set by A6, A7, A35; ::_thesis: verum
end;
then reconsider LsF = Ls as Function-yielding Function ;
A36: rng (doms Ls) =  {  [:(ConsecutiveSet (A,O2)),(ConsecutiveSet (A,O2)):] where O2 is Element of o1 : verum  }
proof
thus  rng (doms Ls) c=  {  [:(ConsecutiveSet (A,O2)),(ConsecutiveSet (A,O2)):] where O2 is Element of o1 : verum  }  :: according to XBOOLE_0:def_10 ::_thesis:  {  [:(ConsecutiveSet (A,O2)),(ConsecutiveSet (A,O2)):] where O2 is Element of o1 : verum  }  c= rng (doms Ls)
proof
let Z be set ; :: according to TARSKI:def_3 ::_thesis: ( not Z in rng (doms Ls) or Z in  {  [:(ConsecutiveSet (A,O2)),(ConsecutiveSet (A,O2)):] where O2 is Element of o1 : verum  }  )
assume  Z in rng (doms Ls) ; ::_thesis: Z in  {  [:(ConsecutiveSet (A,O2)),(ConsecutiveSet (A,O2)):] where O2 is Element of o1 : verum  }
then consider o being set  such that
A37: o in dom (doms Ls) and
A38: Z = (doms Ls) . o by FUNCT_1:def_3;
A39: o in dom LsF by A37, FUNCT_6:59;
then reconsider o9 = o as Element of o1 by A7;
 Ls . o9 = ConsecutiveDelta (q,o9) by A7;
then reconsider ls = Ls . o9 as BiFunction of (ConsecutiveSet (A,o9)),L by A6;
Z = dom ls by A38, A39, FUNCT_6:22
.= [:(ConsecutiveSet (A,o9)),(ConsecutiveSet (A,o9)):] by FUNCT_2:def_1 ;
hence  Z in  {  [:(ConsecutiveSet (A,O2)),(ConsecutiveSet (A,O2)):] where O2 is Element of o1 : verum  }  ; ::_thesis: verum
end;
let Z be set ; :: according to TARSKI:def_3 ::_thesis: ( not Z in  {  [:(ConsecutiveSet (A,O2)),(ConsecutiveSet (A,O2)):] where O2 is Element of o1 : verum  }  or Z in rng (doms Ls) )
assume  Z in  {  [:(ConsecutiveSet (A,O2)),(ConsecutiveSet (A,O2)):] where O2 is Element of o1 : verum  }  ; ::_thesis: Z in rng (doms Ls)
then consider o being Element of o1 such that
A40: Z = [:(ConsecutiveSet (A,o)),(ConsecutiveSet (A,o)):] ;
 Ls . o = ConsecutiveDelta (q,o) by A7;
then reconsider ls = Ls . o as BiFunction of (ConsecutiveSet (A,o)),L by A6;
 o in dom LsF by A7;
then A41: o in dom (doms LsF) by FUNCT_6:59;
Z = dom ls by A40, FUNCT_2:def_1
.= (doms Ls) . o by A7, FUNCT_6:22 ;
hence  Z in rng (doms Ls) by A41, FUNCT_1:def_3; ::_thesis: verum
end;
 {} in O1 by A4, ORDINAL3:8;
then reconsider RTs = rng Ts as non empty set by A34, FUNCT_1:3;
reconsider f = union (rng Ls) as Function by A33;
A42: dom f = union (rng (doms LsF)) by Th1;
A43:  {  [:(ConsecutiveSet (A,O2)),(ConsecutiveSet (A,O2)):] where O2 is Element of o1 : verum  }  =  {  [:a,a:] where a is Element of RTs : a in RTs  }
proof
set XX =  {  [:a,a:] where a is Element of RTs : a in RTs  }  ;
thus   {  [:(ConsecutiveSet (A,O2)),(ConsecutiveSet (A,O2)):] where O2 is Element of o1 : verum  }  c=  {  [:a,a:] where a is Element of RTs : a in RTs  }  :: according to XBOOLE_0:def_10 ::_thesis:  {  [:a,a:] where a is Element of RTs : a in RTs  }  c=  {  [:(ConsecutiveSet (A,O2)),(ConsecutiveSet (A,O2)):] where O2 is Element of o1 : verum  }
proof
let Z be set ; :: according to TARSKI:def_3 ::_thesis: ( not Z in  {  [:(ConsecutiveSet (A,O2)),(ConsecutiveSet (A,O2)):] where O2 is Element of o1 : verum  }  or Z in  {  [:a,a:] where a is Element of RTs : a in RTs  }  )
assume  Z in  {  [:(ConsecutiveSet (A,O2)),(ConsecutiveSet (A,O2)):] where O2 is Element of o1 : verum  }  ; ::_thesis: Z in  {  [:a,a:] where a is Element of RTs : a in RTs  }
then consider o being Element of o1 such that
A44: Z = [:(ConsecutiveSet (A,o)),(ConsecutiveSet (A,o)):] ;
 Ts . o = ConsecutiveSet (A,o) by A34;
then reconsider CoS = ConsecutiveSet (A,o) as Element of RTs by A34, FUNCT_1:def_3;
 Z = [:CoS,CoS:] by A44;
hence  Z in  {  [:a,a:] where a is Element of RTs : a in RTs  }  ; ::_thesis: verum
end;
let Z be set ; :: according to TARSKI:def_3 ::_thesis: ( not Z in  {  [:a,a:] where a is Element of RTs : a in RTs  }  or Z in  {  [:(ConsecutiveSet (A,O2)),(ConsecutiveSet (A,O2)):] where O2 is Element of o1 : verum  }  )
assume  Z in  {  [:a,a:] where a is Element of RTs : a in RTs  }  ; ::_thesis: Z in  {  [:(ConsecutiveSet (A,O2)),(ConsecutiveSet (A,O2)):] where O2 is Element of o1 : verum  }
then consider a being Element of RTs such that
A45: Z = [:a,a:] and
 a in RTs ;
consider o being set  such that
A46: o in dom Ts and
A47: a = Ts . o by FUNCT_1:def_3;
reconsider o9 = o as Ordinal by A46;
 a = ConsecutiveSet (A,o9) by A34, A46, A47;
hence  Z in  {  [:(ConsecutiveSet (A,O2)),(ConsecutiveSet (A,O2)):] where O2 is Element of o1 : verum  }  by A34, A45, A46; ::_thesis: verum
end;
 for x, y being set st x in RTs & y in RTs holds
x,y are_c=-comparable
proof
let x, y be set ; ::_thesis: ( x in RTs & y in RTs implies x,y are_c=-comparable )
assume that
A48: x in RTs and
A49: y in RTs ; ::_thesis: x,y are_c=-comparable
consider o1 being set  such that
A50: o1 in dom Ts and
A51: Ts . o1 = x by A48, FUNCT_1:def_3;
consider o2 being set  such that
A52: o2 in dom Ts and
A53: Ts . o2 = y by A49, FUNCT_1:def_3;
reconsider o19 = o1, o29 = o2 as Ordinal by A50, A52;
A54: Ts . o29 = ConsecutiveSet (A,o29) by A34, A52;
A55: ( o19 c= o29 or o29 c= o19 ) ;
 Ts . o19 = ConsecutiveSet (A,o19) by A34, A50;
then  ( x c= y or y c= x ) by A51, A53, A54, A55, Th29;
hence  x,y are_c=-comparable by XBOOLE_0:def_9; ::_thesis: verum
end;
then A56: RTs is c=-linear by ORDINAL1:def_8;
A57: ConsecutiveDelta (q,O1) = union (rng Ls) by A4, A5, A7, Th28;
[:(ConsecutiveSet (A,O1)),(ConsecutiveSet (A,O1)):] = [:(union (rng Ts)),(ConsecutiveSet (A,O1)):] by A4, A5, A34, Th23
.= [:(union RTs),(union RTs):] by A4, A5, A34, Th23
.= dom f by A42, A36, A56, A43, Th3 ;
hence  S1[O1] by A57, A33, FUNCT_2:def_1, RELSET_1:4; ::_thesis: verum
end;
 ConsecutiveSet (A,{}) = A by Th21;
then A58: S1[ {} ] by Th26;
 for O being Ordinal holds S1[O] from ORDINAL2:sch_1(A58, A1, A3);
hence  ConsecutiveDelta (q,O) is BiFunction of (ConsecutiveSet (A,O)),L ; ::_thesis: verum
end;

definition
let A be non empty set ;
let L be lower-bounded LATTICE;
let d be BiFunction of A,L;
let q be QuadrSeq of d;
let O be Ordinal;
:: original: ConsecutiveDelta
redefine func ConsecutiveDelta (q,O) ->  BiFunction of (ConsecutiveSet (A,O)),L;
coherence
 ConsecutiveDelta (q,O) is BiFunction of (ConsecutiveSet (A,O)),L by Th30;
end;

theorem Th31: :: LATTICE5:31
for A being non empty set
for L being lower-bounded LATTICE
for O being Ordinal
for d being BiFunction of A,L
for q being QuadrSeq of d holds d c= ConsecutiveDelta (q,O)
proof
let A be non empty set ; ::_thesis: for L being lower-bounded LATTICE
for O being Ordinal
for d being BiFunction of A,L
for q being QuadrSeq of d holds d c= ConsecutiveDelta (q,O)
let L be lower-bounded LATTICE; ::_thesis: for O being Ordinal
for d being BiFunction of A,L
for q being QuadrSeq of d holds d c= ConsecutiveDelta (q,O)
let O be Ordinal; ::_thesis: for d being BiFunction of A,L
for q being QuadrSeq of d holds d c= ConsecutiveDelta (q,O)
let d be BiFunction of A,L; ::_thesis: for q being QuadrSeq of d holds d c= ConsecutiveDelta (q,O)
let q be QuadrSeq of d; ::_thesis: d c= ConsecutiveDelta (q,O)
defpred S1[ Ordinal] means d c= ConsecutiveDelta (q,$1);
A1: for O2 being Ordinal st O2 <> {} & O2 is limit_ordinal & ( for O1 being Ordinal st O1 in O2 holds
S1[O1] ) holds
S1[O2]
proof
deffunc H1( Ordinal) ->  BiFunction of (ConsecutiveSet (A,$1)),L = ConsecutiveDelta (q,$1);
let O2 be Ordinal; ::_thesis: ( O2 <> {} & O2 is limit_ordinal & ( for O1 being Ordinal st O1 in O2 holds
S1[O1] ) implies S1[O2] )
assume that
A2: O2 <> {} and
A3: O2 is limit_ordinal and
 for O1 being Ordinal st O1 in O2 holds
d c= ConsecutiveDelta (q,O1) ; ::_thesis: S1[O2]
A4: {} in O2 by A2, ORDINAL3:8;
consider Ls being T-Sequence such that
A5: ( dom Ls = O2 & ( for O1 being Ordinal st O1 in O2 holds
Ls . O1 = H1(O1) ) ) from ORDINAL2:sch_2();
Ls . {} = ConsecutiveDelta (q,{}) by A2, A5, ORDINAL3:8
.= d by Th26 ;
then A6: d in rng Ls by A5, A4, FUNCT_1:def_3;
 ConsecutiveDelta (q,O2) = union (rng Ls) by A2, A3, A5, Th28;
hence  S1[O2] by A6, ZFMISC_1:74; ::_thesis: verum
end;
A7: for O1 being Ordinal st S1[O1] holds
S1[ succ O1]
proof
let O1 be Ordinal; ::_thesis: ( S1[O1] implies S1[ succ O1] )
ConsecutiveDelta (q,(succ O1)) = new_bi_fun ((BiFun ((ConsecutiveDelta (q,O1)),(ConsecutiveSet (A,O1)),L)),(Quadr (q,O1))) by Th27
.= new_bi_fun ((ConsecutiveDelta (q,O1)),(Quadr (q,O1))) by Def15 ;
then A8: ConsecutiveDelta (q,O1) c= ConsecutiveDelta (q,(succ O1)) by Th19;
assume  d c= ConsecutiveDelta (q,O1) ; ::_thesis: S1[ succ O1]
hence  S1[ succ O1] by A8, XBOOLE_1:1; ::_thesis: verum
end;
A9: S1[ {} ] by Th26;
 for O being Ordinal holds S1[O] from ORDINAL2:sch_1(A9, A7, A1);
hence  d c= ConsecutiveDelta (q,O) ; ::_thesis: verum
end;

theorem Th32: :: LATTICE5:32
for A being non empty set
for L being lower-bounded LATTICE
for O1, O2 being Ordinal
for d being BiFunction of A,L
for q being QuadrSeq of d st O1 c= O2 holds
ConsecutiveDelta (q,O1) c= ConsecutiveDelta (q,O2)
proof
let A be non empty set ; ::_thesis: for L being lower-bounded LATTICE
for O1, O2 being Ordinal
for d being BiFunction of A,L
for q being QuadrSeq of d st O1 c= O2 holds
ConsecutiveDelta (q,O1) c= ConsecutiveDelta (q,O2)
let L be lower-bounded LATTICE; ::_thesis: for O1, O2 being Ordinal
for d being BiFunction of A,L
for q being QuadrSeq of d st O1 c= O2 holds
ConsecutiveDelta (q,O1) c= ConsecutiveDelta (q,O2)
let O1, O2 be Ordinal; ::_thesis: for d being BiFunction of A,L
for q being QuadrSeq of d st O1 c= O2 holds
ConsecutiveDelta (q,O1) c= ConsecutiveDelta (q,O2)
let d be BiFunction of A,L; ::_thesis: for q being QuadrSeq of d st O1 c= O2 holds
ConsecutiveDelta (q,O1) c= ConsecutiveDelta (q,O2)
let q be QuadrSeq of d; ::_thesis: ( O1 c= O2 implies ConsecutiveDelta (q,O1) c= ConsecutiveDelta (q,O2) )
defpred S1[ Ordinal] means ( O1 c= $1 implies ConsecutiveDelta (q,O1) c= ConsecutiveDelta (q,$1) );
A1: for O2 being Ordinal st O2 <> {} & O2 is limit_ordinal & ( for O3 being Ordinal st O3 in O2 holds
S1[O3] ) holds
S1[O2]
proof
deffunc H1( Ordinal) ->  BiFunction of (ConsecutiveSet (A,$1)),L = ConsecutiveDelta (q,$1);
let O2 be Ordinal; ::_thesis: ( O2 <> {} & O2 is limit_ordinal & ( for O3 being Ordinal st O3 in O2 holds
S1[O3] ) implies S1[O2] )
assume that
A2: ( O2 <> {} & O2 is limit_ordinal ) and
 for O3 being Ordinal st O3 in O2 & O1 c= O3 holds
ConsecutiveDelta (q,O1) c= ConsecutiveDelta (q,O3) ; ::_thesis: S1[O2]
consider L being T-Sequence such that
A3: ( dom L = O2 & ( for O3 being Ordinal st O3 in O2 holds
L . O3 = H1(O3) ) ) from ORDINAL2:sch_2();
A4: ConsecutiveDelta (q,O2) = union (rng L) by A2, A3, Th28;
assume A5: O1 c= O2 ; ::_thesis: ConsecutiveDelta (q,O1) c= ConsecutiveDelta (q,O2)
percases ( O1 = O2 or O1 <> O2 ) ;
suppose O1 = O2 ; ::_thesis: ConsecutiveDelta (q,O1) c= ConsecutiveDelta (q,O2)
hence  ConsecutiveDelta (q,O1) c= ConsecutiveDelta (q,O2) ; ::_thesis: verum
end;
suppose O1 <> O2 ; ::_thesis: ConsecutiveDelta (q,O1) c= ConsecutiveDelta (q,O2)
then A6: O1 c< O2 by A5, XBOOLE_0:def_8;
then  O1 in O2 by ORDINAL1:11;
then A7: L . O1 in rng L by A3, FUNCT_1:def_3;
 L . O1 = ConsecutiveDelta (q,O1) by A3, A6, ORDINAL1:11;
hence  ConsecutiveDelta (q,O1) c= ConsecutiveDelta (q,O2) by A4, A7, ZFMISC_1:74; ::_thesis: verum
end;
end;
end;
A8: for O2 being Ordinal st S1[O2] holds
S1[ succ O2]
proof
let O2 be Ordinal; ::_thesis: ( S1[O2] implies S1[ succ O2] )
assume A9: ( O1 c= O2 implies ConsecutiveDelta (q,O1) c= ConsecutiveDelta (q,O2) ) ; ::_thesis: S1[ succ O2]
assume A10: O1 c= succ O2 ; ::_thesis: ConsecutiveDelta (q,O1) c= ConsecutiveDelta (q,(succ O2))
percases ( O1 = succ O2 or O1 <> succ O2 ) ;
suppose O1 = succ O2 ; ::_thesis: ConsecutiveDelta (q,O1) c= ConsecutiveDelta (q,(succ O2))
hence  ConsecutiveDelta (q,O1) c= ConsecutiveDelta (q,(succ O2)) ; ::_thesis: verum
end;
suppose O1 <> succ O2 ; ::_thesis: ConsecutiveDelta (q,O1) c= ConsecutiveDelta (q,(succ O2))
then  O1 c< succ O2 by A10, XBOOLE_0:def_8;
then A11: O1 in succ O2 by ORDINAL1:11;
ConsecutiveDelta (q,(succ O2)) = new_bi_fun ((BiFun ((ConsecutiveDelta (q,O2)),(ConsecutiveSet (A,O2)),L)),(Quadr (q,O2))) by Th27
.= new_bi_fun ((ConsecutiveDelta (q,O2)),(Quadr (q,O2))) by Def15 ;
then  ConsecutiveDelta (q,O2) c= ConsecutiveDelta (q,(succ O2)) by Th19;
hence  ConsecutiveDelta (q,O1) c= ConsecutiveDelta (q,(succ O2)) by A9, A11, ORDINAL1:22, XBOOLE_1:1; ::_thesis: verum
end;
end;
end;
A12: S1[ {} ] ;
 for O2 being Ordinal holds S1[O2] from ORDINAL2:sch_1(A12, A8, A1);
hence  ( O1 c= O2 implies ConsecutiveDelta (q,O1) c= ConsecutiveDelta (q,O2) ) ; ::_thesis: verum
end;

theorem Th33: :: LATTICE5:33
for A being non empty set
for L being lower-bounded LATTICE
for O being Ordinal
for d being BiFunction of A,L st d is zeroed holds
for q being QuadrSeq of d holds ConsecutiveDelta (q,O) is zeroed
proof
let A be non empty set ; ::_thesis: for L being lower-bounded LATTICE
for O being Ordinal
for d being BiFunction of A,L st d is zeroed holds
for q being QuadrSeq of d holds ConsecutiveDelta (q,O) is zeroed
let L be lower-bounded LATTICE; ::_thesis: for O being Ordinal
for d being BiFunction of A,L st d is zeroed holds
for q being QuadrSeq of d holds ConsecutiveDelta (q,O) is zeroed
let O be Ordinal; ::_thesis: for d being BiFunction of A,L st d is zeroed holds
for q being QuadrSeq of d holds ConsecutiveDelta (q,O) is zeroed
let d be BiFunction of A,L; ::_thesis: ( d is zeroed implies for q being QuadrSeq of d holds ConsecutiveDelta (q,O) is zeroed )
assume A1: d is zeroed ; ::_thesis: for q being QuadrSeq of d holds ConsecutiveDelta (q,O) is zeroed
let q be QuadrSeq of d; ::_thesis: ConsecutiveDelta (q,O) is zeroed
defpred S1[ Ordinal] means  ConsecutiveDelta (q,$1) is zeroed ;
A2: for O1 being Ordinal st S1[O1] holds
S1[ succ O1]
proof
let O1 be Ordinal; ::_thesis: ( S1[O1] implies S1[ succ O1] )
assume  ConsecutiveDelta (q,O1) is zeroed ; ::_thesis: S1[ succ O1]
then A3: new_bi_fun ((ConsecutiveDelta (q,O1)),(Quadr (q,O1))) is zeroed by Th16;
let z be Element of ConsecutiveSet (A,(succ O1)); :: according to LATTICE5:def_6 ::_thesis: (ConsecutiveDelta (q,(succ O1))) . (z,z) = Bottom L
reconsider z9 = z as Element of new_set (ConsecutiveSet (A,O1)) by Th22;
ConsecutiveDelta (q,(succ O1)) = new_bi_fun ((BiFun ((ConsecutiveDelta (q,O1)),(ConsecutiveSet (A,O1)),L)),(Quadr (q,O1))) by Th27
.= new_bi_fun ((ConsecutiveDelta (q,O1)),(Quadr (q,O1))) by Def15 ;
hence (ConsecutiveDelta (q,(succ O1))) . (z,z) = (new_bi_fun ((ConsecutiveDelta (q,O1)),(Quadr (q,O1)))) . (z9,z9)
.= Bottom L by A3, Def6 ;
::_thesis: verum
end;
A4: for O2 being Ordinal st O2 <> {} & O2 is limit_ordinal & ( for O1 being Ordinal st O1 in O2 holds
S1[O1] ) holds
S1[O2]
proof
deffunc H1( Ordinal) ->  BiFunction of (ConsecutiveSet (A,$1)),L = ConsecutiveDelta (q,$1);
let O2 be Ordinal; ::_thesis: ( O2 <> {} & O2 is limit_ordinal & ( for O1 being Ordinal st O1 in O2 holds
S1[O1] ) implies S1[O2] )
assume that
A5: ( O2 <> {} & O2 is limit_ordinal ) and
A6: for O1 being Ordinal st O1 in O2 holds
ConsecutiveDelta (q,O1) is zeroed ; ::_thesis: S1[O2]
set CS = ConsecutiveSet (A,O2);
consider Ls being T-Sequence such that
A7: ( dom Ls = O2 & ( for O1 being Ordinal st O1 in O2 holds
Ls . O1 = H1(O1) ) ) from ORDINAL2:sch_2();
 ConsecutiveDelta (q,O2) = union (rng Ls) by A5, A7, Th28;
then reconsider f = union (rng Ls) as BiFunction of (ConsecutiveSet (A,O2)),L ;
deffunc H2( Ordinal) ->  set = ConsecutiveSet (A,$1);
consider Ts being T-Sequence such that
A8: ( dom Ts = O2 & ( for O1 being Ordinal st O1 in O2 holds
Ts . O1 = H2(O1) ) ) from ORDINAL2:sch_2();
A9: ConsecutiveSet (A,O2) = union (rng Ts) by A5, A8, Th23;
 f is zeroed
proof
let x be Element of ConsecutiveSet (A,O2); :: according to LATTICE5:def_6 ::_thesis: f . (x,x) = Bottom L
consider y being set  such that
A10: x in y and
A11: y in rng Ts by A9, TARSKI:def_4;
consider o being set  such that
A12: o in dom Ts and
A13: y = Ts . o by A11, FUNCT_1:def_3;
reconsider o = o as Ordinal by A12;
A14: Ls . o = ConsecutiveDelta (q,o) by A7, A8, A12;
then reconsider h = Ls . o as BiFunction of (ConsecutiveSet (A,o)),L ;
reconsider x9 = x as Element of ConsecutiveSet (A,o) by A8, A10, A12, A13;
A15: dom h = [:(ConsecutiveSet (A,o)),(ConsecutiveSet (A,o)):] by FUNCT_2:def_1;
A16: h is zeroed
proof
let z be Element of ConsecutiveSet (A,o); :: according to LATTICE5:def_6 ::_thesis: h . (z,z) = Bottom L
A17: ConsecutiveDelta (q,o) is zeroed by A6, A8, A12;
thus h . (z,z) = (ConsecutiveDelta (q,o)) . (z,z) by A7, A8, A12
.= Bottom L by A17, Def6 ; ::_thesis: verum
end;
 ConsecutiveDelta (q,o) in rng Ls by A7, A8, A12, A14, FUNCT_1:def_3;
then A18: h c= f by A14, ZFMISC_1:74;
 x in ConsecutiveSet (A,o) by A8, A10, A12, A13;
then  [x,x] in dom h by A15, ZFMISC_1:87;
hence f . (x,x) = h . (x9,x9) by A18, GRFUNC_1:2
.= Bottom L by A16, Def6 ;
::_thesis: verum
end;
hence  S1[O2] by A5, A7, Th28; ::_thesis: verum
end;
A19: S1[ {} ]
proof
let z be Element of ConsecutiveSet (A,{}); :: according to LATTICE5:def_6 ::_thesis: (ConsecutiveDelta (q,{})) . (z,z) = Bottom L
reconsider z9 = z as Element of A by Th21;
thus (ConsecutiveDelta (q,{})) . (z,z) = d . (z9,z9) by Th26
.= Bottom L by A1, Def6 ; ::_thesis: verum
end;
 for O being Ordinal holds S1[O] from ORDINAL2:sch_1(A19, A2, A4);
hence  ConsecutiveDelta (q,O) is zeroed ; ::_thesis: verum
end;

theorem Th34: :: LATTICE5:34
for A being non empty set
for L being lower-bounded LATTICE
for O being Ordinal
for d being BiFunction of A,L st d is symmetric holds
for q being QuadrSeq of d holds ConsecutiveDelta (q,O) is symmetric
proof
let A be non empty set ; ::_thesis: for L being lower-bounded LATTICE
for O being Ordinal
for d being BiFunction of A,L st d is symmetric holds
for q being QuadrSeq of d holds ConsecutiveDelta (q,O) is symmetric
let L be lower-bounded LATTICE; ::_thesis: for O being Ordinal
for d being BiFunction of A,L st d is symmetric holds
for q being QuadrSeq of d holds ConsecutiveDelta (q,O) is symmetric
let O be Ordinal; ::_thesis: for d being BiFunction of A,L st d is symmetric holds
for q being QuadrSeq of d holds ConsecutiveDelta (q,O) is symmetric
let d be BiFunction of A,L; ::_thesis: ( d is symmetric implies for q being QuadrSeq of d holds ConsecutiveDelta (q,O) is symmetric )
assume A1: d is symmetric ; ::_thesis: for q being QuadrSeq of d holds ConsecutiveDelta (q,O) is symmetric
let q be QuadrSeq of d; ::_thesis: ConsecutiveDelta (q,O) is symmetric
defpred S1[ Ordinal] means  ConsecutiveDelta (q,$1) is symmetric ;
A2: for O1 being Ordinal st S1[O1] holds
S1[ succ O1]
proof
let O1 be Ordinal; ::_thesis: ( S1[O1] implies S1[ succ O1] )
assume  ConsecutiveDelta (q,O1) is symmetric ; ::_thesis: S1[ succ O1]
then A3: new_bi_fun ((ConsecutiveDelta (q,O1)),(Quadr (q,O1))) is symmetric by Th17;
let x, y be Element of ConsecutiveSet (A,(succ O1)); :: according to LATTICE5:def_5 ::_thesis: (ConsecutiveDelta (q,(succ O1))) . (x,y) = (ConsecutiveDelta (q,(succ O1))) . (y,x)
reconsider x9 = x, y9 = y as Element of new_set (ConsecutiveSet (A,O1)) by Th22;
A4: ConsecutiveDelta (q,(succ O1)) = new_bi_fun ((BiFun ((ConsecutiveDelta (q,O1)),(ConsecutiveSet (A,O1)),L)),(Quadr (q,O1))) by Th27
.= new_bi_fun ((ConsecutiveDelta (q,O1)),(Quadr (q,O1))) by Def15 ;
hence (ConsecutiveDelta (q,(succ O1))) . (x,y) = (new_bi_fun ((ConsecutiveDelta (q,O1)),(Quadr (q,O1)))) . (y9,x9) by A3, Def5
.= (ConsecutiveDelta (q,(succ O1))) . (y,x) by A4 ;
::_thesis: verum
end;
A5: for O2 being Ordinal st O2 <> {} & O2 is limit_ordinal & ( for O1 being Ordinal st O1 in O2 holds
S1[O1] ) holds
S1[O2]
proof
deffunc H1( Ordinal) ->  BiFunction of (ConsecutiveSet (A,$1)),L = ConsecutiveDelta (q,$1);
let O2 be Ordinal; ::_thesis: ( O2 <> {} & O2 is limit_ordinal & ( for O1 being Ordinal st O1 in O2 holds
S1[O1] ) implies S1[O2] )
assume that
A6: ( O2 <> {} & O2 is limit_ordinal ) and
A7: for O1 being Ordinal st O1 in O2 holds
ConsecutiveDelta (q,O1) is symmetric ; ::_thesis: S1[O2]
set CS = ConsecutiveSet (A,O2);
consider Ls being T-Sequence such that
A8: ( dom Ls = O2 & ( for O1 being Ordinal st O1 in O2 holds
Ls . O1 = H1(O1) ) ) from ORDINAL2:sch_2();
 ConsecutiveDelta (q,O2) = union (rng Ls) by A6, A8, Th28;
then reconsider f = union (rng Ls) as BiFunction of (ConsecutiveSet (A,O2)),L ;
deffunc H2( Ordinal) ->  set = ConsecutiveSet (A,$1);
consider Ts being T-Sequence such that
A9: ( dom Ts = O2 & ( for O1 being Ordinal st O1 in O2 holds
Ts . O1 = H2(O1) ) ) from ORDINAL2:sch_2();
A10: ConsecutiveSet (A,O2) = union (rng Ts) by A6, A9, Th23;
 f is symmetric
proof
let x, y be Element of ConsecutiveSet (A,O2); :: according to LATTICE5:def_5 ::_thesis: f . (x,y) = f . (y,x)
consider x1 being set  such that
A11: x in x1 and
A12: x1 in rng Ts by A10, TARSKI:def_4;
consider o1 being set  such that
A13: o1 in dom Ts and
A14: x1 = Ts . o1 by A12, FUNCT_1:def_3;
consider y1 being set  such that
A15: y in y1 and
A16: y1 in rng Ts by A10, TARSKI:def_4;
consider o2 being set  such that
A17: o2 in dom Ts and
A18: y1 = Ts . o2 by A16, FUNCT_1:def_3;
reconsider o1 = o1, o2 = o2 as Ordinal by A13, A17;
A19: x in ConsecutiveSet (A,o1) by A9, A11, A13, A14;
A20: Ls . o1 = ConsecutiveDelta (q,o1) by A8, A9, A13;
then reconsider h1 = Ls . o1 as BiFunction of (ConsecutiveSet (A,o1)),L ;
A21: h1 is symmetric
proof
let x, y be Element of ConsecutiveSet (A,o1); :: according to LATTICE5:def_5 ::_thesis: h1 . (x,y) = h1 . (y,x)
A22: ConsecutiveDelta (q,o1) is symmetric by A7, A9, A13;
thus h1 . (x,y) = (ConsecutiveDelta (q,o1)) . (x,y) by A8, A9, A13
.= (ConsecutiveDelta (q,o1)) . (y,x) by A22, Def5
.= h1 . (y,x) by A8, A9, A13 ; ::_thesis: verum
end;
A23: dom h1 = [:(ConsecutiveSet (A,o1)),(ConsecutiveSet (A,o1)):] by FUNCT_2:def_1;
A24: y in ConsecutiveSet (A,o2) by A9, A15, A17, A18;
A25: Ls . o2 = ConsecutiveDelta (q,o2) by A8, A9, A17;
then reconsider h2 = Ls . o2 as BiFunction of (ConsecutiveSet (A,o2)),L ;
A26: h2 is symmetric
proof
let x, y be Element of ConsecutiveSet (A,o2); :: according to LATTICE5:def_5 ::_thesis: h2 . (x,y) = h2 . (y,x)
A27: ConsecutiveDelta (q,o2) is symmetric by A7, A9, A17;
thus h2 . (x,y) = (ConsecutiveDelta (q,o2)) . (x,y) by A8, A9, A17
.= (ConsecutiveDelta (q,o2)) . (y,x) by A27, Def5
.= h2 . (y,x) by A8, A9, A17 ; ::_thesis: verum
end;
A28: dom h2 = [:(ConsecutiveSet (A,o2)),(ConsecutiveSet (A,o2)):] by FUNCT_2:def_1;
percases ( o1 c= o2 or o2 c= o1 ) ;
suppose o1 c= o2 ; ::_thesis: f . (x,y) = f . (y,x)
then A29: ConsecutiveSet (A,o1) c= ConsecutiveSet (A,o2) by Th29;
then A30: [y,x] in dom h2 by A19, A24, A28, ZFMISC_1:87;
 ConsecutiveDelta (q,o2) in rng Ls by A8, A9, A17, A25, FUNCT_1:def_3;
then A31: h2 c= f by A25, ZFMISC_1:74;
reconsider x9 = x, y9 = y as Element of ConsecutiveSet (A,o2) by A9, A15, A17, A18, A19, A29;
 [x,y] in dom h2 by A19, A24, A28, A29, ZFMISC_1:87;
hence f . (x,y) = h2 . (x9,y9) by A31, GRFUNC_1:2
.= h2 . (y9,x9) by A26, Def5
.= f . (y,x) by A31, A30, GRFUNC_1:2 ;
::_thesis: verum
end;
suppose o2 c= o1 ; ::_thesis: f . (x,y) = f . (y,x)
then A32: ConsecutiveSet (A,o2) c= ConsecutiveSet (A,o1) by Th29;
then A33: [y,x] in dom h1 by A19, A24, A23, ZFMISC_1:87;
 ConsecutiveDelta (q,o1) in rng Ls by A8, A9, A13, A20, FUNCT_1:def_3;
then A34: h1 c= f by A20, ZFMISC_1:74;
reconsider x9 = x, y9 = y as Element of ConsecutiveSet (A,o1) by A9, A11, A13, A14, A24, A32;
 [x,y] in dom h1 by A19, A24, A23, A32, ZFMISC_1:87;
hence f . (x,y) = h1 . (x9,y9) by A34, GRFUNC_1:2
.= h1 . (y9,x9) by A21, Def5
.= f . (y,x) by A34, A33, GRFUNC_1:2 ;
::_thesis: verum
end;
end;
end;
hence  S1[O2] by A6, A8, Th28; ::_thesis: verum
end;
A35: S1[ {} ]
proof
let x, y be Element of ConsecutiveSet (A,{}); :: according to LATTICE5:def_5 ::_thesis: (ConsecutiveDelta (q,{})) . (x,y) = (ConsecutiveDelta (q,{})) . (y,x)
reconsider x9 = x, y9 = y as Element of A by Th21;
thus (ConsecutiveDelta (q,{})) . (x,y) = d . (x9,y9) by Th26
.= d . (y9,x9) by A1, Def5
.= (ConsecutiveDelta (q,{})) . (y,x) by Th26 ; ::_thesis: verum
end;
 for O being Ordinal holds S1[O] from ORDINAL2:sch_1(A35, A2, A5);
hence  ConsecutiveDelta (q,O) is symmetric ; ::_thesis: verum
end;

theorem Th35: :: LATTICE5:35
for A being non empty set
for L being lower-bounded LATTICE
for O being Ordinal
for d being BiFunction of A,L st d is symmetric & d is u.t.i. holds
for q being QuadrSeq of d st O c= DistEsti d holds
ConsecutiveDelta (q,O) is u.t.i.
proof
let A be non empty set ; ::_thesis: for L being lower-bounded LATTICE
for O being Ordinal
for d being BiFunction of A,L st d is symmetric & d is u.t.i. holds
for q being QuadrSeq of d st O c= DistEsti d holds
ConsecutiveDelta (q,O) is u.t.i.
let L be lower-bounded LATTICE; ::_thesis: for O being Ordinal
for d being BiFunction of A,L st d is symmetric & d is u.t.i. holds
for q being QuadrSeq of d st O c= DistEsti d holds
ConsecutiveDelta (q,O) is u.t.i.
let O be Ordinal; ::_thesis: for d being BiFunction of A,L st d is symmetric & d is u.t.i. holds
for q being QuadrSeq of d st O c= DistEsti d holds
ConsecutiveDelta (q,O) is u.t.i.
let d be BiFunction of A,L; ::_thesis: ( d is symmetric & d is u.t.i. implies for q being QuadrSeq of d st O c= DistEsti d holds
ConsecutiveDelta (q,O) is u.t.i. )
assume that
A1: d is symmetric and
A2: d is u.t.i. ; ::_thesis: for q being QuadrSeq of d st O c= DistEsti d holds
ConsecutiveDelta (q,O) is u.t.i.
let q be QuadrSeq of d; ::_thesis: ( O c= DistEsti d implies ConsecutiveDelta (q,O) is u.t.i. )
defpred S1[ Ordinal] means ( $1 c= DistEsti d implies ConsecutiveDelta (q,$1) is u.t.i. );
A3: for O1 being Ordinal st S1[O1] holds
S1[ succ O1]
proof
let O1 be Ordinal; ::_thesis: ( S1[O1] implies S1[ succ O1] )
assume that
A4: ( O1 c= DistEsti d implies ConsecutiveDelta (q,O1) is u.t.i. ) and
A5: succ O1 c= DistEsti d ; ::_thesis: ConsecutiveDelta (q,(succ O1)) is u.t.i.
A6: O1 in DistEsti d by A5, ORDINAL1:21;
then A7: O1 in dom q by Th25;
then  q . O1 in rng q by FUNCT_1:def_3;
then A8: q . O1 in  {  [u,v,a9,b9] where u, v is Element of A, a9, b9 is Element of L : d . (u,v) <= a9 "\/" b9  }  by Def13;
let x, y, z be Element of ConsecutiveSet (A,(succ O1)); :: according to LATTICE5:def_7 ::_thesis: ((ConsecutiveDelta (q,(succ O1))) . (x,y)) "\/" ((ConsecutiveDelta (q,(succ O1))) . (y,z)) >= (ConsecutiveDelta (q,(succ O1))) . (x,z)
A9: ConsecutiveDelta (q,O1) is symmetric by A1, Th34;
reconsider x9 = x, y9 = y, z9 = z as Element of new_set (ConsecutiveSet (A,O1)) by Th22;
set f = new_bi_fun ((ConsecutiveDelta (q,O1)),(Quadr (q,O1)));
set X = (Quadr (q,O1)) `1_4 ;
set Y = (Quadr (q,O1)) `2_4 ;
reconsider a = (Quadr (q,O1)) `3_4 , b = (Quadr (q,O1)) `4_4 as Element of L ;
A10: ( dom d = [:A,A:] & d c= ConsecutiveDelta (q,O1) ) by Th31, FUNCT_2:def_1;
consider u, v being Element of A, a9, b9 being Element of L such that
A11: q . O1 = [u,v,a9,b9] and
A12: d . (u,v) <= a9 "\/" b9 by A8;
A13: Quadr (q,O1) = [u,v,a9,b9] by A7, A11, Def14;
then A14: ( u = (Quadr (q,O1)) `1_4 & v = (Quadr (q,O1)) `2_4 ) by MCART_1:def_8, MCART_1:def_9;
A15: ( a9 = a & b9 = b ) by A13, MCART_1:def_10, MCART_1:def_11;
d . (u,v) = d . [u,v]
.= (ConsecutiveDelta (q,O1)) . (((Quadr (q,O1)) `1_4),((Quadr (q,O1)) `2_4)) by A14, A10, GRFUNC_1:2 ;
then  new_bi_fun ((ConsecutiveDelta (q,O1)),(Quadr (q,O1))) is u.t.i. by A4, A6, A9, A12, A15, Th18, ORDINAL1:def_2;
then A16: (new_bi_fun ((ConsecutiveDelta (q,O1)),(Quadr (q,O1)))) . (x9,z9) <= ((new_bi_fun ((ConsecutiveDelta (q,O1)),(Quadr (q,O1)))) . (x9,y9)) "\/" ((new_bi_fun ((ConsecutiveDelta (q,O1)),(Quadr (q,O1)))) . (y9,z9)) by Def7;
ConsecutiveDelta (q,(succ O1)) = new_bi_fun ((BiFun ((ConsecutiveDelta (q,O1)),(ConsecutiveSet (A,O1)),L)),(Quadr (q,O1))) by Th27
.= new_bi_fun ((ConsecutiveDelta (q,O1)),(Quadr (q,O1))) by Def15 ;
hence  (ConsecutiveDelta (q,(succ O1))) . (x,z) <= ((ConsecutiveDelta (q,(succ O1))) . (x,y)) "\/" ((ConsecutiveDelta (q,(succ O1))) . (y,z)) by A16; ::_thesis: verum
end;
A17: for O2 being Ordinal st O2 <> {} & O2 is limit_ordinal & ( for O1 being Ordinal st O1 in O2 holds
S1[O1] ) holds
S1[O2]
proof
deffunc H1( Ordinal) ->  BiFunction of (ConsecutiveSet (A,$1)),L = ConsecutiveDelta (q,$1);
let O2 be Ordinal; ::_thesis: ( O2 <> {} & O2 is limit_ordinal & ( for O1 being Ordinal st O1 in O2 holds
S1[O1] ) implies S1[O2] )
assume that
A18: ( O2 <> {} & O2 is limit_ordinal ) and
A19: for O1 being Ordinal st O1 in O2 & O1 c= DistEsti d holds
ConsecutiveDelta (q,O1) is u.t.i. and
A20: O2 c= DistEsti d ; ::_thesis: ConsecutiveDelta (q,O2) is u.t.i.
set CS = ConsecutiveSet (A,O2);
consider Ls being T-Sequence such that
A21: ( dom Ls = O2 & ( for O1 being Ordinal st O1 in O2 holds
Ls . O1 = H1(O1) ) ) from ORDINAL2:sch_2();
 ConsecutiveDelta (q,O2) = union (rng Ls) by A18, A21, Th28;
then reconsider f = union (rng Ls) as BiFunction of (ConsecutiveSet (A,O2)),L ;
deffunc H2( Ordinal) ->  set = ConsecutiveSet (A,$1);
consider Ts being T-Sequence such that
A22: ( dom Ts = O2 & ( for O1 being Ordinal st O1 in O2 holds
Ts . O1 = H2(O1) ) ) from ORDINAL2:sch_2();
A23: ConsecutiveSet (A,O2) = union (rng Ts) by A18, A22, Th23;
 f is u.t.i.
proof
let x, y, z be Element of ConsecutiveSet (A,O2); :: according to LATTICE5:def_7 ::_thesis: (f . (x,y)) "\/" (f . (y,z)) >= f . (x,z)
consider X being set  such that
A24: x in X and
A25: X in rng Ts by A23, TARSKI:def_4;
consider o1 being set  such that
A26: o1 in dom Ts and
A27: X = Ts . o1 by A25, FUNCT_1:def_3;
consider Y being set  such that
A28: y in Y and
A29: Y in rng Ts by A23, TARSKI:def_4;
consider o2 being set  such that
A30: o2 in dom Ts and
A31: Y = Ts . o2 by A29, FUNCT_1:def_3;
consider Z being set  such that
A32: z in Z and
A33: Z in rng Ts by A23, TARSKI:def_4;
consider o3 being set  such that
A34: o3 in dom Ts and
A35: Z = Ts . o3 by A33, FUNCT_1:def_3;
reconsider o1 = o1, o2 = o2, o3 = o3 as Ordinal by A26, A30, A34;
A36: x in ConsecutiveSet (A,o1) by A22, A24, A26, A27;
A37: Ls . o3 = ConsecutiveDelta (q,o3) by A21, A22, A34;
then reconsider h3 = Ls . o3 as BiFunction of (ConsecutiveSet (A,o3)),L ;
A38: h3 is u.t.i.
proof
let x, y, z be Element of ConsecutiveSet (A,o3); :: according to LATTICE5:def_7 ::_thesis: (h3 . (x,y)) "\/" (h3 . (y,z)) >= h3 . (x,z)
 o3 c= DistEsti d by A20, A22, A34, ORDINAL1:def_2;
then A39: ConsecutiveDelta (q,o3) is u.t.i. by A19, A22, A34;
 ConsecutiveDelta (q,o3) = h3 by A21, A22, A34;
hence  h3 . (x,z) <= (h3 . (x,y)) "\/" (h3 . (y,z)) by A39, Def7; ::_thesis: verum
end;
A40: dom h3 = [:(ConsecutiveSet (A,o3)),(ConsecutiveSet (A,o3)):] by FUNCT_2:def_1;
A41: z in ConsecutiveSet (A,o3) by A22, A32, A34, A35;
A42: Ls . o2 = ConsecutiveDelta (q,o2) by A21, A22, A30;
then reconsider h2 = Ls . o2 as BiFunction of (ConsecutiveSet (A,o2)),L ;
A43: h2 is u.t.i.
proof
let x, y, z be Element of ConsecutiveSet (A,o2); :: according to LATTICE5:def_7 ::_thesis: (h2 . (x,y)) "\/" (h2 . (y,z)) >= h2 . (x,z)
 o2 c= DistEsti d by A20, A22, A30, ORDINAL1:def_2;
then A44: ConsecutiveDelta (q,o2) is u.t.i. by A19, A22, A30;
 ConsecutiveDelta (q,o2) = h2 by A21, A22, A30;
hence  h2 . (x,z) <= (h2 . (x,y)) "\/" (h2 . (y,z)) by A44, Def7; ::_thesis: verum
end;
A45: dom h2 = [:(ConsecutiveSet (A,o2)),(ConsecutiveSet (A,o2)):] by FUNCT_2:def_1;
A46: Ls . o1 = ConsecutiveDelta (q,o1) by A21, A22, A26;
then reconsider h1 = Ls . o1 as BiFunction of (ConsecutiveSet (A,o1)),L ;
A47: h1 is u.t.i.
proof
let x, y, z be Element of ConsecutiveSet (A,o1); :: according to LATTICE5:def_7 ::_thesis: (h1 . (x,y)) "\/" (h1 . (y,z)) >= h1 . (x,z)
 o1 c= DistEsti d by A20, A22, A26, ORDINAL1:def_2;
then A48: ConsecutiveDelta (q,o1) is u.t.i. by A19, A22, A26;
 ConsecutiveDelta (q,o1) = h1 by A21, A22, A26;
hence  h1 . (x,z) <= (h1 . (x,y)) "\/" (h1 . (y,z)) by A48, Def7; ::_thesis: verum
end;
A49: dom h1 = [:(ConsecutiveSet (A,o1)),(ConsecutiveSet (A,o1)):] by FUNCT_2:def_1;
A50: y in ConsecutiveSet (A,o2) by A22, A28, A30, A31;
percases ( o1 c= o3 or o3 c= o1 ) ;
supposeA51: o1 c= o3 ; ::_thesis: (f . (x,y)) "\/" (f . (y,z)) >= f . (x,z)
then A52: ConsecutiveSet (A,o1) c= ConsecutiveSet (A,o3) by Th29;
thus  (f . (x,y)) "\/" (f . (y,z)) >= f . (x,z) ::_thesis: verum
proof
percases ( o2 c= o3 or o3 c= o2 ) ;
supposeA53: o2 c= o3 ; ::_thesis: (f . (x,y)) "\/" (f . (y,z)) >= f . (x,z)
reconsider z9 = z as Element of ConsecutiveSet (A,o3) by A22, A32, A34, A35;
reconsider x9 = x as Element of ConsecutiveSet (A,o3) by A36, A52;
 ConsecutiveDelta (q,o3) in rng Ls by A21, A22, A34, A37, FUNCT_1:def_3;
then A54: h3 c= f by A37, ZFMISC_1:74;
A55: ConsecutiveSet (A,o2) c= ConsecutiveSet (A,o3) by A53, Th29;
then reconsider y9 = y as Element of ConsecutiveSet (A,o3) by A50;
 [y,z] in dom h3 by A50, A41, A40, A55, ZFMISC_1:87;
then A56: f . (y,z) = h3 . (y9,z9) by A54, GRFUNC_1:2;
 [x,z] in dom h3 by A36, A41, A40, A52, ZFMISC_1:87;
then A57: f . (x,z) = h3 . (x9,z9) by A54, GRFUNC_1:2;
 [x,y] in dom h3 by A36, A50, A40, A52, A55, ZFMISC_1:87;
then  f . (x,y) = h3 . (x9,y9) by A54, GRFUNC_1:2;
hence  (f . (x,y)) "\/" (f . (y,z)) >= f . (x,z) by A38, A56, A57, Def7; ::_thesis: verum
end;
supposeA58: o3 c= o2 ; ::_thesis: (f . (x,y)) "\/" (f . (y,z)) >= f . (x,z)
reconsider y9 = y as Element of ConsecutiveSet (A,o2) by A22, A28, A30, A31;
 ConsecutiveDelta (q,o2) in rng Ls by A21, A22, A30, A42, FUNCT_1:def_3;
then A59: h2 c= f by A42, ZFMISC_1:74;
A60: ConsecutiveSet (A,o3) c= ConsecutiveSet (A,o2) by A58, Th29;
then reconsider z9 = z as Element of ConsecutiveSet (A,o2) by A41;
 [y,z] in dom h2 by A50, A41, A45, A60, ZFMISC_1:87;
then A61: f . (y,z) = h2 . (y9,z9) by A59, GRFUNC_1:2;
 ConsecutiveSet (A,o1) c= ConsecutiveSet (A,o3) by A51, Th29;
then A62: ConsecutiveSet (A,o1) c= ConsecutiveSet (A,o2) by A60, XBOOLE_1:1;
then reconsider x9 = x as Element of ConsecutiveSet (A,o2) by A36;
 [x,y] in dom h2 by A36, A50, A45, A62, ZFMISC_1:87;
then A63: f . (x,y) = h2 . (x9,y9) by A59, GRFUNC_1:2;
 [x,z] in dom h2 by A36, A41, A45, A60, A62, ZFMISC_1:87;
then  f . (x,z) = h2 . (x9,z9) by A59, GRFUNC_1:2;
hence  (f . (x,y)) "\/" (f . (y,z)) >= f . (x,z) by A43, A63, A61, Def7; ::_thesis: verum
end;
end;
end;
end;
supposeA64: o3 c= o1 ; ::_thesis: (f . (x,y)) "\/" (f . (y,z)) >= f . (x,z)
then A65: ConsecutiveSet (A,o3) c= ConsecutiveSet (A,o1) by Th29;
thus  (f . (x,y)) "\/" (f . (y,z)) >= f . (x,z) ::_thesis: verum
proof
percases ( o1 c= o2 or o2 c= o1 ) ;
supposeA66: o1 c= o2 ; ::_thesis: (f . (x,y)) "\/" (f . (y,z)) >= f . (x,z)
reconsider y9 = y as Element of ConsecutiveSet (A,o2) by A22, A28, A30, A31;
 ConsecutiveDelta (q,o2) in rng Ls by A21, A22, A30, A42, FUNCT_1:def_3;
then A67: h2 c= f by A42, ZFMISC_1:74;
A68: ConsecutiveSet (A,o1) c= ConsecutiveSet (A,o2) by A66, Th29;
then reconsider x9 = x as Element of ConsecutiveSet (A,o2) by A36;
 [x,y] in dom h2 by A36, A50, A45, A68, ZFMISC_1:87;
then A69: f . (x,y) = h2 . (x9,y9) by A67, GRFUNC_1:2;
 ConsecutiveSet (A,o3) c= ConsecutiveSet (A,o1) by A64, Th29;
then A70: ConsecutiveSet (A,o3) c= ConsecutiveSet (A,o2) by A68, XBOOLE_1:1;
then reconsider z9 = z as Element of ConsecutiveSet (A,o2) by A41;
 [y,z] in dom h2 by A50, A41, A45, A70, ZFMISC_1:87;
then A71: f . (y,z) = h2 . (y9,z9) by A67, GRFUNC_1:2;
 [x,z] in dom h2 by A36, A41, A45, A68, A70, ZFMISC_1:87;
then  f . (x,z) = h2 . (x9,z9) by A67, GRFUNC_1:2;
hence  (f . (x,y)) "\/" (f . (y,z)) >= f . (x,z) by A43, A69, A71, Def7; ::_thesis: verum
end;
supposeA72: o2 c= o1 ; ::_thesis: (f . (x,y)) "\/" (f . (y,z)) >= f . (x,z)
reconsider x9 = x as Element of ConsecutiveSet (A,o1) by A22, A24, A26, A27;
reconsider z9 = z as Element of ConsecutiveSet (A,o1) by A41, A65;
 ConsecutiveDelta (q,o1) in rng Ls by A21, A22, A26, A46, FUNCT_1:def_3;
then A73: h1 c= f by A46, ZFMISC_1:74;
A74: ConsecutiveSet (A,o2) c= ConsecutiveSet (A,o1) by A72, Th29;
then reconsider y9 = y as Element of ConsecutiveSet (A,o1) by A50;
 [x,y] in dom h1 by A36, A50, A49, A74, ZFMISC_1:87;
then A75: f . (x,y) = h1 . (x9,y9) by A73, GRFUNC_1:2;
 [x,z] in dom h1 by A36, A41, A49, A65, ZFMISC_1:87;
then A76: f . (x,z) = h1 . (x9,z9) by A73, GRFUNC_1:2;
 [y,z] in dom h1 by A50, A41, A49, A65, A74, ZFMISC_1:87;
then  f . (y,z) = h1 . (y9,z9) by A73, GRFUNC_1:2;
hence  (f . (x,y)) "\/" (f . (y,z)) >= f . (x,z) by A47, A75, A76, Def7; ::_thesis: verum
end;
end;
end;
end;
end;
end;
hence  ConsecutiveDelta (q,O2) is u.t.i. by A18, A21, Th28; ::_thesis: verum
end;
A77: S1[ {} ]
proof
assume  {} c= DistEsti d ; ::_thesis: ConsecutiveDelta (q,{}) is u.t.i.
let x, y, z be Element of ConsecutiveSet (A,{}); :: according to LATTICE5:def_7 ::_thesis: ((ConsecutiveDelta (q,{})) . (x,y)) "\/" ((ConsecutiveDelta (q,{})) . (y,z)) >= (ConsecutiveDelta (q,{})) . (x,z)
reconsider x9 = x, y9 = y, z9 = z as Element of A by Th21;
 ( ConsecutiveDelta (q,{}) = d & d . (x9,z9) <= (d . (x9,y9)) "\/" (d . (y9,z9)) ) by A2, Def7, Th26;
hence  (ConsecutiveDelta (q,{})) . (x,z) <= ((ConsecutiveDelta (q,{})) . (x,y)) "\/" ((ConsecutiveDelta (q,{})) . (y,z)) ; ::_thesis: verum
end;
 for O being Ordinal holds S1[O] from ORDINAL2:sch_1(A77, A3, A17);
hence  ( O c= DistEsti d implies ConsecutiveDelta (q,O) is u.t.i. ) ; ::_thesis: verum
end;

theorem :: LATTICE5:36
for A being non empty set
for L being lower-bounded LATTICE
for O being Ordinal
for d being distance_function of A,L
for q being QuadrSeq of d st O c= DistEsti d holds
ConsecutiveDelta (q,O) is distance_function of (ConsecutiveSet (A,O)),L by Th33, Th34, Th35;

definition
let A be non empty set ;
let L be lower-bounded LATTICE;
let d be BiFunction of A,L;
func NextSet d ->  set  equals :: LATTICE5:def 17
 ConsecutiveSet (A,(DistEsti d));
correctness
coherence
ConsecutiveSet (A,(DistEsti d)) is set ;
;
end;

:: deftheorem  defines NextSet LATTICE5:def_17_:_
 for A being non empty set
for L being lower-bounded LATTICE
for d being BiFunction of A,L holds NextSet d = ConsecutiveSet (A,(DistEsti d));

registration
let A be non empty set ;
let L be lower-bounded LATTICE;
let d be BiFunction of A,L;
cluster NextSet d ->  non empty ;
coherence
 not NextSet d is empty ;
end;

definition
let A be non empty set ;
let L be lower-bounded LATTICE;
let d be BiFunction of A,L;
let q be QuadrSeq of d;
func NextDelta q ->  set  equals :: LATTICE5:def 18
 ConsecutiveDelta (q,(DistEsti d));
correctness
coherence
ConsecutiveDelta (q,(DistEsti d)) is set ;
;
end;

:: deftheorem  defines NextDelta LATTICE5:def_18_:_
 for A being non empty set
for L being lower-bounded LATTICE
for d being BiFunction of A,L
for q being QuadrSeq of d holds NextDelta q = ConsecutiveDelta (q,(DistEsti d));

definition
let A be non empty set ;
let L be lower-bounded LATTICE;
let d be distance_function of A,L;
let q be QuadrSeq of d;
:: original: NextDelta
redefine func NextDelta q ->  distance_function of (NextSet d),L;
coherence
 NextDelta q is distance_function of (NextSet d),L by Th33, Th34, Th35;
end;

definition
let A be non empty set ;
let L be lower-bounded LATTICE;
let d be distance_function of A,L;
let Aq be non empty set ;
let dq be distance_function of Aq,L;
predAq,dq is_extension_of A,d means :Def19: :: LATTICE5:def 19
 ex q being QuadrSeq of d st
( Aq = NextSet d & dq = NextDelta q );
end;

:: deftheorem Def19 defines is_extension_of LATTICE5:def_19_:_
 for A being non empty set
for L being lower-bounded LATTICE
for d being distance_function of A,L
for Aq being non empty set
for dq being distance_function of Aq,L holds
( Aq,dq is_extension_of A,d iff ex q being QuadrSeq of d st
( Aq = NextSet d & dq = NextDelta q ) );

theorem Th37: :: LATTICE5:37
for A being non empty set
for L being lower-bounded LATTICE
for d being distance_function of A,L
for Aq being non empty set
for dq being distance_function of Aq,L st Aq,dq is_extension_of A,d holds
for x, y being Element of A
for a, b being Element of L st d . (x,y) <= a "\/" b holds
ex z1, z2, z3 being Element of Aq st
( dq . (x,z1) = a & dq . (z2,z3) = a & dq . (z1,z2) = b & dq . (z3,y) = b )
proof
let A be non empty set ; ::_thesis: for L being lower-bounded LATTICE
for d being distance_function of A,L
for Aq being non empty set
for dq being distance_function of Aq,L st Aq,dq is_extension_of A,d holds
for x, y being Element of A
for a, b being Element of L st d . (x,y) <= a "\/" b holds
ex z1, z2, z3 being Element of Aq st
( dq . (x,z1) = a & dq . (z2,z3) = a & dq . (z1,z2) = b & dq . (z3,y) = b )
let L be lower-bounded LATTICE; ::_thesis: for d being distance_function of A,L
for Aq being non empty set
for dq being distance_function of Aq,L st Aq,dq is_extension_of A,d holds
for x, y being Element of A
for a, b being Element of L st d . (x,y) <= a "\/" b holds
ex z1, z2, z3 being Element of Aq st
( dq . (x,z1) = a & dq . (z2,z3) = a & dq . (z1,z2) = b & dq . (z3,y) = b )
let d be distance_function of A,L; ::_thesis: for Aq being non empty set
for dq being distance_function of Aq,L st Aq,dq is_extension_of A,d holds
for x, y being Element of A
for a, b being Element of L st d . (x,y) <= a "\/" b holds
ex z1, z2, z3 being Element of Aq st
( dq . (x,z1) = a & dq . (z2,z3) = a & dq . (z1,z2) = b & dq . (z3,y) = b )
let Aq be non empty set ; ::_thesis: for dq being distance_function of Aq,L st Aq,dq is_extension_of A,d holds
for x, y being Element of A
for a, b being Element of L st d . (x,y) <= a "\/" b holds
ex z1, z2, z3 being Element of Aq st
( dq . (x,z1) = a & dq . (z2,z3) = a & dq . (z1,z2) = b & dq . (z3,y) = b )
let dq be distance_function of Aq,L; ::_thesis: ( Aq,dq is_extension_of A,d implies for x, y being Element of A
for a, b being Element of L st d . (x,y) <= a "\/" b holds
ex z1, z2, z3 being Element of Aq st
( dq . (x,z1) = a & dq . (z2,z3) = a & dq . (z1,z2) = b & dq . (z3,y) = b ) )
assume  Aq,dq is_extension_of A,d ; ::_thesis: for x, y being Element of A
for a, b being Element of L st d . (x,y) <= a "\/" b holds
ex z1, z2, z3 being Element of Aq st
( dq . (x,z1) = a & dq . (z2,z3) = a & dq . (z1,z2) = b & dq . (z3,y) = b )
then consider q being QuadrSeq of d such that
A1: Aq = NextSet d and
A2: dq = NextDelta q by Def19;
let x, y be Element of A; ::_thesis: for a, b being Element of L st d . (x,y) <= a "\/" b holds
ex z1, z2, z3 being Element of Aq st
( dq . (x,z1) = a & dq . (z2,z3) = a & dq . (z1,z2) = b & dq . (z3,y) = b )
let a, b be Element of L; ::_thesis: ( d . (x,y) <= a "\/" b implies ex z1, z2, z3 being Element of Aq st
( dq . (x,z1) = a & dq . (z2,z3) = a & dq . (z1,z2) = b & dq . (z3,y) = b ) )
assume A3: d . (x,y) <= a "\/" b ; ::_thesis: ex z1, z2, z3 being Element of Aq st
( dq . (x,z1) = a & dq . (z2,z3) = a & dq . (z1,z2) = b & dq . (z3,y) = b )
 rng q =  {  [x9,y9,a9,b9] where x9, y9 is Element of A, a9, b9 is Element of L : d . (x9,y9) <= a9 "\/" b9  }  by Def13;
then  [x,y,a,b] in rng q by A3;
then consider o being set  such that
A4: o in dom q and
A5: q . o = [x,y,a,b] by FUNCT_1:def_3;
reconsider o = o as Ordinal by A4;
A6: q . o = Quadr (q,o) by A4, Def14;
then A7: x = (Quadr (q,o)) `1_4 by A5, MCART_1:74;
A8: b = (Quadr (q,o)) `4_4 by A5, A6, MCART_1:74;
A9: y = (Quadr (q,o)) `2_4 by A5, A6, MCART_1:74;
A10: a = (Quadr (q,o)) `3_4 by A5, A6, MCART_1:74;
reconsider B = ConsecutiveSet (A,o) as non empty set ;
 {B} in {{B},{{B}},{{{B}}}} by ENUMSET1:def_1;
then A11: {B} in B \/ {{B},{{B}},{{{B}}}} by XBOOLE_0:def_3;
reconsider cd = ConsecutiveDelta (q,o) as BiFunction of B,L ;
reconsider Q = Quadr (q,o) as Element of [:B,B, the carrier of L, the carrier of L:] ;
A12: ( x in A & y in A ) ;
A13: {{B}} in {{B},{{B}},{{{B}}}} by ENUMSET1:def_1;
then A14: {{B}} in new_set B by XBOOLE_0:def_3;
 A c= B by Th24;
then reconsider xo = x, yo = y as Element of B by A12;
A15: B c= new_set B by XBOOLE_1:7;
 ( xo in B & yo in B ) ;
then reconsider x1 = xo, y1 = yo as Element of new_set B by A15;
A16: cd is zeroed by Th33;
A17: {{{B}}} in {{B},{{B}},{{{B}}}} by ENUMSET1:def_1;
then A18: {{{B}}} in new_set B by XBOOLE_0:def_3;
 o in DistEsti d by A4, Th25;
then A19: succ o c= DistEsti d by ORDINAL1:21;
then A20: ConsecutiveDelta (q,(succ o)) c= ConsecutiveDelta (q,(DistEsti d)) by Th32;
 ConsecutiveSet (A,(succ o)) = new_set B by Th22;
then  new_set B c= ConsecutiveSet (A,(DistEsti d)) by A19, Th29;
then reconsider z1 = {B}, z2 = {{B}}, z3 = {{{B}}} as Element of Aq by A1, A11, A14, A18;
take  z1 ; ::_thesis: ex z2, z3 being Element of Aq st
( dq . (x,z1) = a & dq . (z2,z3) = a & dq . (z1,z2) = b & dq . (z3,y) = b )
take  z2 ; ::_thesis: ex z3 being Element of Aq st
( dq . (x,z1) = a & dq . (z2,z3) = a & dq . (z1,z2) = b & dq . (z3,y) = b )
take  z3 ; ::_thesis: ( dq . (x,z1) = a & dq . (z2,z3) = a & dq . (z1,z2) = b & dq . (z3,y) = b )
A21: ConsecutiveDelta (q,(succ o)) = new_bi_fun ((BiFun ((ConsecutiveDelta (q,o)),(ConsecutiveSet (A,o)),L)),(Quadr (q,o))) by Th27
.= new_bi_fun (cd,Q) by Def15 ;
A22: dom (new_bi_fun (cd,Q)) = [:(new_set B),(new_set B):] by FUNCT_2:def_1;
then  [x1,{B}] in dom (new_bi_fun (cd,Q)) by A11, ZFMISC_1:87;
hence dq . (x,z1) = (new_bi_fun (cd,Q)) . (x1,{B}) by A2, A20, A21, GRFUNC_1:2
.= (cd . (xo,xo)) "\/" a by A7, A10, Def10
.= (Bottom L) "\/" a by A16, Def6
.= a by WAYBEL_1:3 ;
::_thesis: ( dq . (z2,z3) = a & dq . (z1,z2) = b & dq . (z3,y) = b )
 {{B}} in B \/ {{B},{{B}},{{{B}}}} by A13, XBOOLE_0:def_3;
then  [{{B}},{{{B}}}] in dom (new_bi_fun (cd,Q)) by A18, A22, ZFMISC_1:87;
hence dq . (z2,z3) = (new_bi_fun (cd,Q)) . ({{B}},{{{B}}}) by A2, A20, A21, GRFUNC_1:2
.= a by A10, Def10 ;
::_thesis: ( dq . (z1,z2) = b & dq . (z3,y) = b )
 [{B},{{B}}] in dom (new_bi_fun (cd,Q)) by A11, A14, A22, ZFMISC_1:87;
hence dq . (z1,z2) = (new_bi_fun (cd,Q)) . ({B},{{B}}) by A2, A20, A21, GRFUNC_1:2
.= b by A8, Def10 ;
::_thesis: dq . (z3,y) = b
 {{{B}}} in B \/ {{B},{{B}},{{{B}}}} by A17, XBOOLE_0:def_3;
then  [{{{B}}},y1] in dom (new_bi_fun (cd,Q)) by A22, ZFMISC_1:87;
hence dq . (z3,y) = (new_bi_fun (cd,Q)) . ({{{B}}},y1) by A2, A20, A21, GRFUNC_1:2
.= (cd . (yo,yo)) "\/" b by A9, A8, Def10
.= (Bottom L) "\/" b by A16, Def6
.= b by WAYBEL_1:3 ;
::_thesis: verum
end;

definition
let A be non empty set ;
let L be lower-bounded LATTICE;
let d be distance_function of A,L;
mode ExtensionSeq of A,d ->  Function means :Def20: :: LATTICE5:def 20
( dom it = NAT & it . 0 = [A,d] & ( for n being Element of NAT ex A9 being non empty set ex d9 being distance_function of A9,L ex Aq being non empty set ex dq being distance_function of Aq,L st
( Aq,dq is_extension_of A9,d9 & it . n = [A9,d9] & it . (n + 1) = [Aq,dq] ) ) );
existence
 ex b1 being Function st
( dom b1 = NAT & b1 . 0 = [A,d] & ( for n being Element of NAT ex A9 being non empty set ex d9 being distance_function of A9,L ex Aq being non empty set ex dq being distance_function of Aq,L st
( Aq,dq is_extension_of A9,d9 & b1 . n = [A9,d9] & b1 . (n + 1) = [Aq,dq] ) ) )
proof
defpred S1[ set , set , set ] means ( ex A9 being non empty set ex d9 being distance_function of A9,L ex Aq being non empty set ex dq being distance_function of Aq,L st
( Aq,dq is_extension_of A9,d9 & $2 = [A9,d9] & $3 = [Aq,dq] ) or ( $3 = 0 & ( for A9 being non empty set
for d9 being distance_function of A9,L
for Aq being non empty set
for dq being distance_function of Aq,L holds
( not Aq,dq is_extension_of A9,d9 or not $2 = [A9,d9] ) ) ) );
A1: for n being Element of NAT
for x being set ex y being set st S1[n,x,y]
proof
let n be Element of NAT ; ::_thesis: for x being set ex y being set st S1[n,x,y]
let x be set ; ::_thesis: ex y being set st S1[n,x,y]
percases ( ex A9 being non empty set ex d9 being distance_function of A9,L ex Aq being non empty set ex dq being distance_function of Aq,L st
( Aq,dq is_extension_of A9,d9 & x = [A9,d9] ) or for A9 being non empty set
for d9 being distance_function of A9,L
for Aq being non empty set
for dq being distance_function of Aq,L holds
( not Aq,dq is_extension_of A9,d9 or not x = [A9,d9] ) ) ;
suppose ex A9 being non empty set ex d9 being distance_function of A9,L ex Aq being non empty set ex dq being distance_function of Aq,L st
( Aq,dq is_extension_of A9,d9 & x = [A9,d9] ) ; ::_thesis: ex y being set st S1[n,x,y]
then consider A9 being non empty set , d9 being distance_function of A9,L, Aq being non empty set , dq being distance_function of Aq,L such that
A2: ( Aq,dq is_extension_of A9,d9 & x = [A9,d9] ) ;
take  [Aq,dq] ; ::_thesis: S1[n,x,[Aq,dq]]
thus  S1[n,x,[Aq,dq]] by A2; ::_thesis: verum
end;
supposeA3: for A9 being non empty set
for d9 being distance_function of A9,L
for Aq being non empty set
for dq being distance_function of Aq,L holds
( not Aq,dq is_extension_of A9,d9 or not x = [A9,d9] ) ; ::_thesis: ex y being set st S1[n,x,y]
take  0 ; ::_thesis: S1[n,x, 0 ]
thus  S1[n,x, 0 ] by A3; ::_thesis: verum
end;
end;
end;
consider f being Function such that
A4: dom f = NAT and
A5: f . 0 = [A,d] and
A6: for n being Element of NAT holds S1[n,f . n,f . (n + 1)] from RECDEF_1:sch_1(A1);
take  f ; ::_thesis: ( dom f = NAT & f . 0 = [A,d] & ( for n being Element of NAT ex A9 being non empty set ex d9 being distance_function of A9,L ex Aq being non empty set ex dq being distance_function of Aq,L st
( Aq,dq is_extension_of A9,d9 & f . n = [A9,d9] & f . (n + 1) = [Aq,dq] ) ) )
thus  dom f = NAT by A4; ::_thesis: ( f . 0 = [A,d] & ( for n being Element of NAT ex A9 being non empty set ex d9 being distance_function of A9,L ex Aq being non empty set ex dq being distance_function of Aq,L st
( Aq,dq is_extension_of A9,d9 & f . n = [A9,d9] & f . (n + 1) = [Aq,dq] ) ) )
thus  f . 0 = [A,d] by A5; ::_thesis: for n being Element of NAT ex A9 being non empty set ex d9 being distance_function of A9,L ex Aq being non empty set ex dq being distance_function of Aq,L st
( Aq,dq is_extension_of A9,d9 & f . n = [A9,d9] & f . (n + 1) = [Aq,dq] )
defpred S2[ Element of NAT ] means  ex A9 being non empty set ex d9 being distance_function of A9,L ex Aq being non empty set ex dq being distance_function of Aq,L st
( Aq,dq is_extension_of A9,d9 & f . $1 = [A9,d9] & f . ($1 + 1) = [Aq,dq] );
A7: for k being Element of NAT st S2[k] holds
S2[k + 1]
proof
let k be Element of NAT ; ::_thesis: ( S2[k] implies S2[k + 1] )
given A9 being non empty set , d9 being distance_function of A9,L, Aq being non empty set , dq being distance_function of Aq,L such that  Aq,dq is_extension_of A9,d9 and
 f . k = [A9,d9] and
A8: f . (k + 1) = [Aq,dq] ; ::_thesis: S2[k + 1]
 ex A1 being non empty set ex d1 being distance_function of A1,L ex AQ being non empty set ex DQ being distance_function of AQ,L st
( AQ,DQ is_extension_of A1,d1 & f . (k + 1) = [A1,d1] )
proof
set Q = the QuadrSeq of dq;
set AQ = NextSet dq;
take  Aq ; ::_thesis: ex d1 being distance_function of Aq,L ex AQ being non empty set ex DQ being distance_function of AQ,L st
( AQ,DQ is_extension_of Aq,d1 & f . (k + 1) = [Aq,d1] )
take  dq ; ::_thesis: ex AQ being non empty set ex DQ being distance_function of AQ,L st
( AQ,DQ is_extension_of Aq,dq & f . (k + 1) = [Aq,dq] )
set DQ = NextDelta the QuadrSeq of dq;
take  NextSet dq ; ::_thesis: ex DQ being distance_function of (NextSet dq),L st
( NextSet dq,DQ is_extension_of Aq,dq & f . (k + 1) = [Aq,dq] )
take  NextDelta the QuadrSeq of dq ; ::_thesis: ( NextSet dq, NextDelta the QuadrSeq of dq is_extension_of Aq,dq & f . (k + 1) = [Aq,dq] )
thus  NextSet dq, NextDelta the QuadrSeq of dq is_extension_of Aq,dq by Def19; ::_thesis: f . (k + 1) = [Aq,dq]
thus  f . (k + 1) = [Aq,dq] by A8; ::_thesis: verum
end;
hence  S2[k + 1] by A6; ::_thesis: verum
end;
 ex A9 being non empty set ex d9 being distance_function of A9,L ex Aq being non empty set ex dq being distance_function of Aq,L st
( Aq,dq is_extension_of A9,d9 & f . 0 = [A9,d9] )
proof
set Aq = NextSet d;
set q = the QuadrSeq of d;
take  A ; ::_thesis: ex d9 being distance_function of A,L ex Aq being non empty set ex dq being distance_function of Aq,L st
( Aq,dq is_extension_of A,d9 & f . 0 = [A,d9] )
take  d ; ::_thesis: ex Aq being non empty set ex dq being distance_function of Aq,L st
( Aq,dq is_extension_of A,d & f . 0 = [A,d] )
consider dq being distance_function of (NextSet d),L such that
A9: dq = NextDelta the QuadrSeq of d ;
take  NextSet d ; ::_thesis: ex dq being distance_function of (NextSet d),L st
( NextSet d,dq is_extension_of A,d & f . 0 = [A,d] )
take  dq ; ::_thesis: ( NextSet d,dq is_extension_of A,d & f . 0 = [A,d] )
thus  NextSet d,dq is_extension_of A,d by A9, Def19; ::_thesis: f . 0 = [A,d]
thus  f . 0 = [A,d] by A5; ::_thesis: verum
end;
then A10: S2[ 0 ] by A6;
thus  for k being Element of NAT holds S2[k] from NAT_1:sch_1(A10, A7); ::_thesis: verum
end;
end;

:: deftheorem Def20 defines ExtensionSeq LATTICE5:def_20_:_
 for A being non empty set
for L being lower-bounded LATTICE
for d being distance_function of A,L
for b4 being Function holds
( b4 is ExtensionSeq of A,d iff ( dom b4 = NAT & b4 . 0 = [A,d] & ( for n being Element of NAT ex A9 being non empty set ex d9 being distance_function of A9,L ex Aq being non empty set ex dq being distance_function of Aq,L st
( Aq,dq is_extension_of A9,d9 & b4 . n = [A9,d9] & b4 . (n + 1) = [Aq,dq] ) ) ) );

theorem Th38: :: LATTICE5:38
for A being non empty set
for L being lower-bounded LATTICE
for d being distance_function of A,L
for S being ExtensionSeq of A,d
for k, l being Element of NAT st k <= l holds
(S . k) `1 c= (S . l) `1
proof
let A be non empty set ; ::_thesis: for L being lower-bounded LATTICE
for d being distance_function of A,L
for S being ExtensionSeq of A,d
for k, l being Element of NAT st k <= l holds
(S . k) `1 c= (S . l) `1
let L be lower-bounded LATTICE; ::_thesis: for d being distance_function of A,L
for S being ExtensionSeq of A,d
for k, l being Element of NAT st k <= l holds
(S . k) `1 c= (S . l) `1
let d be distance_function of A,L; ::_thesis: for S being ExtensionSeq of A,d
for k, l being Element of NAT st k <= l holds
(S . k) `1 c= (S . l) `1
let S be ExtensionSeq of A,d; ::_thesis: for k, l being Element of NAT st k <= l holds
(S . k) `1 c= (S . l) `1
let k be Element of NAT ; ::_thesis: for l being Element of NAT st k <= l holds
(S . k) `1 c= (S . l) `1
defpred S1[ Element of NAT ] means ( k <= $1 implies (S . k) `1 c= (S . $1) `1 );
A1: for i being Element of NAT st S1[i] holds
S1[i + 1]
proof
let i be Element of NAT ; ::_thesis: ( S1[i] implies S1[i + 1] )
assume that
A2: ( k <= i implies (S . k) `1 c= (S . i) `1 ) and
A3: k <= i + 1 ; ::_thesis: (S . k) `1 c= (S . (i + 1)) `1
percases ( k = i + 1 or k <= i ) by A3, NAT_1:8;
suppose k = i + 1 ; ::_thesis: (S . k) `1 c= (S . (i + 1)) `1
hence  (S . k) `1 c= (S . (i + 1)) `1 ; ::_thesis: verum
end;
supposeA4: k <= i ; ::_thesis: (S . k) `1 c= (S . (i + 1)) `1
consider A9 being non empty set , d9 being distance_function of A9,L, Aq being non empty set , dq being distance_function of Aq,L such that
A5: Aq,dq is_extension_of A9,d9 and
A6: S . i = [A9,d9] and
A7: S . (i + 1) = [Aq,dq] by Def20;
 [A9,d9] `1 = A9 ;
then A8: (S . i) `1 c= ConsecutiveSet (A9,(DistEsti d9)) by Th24, A6;
B7: [Aq,dq] `1 = Aq ;
 ex q being QuadrSeq of d9 st
( Aq = NextSet d9 & dq = NextDelta q ) by A5, Def19;
then  (S . (i + 1)) `1 = ConsecutiveSet (A9,(DistEsti d9)) by A7, B7;
hence  (S . k) `1 c= (S . (i + 1)) `1 by A2, A4, A8, XBOOLE_1:1; ::_thesis: verum
end;
end;
end;
A9: S1[ 0 ] by NAT_1:3;
thus  for l being Element of NAT holds S1[l] from NAT_1:sch_1(A9, A1); ::_thesis: verum
end;

theorem Th39: :: LATTICE5:39
for A being non empty set
for L being lower-bounded LATTICE
for d being distance_function of A,L
for S being ExtensionSeq of A,d
for k, l being Element of NAT st k <= l holds
(S . k) `2 c= (S . l) `2
proof
let A be non empty set ; ::_thesis: for L being lower-bounded LATTICE
for d being distance_function of A,L
for S being ExtensionSeq of A,d
for k, l being Element of NAT st k <= l holds
(S . k) `2 c= (S . l) `2
let L be lower-bounded LATTICE; ::_thesis: for d being distance_function of A,L
for S being ExtensionSeq of A,d
for k, l being Element of NAT st k <= l holds
(S . k) `2 c= (S . l) `2
let d be distance_function of A,L; ::_thesis: for S being ExtensionSeq of A,d
for k, l being Element of NAT st k <= l holds
(S . k) `2 c= (S . l) `2
let S be ExtensionSeq of A,d; ::_thesis: for k, l being Element of NAT st k <= l holds
(S . k) `2 c= (S . l) `2
let k be Element of NAT ; ::_thesis: for l being Element of NAT st k <= l holds
(S . k) `2 c= (S . l) `2
defpred S1[ Element of NAT ] means ( k <= $1 implies (S . k) `2 c= (S . $1) `2 );
A1: for i being Element of NAT st S1[i] holds
S1[i + 1]
proof
let i be Element of NAT ; ::_thesis: ( S1[i] implies S1[i + 1] )
assume that
A2: ( k <= i implies (S . k) `2 c= (S . i) `2 ) and
A3: k <= i + 1 ; ::_thesis: (S . k) `2 c= (S . (i + 1)) `2
percases ( k = i + 1 or k <= i ) by A3, NAT_1:8;
suppose k = i + 1 ; ::_thesis: (S . k) `2 c= (S . (i + 1)) `2
hence  (S . k) `2 c= (S . (i + 1)) `2 ; ::_thesis: verum
end;
supposeA4: k <= i ; ::_thesis: (S . k) `2 c= (S . (i + 1)) `2
consider A9 being non empty set , d9 being distance_function of A9,L, Aq being non empty set , dq being distance_function of Aq,L such that
A5: Aq,dq is_extension_of A9,d9 and
A6: S . i = [A9,d9] and
A7: S . (i + 1) = [Aq,dq] by Def20;
consider q being QuadrSeq of d9 such that
 Aq = NextSet d9 and
A8: dq = NextDelta q by A5, Def19;
 [A9,d9] `2 = d9 ;
then A9: (S . i) `2 c= ConsecutiveDelta (q,(DistEsti d9)) by Th31, A6;
B7: [Aq,dq] `2 = dq ;
 (S . (i + 1)) `2 = ConsecutiveDelta (q,(DistEsti d9)) by A7, A8, B7;
hence  (S . k) `2 c= (S . (i + 1)) `2 by A2, A4, A9, XBOOLE_1:1; ::_thesis: verum
end;
end;
end;
A10: S1[ 0 ] by NAT_1:3;
thus  for l being Element of NAT holds S1[l] from NAT_1:sch_1(A10, A1); ::_thesis: verum
end;

definition
let L be lower-bounded LATTICE;
func BasicDF L ->  distance_function of the carrier of L,L means :Def21: :: LATTICE5:def 21
 for x, y being Element of L holds
( ( x <> y implies it . (x,y) = x "\/" y ) & ( x = y implies it . (x,y) = Bottom L ) );
existence
 ex b1 being distance_function of the carrier of L,L st
for x, y being Element of L holds
( ( x <> y implies b1 . (x,y) = x "\/" y ) & ( x = y implies b1 . (x,y) = Bottom L ) )
proof
defpred S1[ Element of L, Element of L, set ] means ( ( $1 = $2 implies $3 = Bottom L ) & ( $1 <> $2 implies $3 = $1 "\/" $2 ) );
set A = the carrier of L;
A1: for x, y being Element of L ex z being Element of L st S1[x,y,z]
proof
let x, y be Element of L; ::_thesis: ex z being Element of L st S1[x,y,z]
percases ( x = y or x <> y ) ;
supposeA2: x = y ; ::_thesis: ex z being Element of L st S1[x,y,z]
take  Bottom L ; ::_thesis: S1[x,y, Bottom L]
thus  S1[x,y, Bottom L] by A2; ::_thesis: verum
end;
supposeA3: x <> y ; ::_thesis: ex z being Element of L st S1[x,y,z]
take  x "\/" y ; ::_thesis: S1[x,y,x "\/" y]
thus  S1[x,y,x "\/" y] by A3; ::_thesis: verum
end;
end;
end;
consider f being Function of [: the carrier of L, the carrier of L:], the carrier of L such that
A4: for x, y being Element of L holds S1[x,y,f . (x,y)] from BINOP_1:sch_3(A1);
reconsider f = f as BiFunction of the carrier of L,L ;
A5: f is zeroed
proof
let x be Element of the carrier of L; :: according to LATTICE5:def_6 ::_thesis: f . (x,x) = Bottom L
thus  f . (x,x) = Bottom L by A4; ::_thesis: verum
end;
A6: f is u.t.i.
proof
let x, y, z be Element of the carrier of L; :: according to LATTICE5:def_7 ::_thesis: (f . (x,y)) "\/" (f . (y,z)) >= f . (x,z)
reconsider x9 = x, y9 = y, z9 = z as Element of L ;
percases ( x = z or x <> z ) ;
supposeA7: x = z ; ::_thesis: (f . (x,y)) "\/" (f . (y,z)) >= f . (x,z)
 (f . (x,y)) "\/" (f . (y,z)) >= Bottom L by YELLOW_0:44;
hence  (f . (x,y)) "\/" (f . (y,z)) >= f . (x,z) by A4, A7; ::_thesis: verum
end;
supposeA8: x <> z ; ::_thesis: (f . (x,y)) "\/" (f . (y,z)) >= f . (x,z)
thus  (f . (x,y)) "\/" (f . (y,z)) >= f . (x,z) ::_thesis: verum
proof
percases ( x = y or x <> y ) ;
supposeA9: x = y ; ::_thesis: (f . (x,y)) "\/" (f . (y,z)) >= f . (x,z)
 x9 "\/" z9 >= x9 "\/" z9 by ORDERS_2:1;
then  f . (x,z) >= x9 "\/" z9 by A4, A8;
then  (Bottom L) "\/" (f . (x,z)) >= x9 "\/" z9 by WAYBEL_1:3;
then  (f . (x,y)) "\/" (f . (y,z)) >= x9 "\/" z9 by A4, A9;
hence  (f . (x,y)) "\/" (f . (y,z)) >= f . (x,z) by A4, A8; ::_thesis: verum
end;
supposeA10: x <> y ; ::_thesis: (f . (x,y)) "\/" (f . (y,z)) >= f . (x,z)
 (x9 "\/" y9) "\/" (f . (y,z)) >= x9 "\/" z9
proof
percases ( y = z or y <> z ) ;
supposeA11: y = z ; ::_thesis: (x9 "\/" y9) "\/" (f . (y,z)) >= x9 "\/" z9
 x9 "\/" y9 >= x9 "\/" y9 by ORDERS_2:1;
then  (Bottom L) "\/" (x9 "\/" y9) >= x9 "\/" z9 by A11, WAYBEL_1:3;
hence  (x9 "\/" y9) "\/" (f . (y,z)) >= x9 "\/" z9 by A4, A11; ::_thesis: verum
end;
supposeA12: y <> z ; ::_thesis: (x9 "\/" y9) "\/" (f . (y,z)) >= x9 "\/" z9
(x9 "\/" z9) "\/" y9 = (x9 "\/" z9) "\/" (y9 "\/" y9) by YELLOW_5:1
.= x9 "\/" (z9 "\/" (y9 "\/" y9)) by LATTICE3:14
.= x9 "\/" (y9 "\/" (y9 "\/" z9)) by LATTICE3:14
.= (x9 "\/" y9) "\/" (y9 "\/" z9) by LATTICE3:14 ;
then  (x9 "\/" y9) "\/" (y9 "\/" z9) >= x9 "\/" z9 by YELLOW_0:22;
hence  (x9 "\/" y9) "\/" (f . (y,z)) >= x9 "\/" z9 by A4, A12; ::_thesis: verum
end;
end;
end;
then  (f . (x,y)) "\/" (f . (y,z)) >= x9 "\/" z9 by A4, A10;
hence  (f . (x,y)) "\/" (f . (y,z)) >= f . (x,z) by A4, A8; ::_thesis: verum
end;
end;
end;
end;
end;
end;
 f is symmetric
proof
let x, y be Element of the carrier of L; :: according to LATTICE5:def_5 ::_thesis: f . (x,y) = f . (y,x)
reconsider x9 = x, y9 = y as Element of L ;
percases ( x = y or x <> y ) ;
suppose x = y ; ::_thesis: f . (x,y) = f . (y,x)
hence  f . (x,y) = f . (y,x) ; ::_thesis: verum
end;
supposeA13: x <> y ; ::_thesis: f . (x,y) = f . (y,x)
hence f . (x,y) = y9 "\/" x9 by A4
.= f . (y,x) by A4, A13 ;
::_thesis: verum
end;
end;
end;
then reconsider f = f as distance_function of the carrier of L,L by A5, A6;
take  f ; ::_thesis: for x, y being Element of L holds
( ( x <> y implies f . (x,y) = x "\/" y ) & ( x = y implies f . (x,y) = Bottom L ) )
thus  for x, y being Element of L holds
( ( x <> y implies f . (x,y) = x "\/" y ) & ( x = y implies f . (x,y) = Bottom L ) ) by A4; ::_thesis: verum
end;
uniqueness
 for b1, b2 being distance_function of the carrier of L,L st ( for x, y being Element of L holds
( ( x <> y implies b1 . (x,y) = x "\/" y ) & ( x = y implies b1 . (x,y) = Bottom L ) ) ) & ( for x, y being Element of L holds
( ( x <> y implies b2 . (x,y) = x "\/" y ) & ( x = y implies b2 . (x,y) = Bottom L ) ) ) holds
b1 = b2
proof
let f1, f2 be distance_function of the carrier of L,L; ::_thesis: ( ( for x, y being Element of L holds
( ( x <> y implies f1 . (x,y) = x "\/" y ) & ( x = y implies f1 . (x,y) = Bottom L ) ) ) & ( for x, y being Element of L holds
( ( x <> y implies f2 . (x,y) = x "\/" y ) & ( x = y implies f2 . (x,y) = Bottom L ) ) ) implies f1 = f2 )
assume that
A14: for x, y being Element of L holds
( ( x <> y implies f1 . (x,y) = x "\/" y ) & ( x = y implies f1 . (x,y) = Bottom L ) ) and
A15: for x, y being Element of L holds
( ( x <> y implies f2 . (x,y) = x "\/" y ) & ( x = y implies f2 . (x,y) = Bottom L ) ) ; ::_thesis: f1 = f2
A16: for z being set st z in dom f1 holds
f1 . z = f2 . z
proof
let z be set ; ::_thesis: ( z in dom f1 implies f1 . z = f2 . z )
assume A17: z in dom f1 ; ::_thesis: f1 . z = f2 . z
then consider x, y being set  such that
A18: z = [x,y] by RELAT_1:def_1;
reconsider x = x, y = y as Element of L by A17, A18, ZFMISC_1:87;
percases ( x = y or x <> y ) ;
supposeA19: x = y ; ::_thesis: f1 . z = f2 . z
thus f1 . z = f1 . (x,y) by A18
.= Bottom L by A14, A19
.= f2 . (x,y) by A15, A19
.= f2 . z by A18 ; ::_thesis: verum
end;
supposeA20: x <> y ; ::_thesis: f1 . z = f2 . z
thus f1 . z = f1 . (x,y) by A18
.= x "\/" y by A14, A20
.= f2 . (x,y) by A15, A20
.= f2 . z by A18 ; ::_thesis: verum
end;
end;
end;
dom f1 = [: the carrier of L, the carrier of L:] by FUNCT_2:def_1
.= dom f2 by FUNCT_2:def_1 ;
hence  f1 = f2 by A16, FUNCT_1:2; ::_thesis: verum
end;
end;

:: deftheorem Def21 defines BasicDF LATTICE5:def_21_:_
 for L being lower-bounded LATTICE
for b2 being distance_function of the carrier of L,L holds
( b2 = BasicDF L iff for x, y being Element of L holds
( ( x <> y implies b2 . (x,y) = x "\/" y ) & ( x = y implies b2 . (x,y) = Bottom L ) ) );

theorem Th40: :: LATTICE5:40
for L being lower-bounded LATTICE holds BasicDF L is onto
proof
let L be lower-bounded LATTICE; ::_thesis: BasicDF L is onto
set X = the carrier of L;
set f = BasicDF L;
 for w being set st w in the carrier of L holds
ex z being set st
( z in [: the carrier of L, the carrier of L:] & w = (BasicDF L) . z )
proof
let w be set ; ::_thesis: ( w in the carrier of L implies ex z being set st
( z in [: the carrier of L, the carrier of L:] & w = (BasicDF L) . z ) )
assume A1: w in the carrier of L ; ::_thesis: ex z being set st
( z in [: the carrier of L, the carrier of L:] & w = (BasicDF L) . z )
then reconsider w9 = w as Element of L ;
reconsider w99 = w as Element of L by A1;
percases ( w = Bottom L or w <> Bottom L ) ;
supposeA2: w = Bottom L ; ::_thesis: ex z being set st
( z in [: the carrier of L, the carrier of L:] & w = (BasicDF L) . z )
take z = [w,w]; ::_thesis: ( z in [: the carrier of L, the carrier of L:] & w = (BasicDF L) . z )
thus  z in [: the carrier of L, the carrier of L:] by A1, ZFMISC_1:87; ::_thesis: w = (BasicDF L) . z
thus (BasicDF L) . z = (BasicDF L) . (w9,w9)
.= w by A2, Def21 ; ::_thesis: verum
end;
supposeA3: w <> Bottom L ; ::_thesis: ex z being set st
( z in [: the carrier of L, the carrier of L:] & w = (BasicDF L) . z )
take z = [(Bottom L),w]; ::_thesis: ( z in [: the carrier of L, the carrier of L:] & w = (BasicDF L) . z )
thus  z in [: the carrier of L, the carrier of L:] by A1, ZFMISC_1:87; ::_thesis: w = (BasicDF L) . z
thus (BasicDF L) . z = (BasicDF L) . ((Bottom L),w9)
.= (Bottom L) "\/" w99 by A3, Def21
.= w by WAYBEL_1:3 ; ::_thesis: verum
end;
end;
end;
then  rng (BasicDF L) = the carrier of L by FUNCT_2:10;
hence  BasicDF L is onto by FUNCT_2:def_3; ::_thesis: verum
end;

Lm2: now__::_thesis:_for_j_being_Element_of_NAT_st_1_<=_j_&_j_<_5_&_not_j_=_1_&_not_j_=_2_&_not_j_=_3_holds_
j_=_4
let j be Element of NAT ; ::_thesis: ( 1 <= j & j < 5 & not j = 1 & not j = 2 & not j = 3 implies j = 4 )
assume that
A1: 1 <= j and
A2: j < 5 ; ::_thesis: ( j = 1 or j = 2 or j = 3 or j = 4 )
 j < 4 + 1 by A2;
then  j <= 4 by NAT_1:13;
then  ( j = 0 or j = 1 or j = 2 or j = 3 or j = 4 ) by NAT_1:28;
hence  ( j = 1 or j = 2 or j = 3 or j = 4 ) by A1; ::_thesis: verum
end;

Lm3: now__::_thesis:_for_m_being_Element_of_NAT_holds_
(_not_m_in_Seg_5_or_m_=_1_or_m_=_2_or_m_=_3_or_m_=_4_or_m_=_5_)
let m be Element of NAT ; ::_thesis: ( not m in Seg 5 or m = 1 or m = 2 or m = 3 or m = 4 or m = 5 )
assume A1: m in Seg 5 ; ::_thesis: ( m = 1 or m = 2 or m = 3 or m = 4 or m = 5 )
then  m <= 5 by FINSEQ_1:1;
then  ( m = 0 or m = 1 or m = 2 or m = 3 or m = 4 or m = 5 ) by NAT_1:29;
hence  ( m = 1 or m = 2 or m = 3 or m = 4 or m = 5 ) by A1, FINSEQ_1:1; ::_thesis: verum
end;

Lm4: now__::_thesis:_for_A_being_non_empty_set_
for_L_being_lower-bounded_LATTICE
for_d_being_distance_function_of_A,L_holds_succ_{}_c=_DistEsti_d
let A be non empty set ; ::_thesis: for L being lower-bounded LATTICE
for d being distance_function of A,L holds succ {} c= DistEsti d
let L be lower-bounded LATTICE; ::_thesis: for d being distance_function of A,L holds succ {} c= DistEsti d
let d be distance_function of A,L; ::_thesis: succ {} c= DistEsti d
 ( succ {} c= DistEsti d or DistEsti d in succ {} ) by ORDINAL1:16;
then  ( succ {} c= DistEsti d or DistEsti d c= {} ) by ORDINAL1:22;
hence  succ {} c= DistEsti d by Th20, XBOOLE_1:3; ::_thesis: verum
end;

theorem Th41: :: LATTICE5:41
for L being lower-bounded LATTICE
for S being ExtensionSeq of the carrier of L, BasicDF L
for FS being non empty set st FS = union  {  ((S . i) `1) where i is Element of NAT : verum  }  holds
union  {  ((S . i) `2) where i is Element of NAT : verum  }  is distance_function of FS,L
proof
let L be lower-bounded LATTICE; ::_thesis: for S being ExtensionSeq of the carrier of L, BasicDF L
for FS being non empty set st FS = union  {  ((S . i) `1) where i is Element of NAT : verum  }  holds
union  {  ((S . i) `2) where i is Element of NAT : verum  }  is distance_function of FS,L
let S be ExtensionSeq of the carrier of L, BasicDF L; ::_thesis: for FS being non empty set st FS = union  {  ((S . i) `1) where i is Element of NAT : verum  }  holds
union  {  ((S . i) `2) where i is Element of NAT : verum  }  is distance_function of FS,L
let FS be non empty set ; ::_thesis: ( FS = union  {  ((S . i) `1) where i is Element of NAT : verum  }  implies union  {  ((S . i) `2) where i is Element of NAT : verum  }  is distance_function of FS,L )
assume A1: FS = union  {  ((S . i) `1) where i is Element of NAT : verum  }  ; ::_thesis: union  {  ((S . i) `2) where i is Element of NAT : verum  }  is distance_function of FS,L
reconsider FS = FS as non empty set ;
set A = the carrier of L;
set FD = union  {  ((S . i) `2) where i is Element of NAT : verum  }  ;
now__::_thesis:_for_x,_y_being_set_st_x_in__{__((S_._i)_`2)_where_i_is_Element_of_NAT_:_verum__}__&_y_in__{__((S_._i)_`2)_where_i_is_Element_of_NAT_:_verum__}__holds_
x,y_are_c=-comparable
let x, y be set ; ::_thesis: ( x in  {  ((S . i) `2) where i is Element of NAT : verum  }  & y in  {  ((S . i) `2) where i is Element of NAT : verum  }  implies x,y are_c=-comparable )
assume that
A2: x in  {  ((S . i) `2) where i is Element of NAT : verum  }  and
A3: y in  {  ((S . i) `2) where i is Element of NAT : verum  }  ; ::_thesis: x,y are_c=-comparable
consider k being Element of NAT  such that
A4: x = (S . k) `2 by A2;
consider l being Element of NAT  such that
A5: y = (S . l) `2 by A3;
 ( k <= l or l <= k ) ;
then  ( x c= y or y c= x ) by A4, A5, Th39;
hence  x,y are_c=-comparable by XBOOLE_0:def_9; ::_thesis: verum
end;
then A6:  {  ((S . i) `2) where i is Element of NAT : verum  }  is c=-linear by ORDINAL1:def_8;
  {  ((S . i) `2) where i is Element of NAT : verum  }  c= PFuncs ([:FS,FS:], the carrier of L)
proof
let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in  {  ((S . i) `2) where i is Element of NAT : verum  }  or z in PFuncs ([:FS,FS:], the carrier of L) )
assume  z in  {  ((S . i) `2) where i is Element of NAT : verum  }  ; ::_thesis: z in PFuncs ([:FS,FS:], the carrier of L)
then consider j being Element of NAT  such that
A7: z = (S . j) `2 ;
consider A9 being non empty set , d9 being distance_function of A9,L, Aq being non empty set , dq being distance_function of Aq,L such that
 Aq,dq is_extension_of A9,d9 and
A8: S . j = [A9,d9] and
 S . (j + 1) = [Aq,dq] by Def20;
C8: d9 = [A9,d9] `2 ;
 A9 = [A9,d9] `1 ;
then  A9 in  {  ((S . i) `1) where i is Element of NAT : verum  }  by A8;
then  ( dom d9 = [:A9,A9:] & A9 c= FS ) by A1, FUNCT_2:def_1, ZFMISC_1:74;
then A9: ( rng d9 c= the carrier of L & dom d9 c= [:FS,FS:] ) by ZFMISC_1:96;
 z = d9 by A7, A8, C8;
hence  z in PFuncs ([:FS,FS:], the carrier of L) by A9, PARTFUN1:def_3; ::_thesis: verum
end;
then  union  {  ((S . i) `2) where i is Element of NAT : verum  }  in PFuncs ([:FS,FS:], the carrier of L) by A6, TREES_2:40;
then A10: ex g being Function st
( union  {  ((S . i) `2) where i is Element of NAT : verum  }  = g & dom g c= [:FS,FS:] & rng g c= the carrier of L ) by PARTFUN1:def_3;
 (S . 0) `1 in  {  ((S . i) `1) where i is Element of NAT : verum  }  ;
then reconsider X =  {  ((S . i) `1) where i is Element of NAT : verum  }  as non empty set ;
set LL =  {  [:I,I:] where I is Element of X : I in X  }  ;
set PP =  {  [:((S . i) `1),((S . i) `1):] where i is Element of NAT : verum  }  ;
defpred S1[ set , set ] means $2 = (S . $1) `2 ;
A11:  {  [:I,I:] where I is Element of X : I in X  }  =  {  [:((S . i) `1),((S . i) `1):] where i is Element of NAT : verum  }
proof
thus   {  [:I,I:] where I is Element of X : I in X  }  c=  {  [:((S . i) `1),((S . i) `1):] where i is Element of NAT : verum  }  :: according to XBOOLE_0:def_10 ::_thesis:  {  [:((S . i) `1),((S . i) `1):] where i is Element of NAT : verum  }  c=  {  [:I,I:] where I is Element of X : I in X  }
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in  {  [:I,I:] where I is Element of X : I in X  }  or x in  {  [:((S . i) `1),((S . i) `1):] where i is Element of NAT : verum  }  )
assume  x in  {  [:I,I:] where I is Element of X : I in X  }  ; ::_thesis: x in  {  [:((S . i) `1),((S . i) `1):] where i is Element of NAT : verum  }
then consider J being Element of X such that
A12: x = [:J,J:] and
A13: J in X ;
 ex j being Element of NAT st J = (S . j) `1 by A13;
hence  x in  {  [:((S . i) `1),((S . i) `1):] where i is Element of NAT : verum  }  by A12; ::_thesis: verum
end;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in  {  [:((S . i) `1),((S . i) `1):] where i is Element of NAT : verum  }  or x in  {  [:I,I:] where I is Element of X : I in X  }  )
assume  x in  {  [:((S . i) `1),((S . i) `1):] where i is Element of NAT : verum  }  ; ::_thesis: x in  {  [:I,I:] where I is Element of X : I in X  }
then consider j being Element of NAT  such that
A14: x = [:((S . j) `1),((S . j) `1):] ;
 (S . j) `1 in X ;
hence  x in  {  [:I,I:] where I is Element of X : I in X  }  by A14; ::_thesis: verum
end;
reconsider FD = union  {  ((S . i) `2) where i is Element of NAT : verum  }  as Function by A10;
A15: for x being set st x in NAT holds
ex y being set st S1[x,y] ;
consider F being Function such that
A16: dom F = NAT and
A17: for x being set st x in NAT holds
S1[x,F . x] from CLASSES1:sch_1(A15);
A18: rng F =  {  ((S . i) `2) where i is Element of NAT : verum  }
proof
thus  rng F c=  {  ((S . i) `2) where i is Element of NAT : verum  }  :: according to XBOOLE_0:def_10 ::_thesis:  {  ((S . i) `2) where i is Element of NAT : verum  }  c= rng F
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in rng F or x in  {  ((S . i) `2) where i is Element of NAT : verum  }  )
assume  x in rng F ; ::_thesis: x in  {  ((S . i) `2) where i is Element of NAT : verum  }
then consider j being set  such that
A19: j in dom F and
A20: F . j = x by FUNCT_1:def_3;
reconsider j = j as Element of NAT by A16, A19;
 x = (S . j) `2 by A17, A20;
hence  x in  {  ((S . i) `2) where i is Element of NAT : verum  }  ; ::_thesis: verum
end;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in  {  ((S . i) `2) where i is Element of NAT : verum  }  or x in rng F )
assume  x in  {  ((S . i) `2) where i is Element of NAT : verum  }  ; ::_thesis: x in rng F
then consider j being Element of NAT  such that
A21: x = (S . j) `2 ;
 x = F . j by A17, A21;
hence  x in rng F by A16, FUNCT_1:def_3; ::_thesis: verum
end;
 F is Function-yielding
proof
let x be set ; :: according to FUNCOP_1:def_6 ::_thesis: ( not x in proj1 F or F . x is set )
assume  x in dom F ; ::_thesis: F . x is set
then reconsider j = x as Element of NAT by A16;
consider A1 being non empty set , d1 being distance_function of A1,L, Aq1 being non empty set , dq1 being distance_function of Aq1,L such that
 Aq1,dq1 is_extension_of A1,d1 and
A22: S . j = [A1,d1] and
 S . (j + 1) = [Aq1,dq1] by Def20;
 [A1,d1] `2 = d1 ;
hence  F . x is set by A17, A22; ::_thesis: verum
end;
then reconsider F = F as Function-yielding Function ;
A23: rng (doms F) =  {  [:((S . i) `1),((S . i) `1):] where i is Element of NAT : verum  }
proof
thus  rng (doms F) c=  {  [:((S . i) `1),((S . i) `1):] where i is Element of NAT : verum  }  :: according to XBOOLE_0:def_10 ::_thesis:  {  [:((S . i) `1),((S . i) `1):] where i is Element of NAT : verum  }  c= rng (doms F)
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in rng (doms F) or x in  {  [:((S . i) `1),((S . i) `1):] where i is Element of NAT : verum  }  )
assume  x in rng (doms F) ; ::_thesis: x in  {  [:((S . i) `1),((S . i) `1):] where i is Element of NAT : verum  }
then consider j being set  such that
A24: j in dom (doms F) and
A25: x = (doms F) . j by FUNCT_1:def_3;
A26: j in dom F by A24, FUNCT_6:59;
reconsider j = j as Element of NAT by A16, A24, FUNCT_6:59;
consider A1 being non empty set , d1 being distance_function of A1,L, Aq1 being non empty set , dq1 being distance_function of Aq1,L such that
 Aq1,dq1 is_extension_of A1,d1 and
A27: S . j = [A1,d1] and
 S . (j + 1) = [Aq1,dq1] by Def20;
A28: [A1,d1] `2 = d1 ;
A29: [A1,d1] `1 = A1 ;
x = dom (F . j) by A25, A26, FUNCT_6:22
.= dom d1 by A17, A28, A27
.= [:((S . j) `1),((S . j) `1):] by A29, A27, FUNCT_2:def_1 ;
hence  x in  {  [:((S . i) `1),((S . i) `1):] where i is Element of NAT : verum  }  ; ::_thesis: verum
end;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in  {  [:((S . i) `1),((S . i) `1):] where i is Element of NAT : verum  }  or x in rng (doms F) )
assume  x in  {  [:((S . i) `1),((S . i) `1):] where i is Element of NAT : verum  }  ; ::_thesis: x in rng (doms F)
then consider j being Element of NAT  such that
A30: x = [:((S . j) `1),((S . j) `1):] ;
consider A1 being non empty set , d1 being distance_function of A1,L, Aq1 being non empty set , dq1 being distance_function of Aq1,L such that
 Aq1,dq1 is_extension_of A1,d1 and
A31: S . j = [A1,d1] and
 S . (j + 1) = [Aq1,dq1] by Def20;
A32: [A1,d1] `2 = d1 ;
 j in NAT ;
then A33: j in dom (doms F) by A16, FUNCT_6:59;
 [A1,d1] `1 = A1 ;
then x = dom d1 by A30, A31, FUNCT_2:def_1
.= dom (F . j) by A17, A32, A31
.= (doms F) . j by A16, FUNCT_6:22 ;
hence  x in rng (doms F) by A33, FUNCT_1:def_3; ::_thesis: verum
end;
now__::_thesis:_for_x,_y_being_set_st_x_in_X_&_y_in_X_holds_
x,y_are_c=-comparable
let x, y be set ; ::_thesis: ( x in X & y in X implies x,y are_c=-comparable )
assume that
A34: x in X and
A35: y in X ; ::_thesis: x,y are_c=-comparable
consider k being Element of NAT  such that
A36: x = (S . k) `1 by A34;
consider l being Element of NAT  such that
A37: y = (S . l) `1 by A35;
 ( k <= l or l <= k ) ;
then  ( x c= y or y c= x ) by A36, A37, Th38;
hence  x,y are_c=-comparable by XBOOLE_0:def_9; ::_thesis: verum
end;
then  X is c=-linear by ORDINAL1:def_8;
then [:FS,FS:] = union (rng (doms F)) by A1, A23, A11, Th3
.= dom FD by A18, Th1 ;
then reconsider FD = FD as BiFunction of FS,L by A10, FUNCT_2:def_1, RELSET_1:4;
A38: FD is symmetric
proof
let x, y be Element of FS; :: according to LATTICE5:def_5 ::_thesis: FD . (x,y) = FD . (y,x)
consider x1 being set  such that
A39: x in x1 and
A40: x1 in X by A1, TARSKI:def_4;
consider k being Element of NAT  such that
A41: x1 = (S . k) `1 by A40;
consider A1 being non empty set , d1 being distance_function of A1,L, Aq1 being non empty set , dq1 being distance_function of Aq1,L such that
 Aq1,dq1 is_extension_of A1,d1 and
A42: S . k = [A1,d1] and
 S . (k + 1) = [Aq1,dq1] by Def20;
A44: [A1,d1] `1 = A1 ;
then A43: x in A1 by A39, A41, A42;
 [A1,d1] `2 = d1 ;
then  d1 in  {  ((S . i) `2) where i is Element of NAT : verum  }  by A42;
then A45: d1 c= FD by ZFMISC_1:74;
consider y1 being set  such that
A46: y in y1 and
A47: y1 in X by A1, TARSKI:def_4;
consider l being Element of NAT  such that
A48: y1 = (S . l) `1 by A47;
consider A2 being non empty set , d2 being distance_function of A2,L, Aq2 being non empty set , dq2 being distance_function of Aq2,L such that
 Aq2,dq2 is_extension_of A2,d2 and
A49: S . l = [A2,d2] and
 S . (l + 1) = [Aq2,dq2] by Def20;
A51: [A2,d2] `1 = A2 ;
then A50: y in A2 by A46, A48, A49;
 [A2,d2] `2 = d2 ;
then  d2 in  {  ((S . i) `2) where i is Element of NAT : verum  }  by A49;
then A52: d2 c= FD by ZFMISC_1:74;
percases ( k <= l or l <= k ) ;
suppose k <= l ; ::_thesis: FD . (x,y) = FD . (y,x)
then  A1 c= A2 by A44, A51, Th38, A42, A49;
then reconsider x9 = x, y9 = y as Element of A2 by A43, A50;
A53: dom d2 = [:A2,A2:] by FUNCT_2:def_1;
hence FD . (x,y) = d2 . [x9,y9] by A52, GRFUNC_1:2
.= d2 . (x9,y9)
.= d2 . (y9,x9) by Def5
.= FD . [y9,x9] by A52, A53, GRFUNC_1:2
.= FD . (y,x) ;
::_thesis: verum
end;
suppose l <= k ; ::_thesis: FD . (x,y) = FD . (y,x)
then  A2 c= A1 by A44, A51, Th38, A49, A42;
then reconsider x9 = x, y9 = y as Element of A1 by A39, A41, A42, A50, A44;
A54: dom d1 = [:A1,A1:] by FUNCT_2:def_1;
hence FD . (x,y) = d1 . [x9,y9] by A45, GRFUNC_1:2
.= d1 . (x9,y9)
.= d1 . (y9,x9) by Def5
.= FD . [y9,x9] by A45, A54, GRFUNC_1:2
.= FD . (y,x) ;
::_thesis: verum
end;
end;
end;
A55: FD is u.t.i.
proof
let x, y, z be Element of FS; :: according to LATTICE5:def_7 ::_thesis: (FD . (x,y)) "\/" (FD . (y,z)) >= FD . (x,z)
consider x1 being set  such that
A56: x in x1 and
A57: x1 in X by A1, TARSKI:def_4;
consider k being Element of NAT  such that
A58: x1 = (S . k) `1 by A57;
consider A1 being non empty set , d1 being distance_function of A1,L, Aq1 being non empty set , dq1 being distance_function of Aq1,L such that
 Aq1,dq1 is_extension_of A1,d1 and
A59: S . k = [A1,d1] and
 S . (k + 1) = [Aq1,dq1] by Def20;
B59: [A1,d1] `1 = A1 ;
then A60: x in A1 by A56, A58, A59;
 [A1,d1] `2 = d1 ;
then  d1 in  {  ((S . i) `2) where i is Element of NAT : verum  }  by A59;
then A61: d1 c= FD by ZFMISC_1:74;
A62: dom d1 = [:A1,A1:] by FUNCT_2:def_1;
A63: (S . k) `1 = A1 by A59, B59;
consider y1 being set  such that
A64: y in y1 and
A65: y1 in X by A1, TARSKI:def_4;
consider l being Element of NAT  such that
A66: y1 = (S . l) `1 by A65;
consider A2 being non empty set , d2 being distance_function of A2,L, Aq2 being non empty set , dq2 being distance_function of Aq2,L such that
 Aq2,dq2 is_extension_of A2,d2 and
A67: S . l = [A2,d2] and
 S . (l + 1) = [Aq2,dq2] by Def20;
 [A2,d2] `1 = A2 ;
then A68: y in A2 by A64, A66, A67;
 [A2,d2] `2 = d2 ;
then  d2 in  {  ((S . i) `2) where i is Element of NAT : verum  }  by A67;
then A69: d2 c= FD by ZFMISC_1:74;
A70: dom d2 = [:A2,A2:] by FUNCT_2:def_1;
A71: [A2,d2] `1 = A2 ;
consider z1 being set  such that
A72: z in z1 and
A73: z1 in X by A1, TARSKI:def_4;
consider n being Element of NAT  such that
A74: z1 = (S . n) `1 by A73;
consider A3 being non empty set , d3 being distance_function of A3,L, Aq3 being non empty set , dq3 being distance_function of Aq3,L such that
 Aq3,dq3 is_extension_of A3,d3 and
A75: S . n = [A3,d3] and
 S . (n + 1) = [Aq3,dq3] by Def20;
A77: [A3,d3] `1 = A3 ;
then A76: z in A3 by A72, A74, A75;
 [A3,d3] `2 = d3 ;
then  d3 in  {  ((S . i) `2) where i is Element of NAT : verum  }  by A75;
then A78: d3 c= FD by ZFMISC_1:74;
A79: dom d3 = [:A3,A3:] by FUNCT_2:def_1;
percases ( k <= l or l <= k ) ;
suppose k <= l ; ::_thesis: (FD . (x,y)) "\/" (FD . (y,z)) >= FD . (x,z)
then A80: A1 c= A2 by A63, A71, Th38, A67;
thus  (FD . (x,y)) "\/" (FD . (y,z)) >= FD . (x,z) ::_thesis: verum
proof
percases ( l <= n or n <= l ) ;
suppose l <= n ; ::_thesis: (FD . (x,y)) "\/" (FD . (y,z)) >= FD . (x,z)
then A81: A2 c= A3 by A71, A77, Th38, A75, A67;
then  A1 c= A3 by A80, XBOOLE_1:1;
then reconsider x9 = x, y9 = y as Element of A3 by A60, A68, A81;
reconsider z9 = z as Element of A3 by A76;
A82: FD . (y,z) = d3 . [y9,z9] by A78, A79, GRFUNC_1:2
.= d3 . (y9,z9) ;
A83: FD . (x,z) = d3 . [x9,z9] by A78, A79, GRFUNC_1:2
.= d3 . (x9,z9) ;
FD . (x,y) = d3 . [x9,y9] by A78, A79, GRFUNC_1:2
.= d3 . (x9,y9) ;
hence  (FD . (x,y)) "\/" (FD . (y,z)) >= FD . (x,z) by A82, A83, Def7; ::_thesis: verum
end;
suppose n <= l ; ::_thesis: (FD . (x,y)) "\/" (FD . (y,z)) >= FD . (x,z)
then X: A3 c= A2 by A71, A77, Th38, A75, A67;
reconsider y9 = y as Element of A2 by A68;
reconsider x9 = x as Element of A2 by A60, A80;
reconsider z9 = z as Element of A2 by A76, X;
A84: FD . (y,z) = d2 . [y9,z9] by A69, A70, GRFUNC_1:2
.= d2 . (y9,z9) ;
A85: FD . (x,z) = d2 . [x9,z9] by A69, A70, GRFUNC_1:2
.= d2 . (x9,z9) ;
FD . (x,y) = d2 . [x9,y9] by A69, A70, GRFUNC_1:2
.= d2 . (x9,y9) ;
hence  (FD . (x,y)) "\/" (FD . (y,z)) >= FD . (x,z) by A84, A85, Def7; ::_thesis: verum
end;
end;
end;
end;
suppose l <= k ; ::_thesis: (FD . (x,y)) "\/" (FD . (y,z)) >= FD . (x,z)
then A86: A2 c= A1 by A63, A71, Th38, A67;
thus  (FD . (x,y)) "\/" (FD . (y,z)) >= FD . (x,z) ::_thesis: verum
proof
percases ( k <= n or n <= k ) ;
suppose k <= n ; ::_thesis: (FD . (x,y)) "\/" (FD . (y,z)) >= FD . (x,z)
then A87: A1 c= A3 by A63, A77, Th38, A75;
then X: A2 c= A3 by A86, XBOOLE_1:1;
reconsider x9 = x as Element of A3 by A60, A87;
reconsider z9 = z as Element of A3 by A72, A74, A75, A77;
reconsider y9 = y as Element of A3 by A68, X;
A88: FD . (y,z) = d3 . [y9,z9] by A78, A79, GRFUNC_1:2
.= d3 . (y9,z9) ;
A89: FD . (x,z) = d3 . [x9,z9] by A78, A79, GRFUNC_1:2
.= d3 . (x9,z9) ;
FD . (x,y) = d3 . [x9,y9] by A78, A79, GRFUNC_1:2
.= d3 . (x9,y9) ;
hence  (FD . (x,y)) "\/" (FD . (y,z)) >= FD . (x,z) by A88, A89, Def7; ::_thesis: verum
end;
suppose n <= k ; ::_thesis: (FD . (x,y)) "\/" (FD . (y,z)) >= FD . (x,z)
then  A3 c= A1 by A63, A77, Th38, A75;
then reconsider x9 = x, y9 = y, z9 = z as Element of A1 by A56, A58, A59, A68, A76, A86, B59;
A90: FD . (y,z) = d1 . [y9,z9] by A61, A62, GRFUNC_1:2
.= d1 . (y9,z9) ;
A91: FD . (x,z) = d1 . [x9,z9] by A61, A62, GRFUNC_1:2
.= d1 . (x9,z9) ;
FD . (x,y) = d1 . [x9,y9] by A61, A62, GRFUNC_1:2
.= d1 . (x9,y9) ;
hence  (FD . (x,y)) "\/" (FD . (y,z)) >= FD . (x,z) by A90, A91, Def7; ::_thesis: verum
end;
end;
end;
end;
end;
end;
 FD is zeroed
proof
let x be Element of FS; :: according to LATTICE5:def_6 ::_thesis: FD . (x,x) = Bottom L
consider y being set  such that
A92: x in y and
A93: y in X by A1, TARSKI:def_4;
consider j being Element of NAT  such that
A94: y = (S . j) `1 by A93;
consider A1 being non empty set , d1 being distance_function of A1,L, Aq1 being non empty set , dq1 being distance_function of Aq1,L such that
 Aq1,dq1 is_extension_of A1,d1 and
A95: S . j = [A1,d1] and
 S . (j + 1) = [Aq1,dq1] by Def20;
 [A1,d1] `1 = A1 ;
then reconsider x9 = x as Element of A1 by A92, A94, A95;
 [A1,d1] `2 = d1 ;
then  d1 in  {  ((S . i) `2) where i is Element of NAT : verum  }  by A95;
then A96: d1 c= FD by ZFMISC_1:74;
 dom d1 = [:A1,A1:] by FUNCT_2:def_1;
hence FD . (x,x) = d1 . [x9,x9] by A96, GRFUNC_1:2
.= d1 . (x9,x9)
.= Bottom L by Def6 ;
::_thesis: verum
end;
hence  union  {  ((S . i) `2) where i is Element of NAT : verum  }  is distance_function of FS,L by A38, A55; ::_thesis: verum
end;

theorem Th42: :: LATTICE5:42
for L being lower-bounded LATTICE
for S being ExtensionSeq of the carrier of L, BasicDF L
for FS being non empty set
for FD being distance_function of FS,L
for x, y being Element of FS
for a, b being Element of L st FS = union  {  ((S . i) `1) where i is Element of NAT : verum  }  & FD = union  {  ((S . i) `2) where i is Element of NAT : verum  }  & FD . (x,y) <= a "\/" b holds
ex z1, z2, z3 being Element of FS st
( FD . (x,z1) = a & FD . (z2,z3) = a & FD . (z1,z2) = b & FD . (z3,y) = b )
proof
let L be lower-bounded LATTICE; ::_thesis: for S being ExtensionSeq of the carrier of L, BasicDF L
for FS being non empty set
for FD being distance_function of FS,L
for x, y being Element of FS
for a, b being Element of L st FS = union  {  ((S . i) `1) where i is Element of NAT : verum  }  & FD = union  {  ((S . i) `2) where i is Element of NAT : verum  }  & FD . (x,y) <= a "\/" b holds
ex z1, z2, z3 being Element of FS st
( FD . (x,z1) = a & FD . (z2,z3) = a & FD . (z1,z2) = b & FD . (z3,y) = b )
let S be ExtensionSeq of the carrier of L, BasicDF L; ::_thesis: for FS being non empty set
for FD being distance_function of FS,L
for x, y being Element of FS
for a, b being Element of L st FS = union  {  ((S . i) `1) where i is Element of NAT : verum  }  & FD = union  {  ((S . i) `2) where i is Element of NAT : verum  }  & FD . (x,y) <= a "\/" b holds
ex z1, z2, z3 being Element of FS st
( FD . (x,z1) = a & FD . (z2,z3) = a & FD . (z1,z2) = b & FD . (z3,y) = b )
let FS be non empty set ; ::_thesis: for FD being distance_function of FS,L
for x, y being Element of FS
for a, b being Element of L st FS = union  {  ((S . i) `1) where i is Element of NAT : verum  }  & FD = union  {  ((S . i) `2) where i is Element of NAT : verum  }  & FD . (x,y) <= a "\/" b holds
ex z1, z2, z3 being Element of FS st
( FD . (x,z1) = a & FD . (z2,z3) = a & FD . (z1,z2) = b & FD . (z3,y) = b )
let FD be distance_function of FS,L; ::_thesis: for x, y being Element of FS
for a, b being Element of L st FS = union  {  ((S . i) `1) where i is Element of NAT : verum  }  & FD = union  {  ((S . i) `2) where i is Element of NAT : verum  }  & FD . (x,y) <= a "\/" b holds
ex z1, z2, z3 being Element of FS st
( FD . (x,z1) = a & FD . (z2,z3) = a & FD . (z1,z2) = b & FD . (z3,y) = b )
let x, y be Element of FS; ::_thesis: for a, b being Element of L st FS = union  {  ((S . i) `1) where i is Element of NAT : verum  }  & FD = union  {  ((S . i) `2) where i is Element of NAT : verum  }  & FD . (x,y) <= a "\/" b holds
ex z1, z2, z3 being Element of FS st
( FD . (x,z1) = a & FD . (z2,z3) = a & FD . (z1,z2) = b & FD . (z3,y) = b )
let a, b be Element of L; ::_thesis: ( FS = union  {  ((S . i) `1) where i is Element of NAT : verum  }  & FD = union  {  ((S . i) `2) where i is Element of NAT : verum  }  & FD . (x,y) <= a "\/" b implies ex z1, z2, z3 being Element of FS st
( FD . (x,z1) = a & FD . (z2,z3) = a & FD . (z1,z2) = b & FD . (z3,y) = b ) )
assume that
A1: FS = union  {  ((S . i) `1) where i is Element of NAT : verum  }  and
A2: FD = union  {  ((S . i) `2) where i is Element of NAT : verum  }  and
A3: FD . (x,y) <= a "\/" b ; ::_thesis: ex z1, z2, z3 being Element of FS st
( FD . (x,z1) = a & FD . (z2,z3) = a & FD . (z1,z2) = b & FD . (z3,y) = b )
 (S . 0) `1 in  {  ((S . i) `1) where i is Element of NAT : verum  }  ;
then reconsider X =  {  ((S . i) `1) where i is Element of NAT : verum  }  as non empty set ;
consider x1 being set  such that
A4: x in x1 and
A5: x1 in X by A1, TARSKI:def_4;
consider k being Element of NAT  such that
A6: x1 = (S . k) `1 by A5;
consider A1 being non empty set , d1 being distance_function of A1,L, Aq1 being non empty set , dq1 being distance_function of Aq1,L such that
A7: Aq1,dq1 is_extension_of A1,d1 and
A8: S . k = [A1,d1] and
A9: S . (k + 1) = [Aq1,dq1] by Def20;
A12: [A1,d1] `1 = A1 ;
then A10: x in A1 by A4, A6, A8;
 [A1,d1] `2 = d1 ;
then  d1 in  {  ((S . i) `2) where i is Element of NAT : verum  }  by A8;
then A11: d1 c= FD by A2, ZFMISC_1:74;
A13: [Aq1,dq1] `1 = Aq1 ;
then  Aq1 in  {  ((S . i) `1) where i is Element of NAT : verum  }  by A9;
then A14: Aq1 c= FS by A1, ZFMISC_1:74;
 [Aq1,dq1] `2 = dq1 ;
then  dq1 in  {  ((S . i) `2) where i is Element of NAT : verum  }  by A9;
then A15: dq1 c= FD by A2, ZFMISC_1:74;
consider y1 being set  such that
A16: y in y1 and
A17: y1 in X by A1, TARSKI:def_4;
consider l being Element of NAT  such that
A18: y1 = (S . l) `1 by A17;
consider A2 being non empty set , d2 being distance_function of A2,L, Aq2 being non empty set , dq2 being distance_function of Aq2,L such that
A19: Aq2,dq2 is_extension_of A2,d2 and
A20: S . l = [A2,d2] and
A21: S . (l + 1) = [Aq2,dq2] by Def20;
A24: [A2,d2] `1 = A2 ;
then A22: y in A2 by A16, A18, A20;
 [A2,d2] `2 = d2 ;
then  d2 in  {  ((S . i) `2) where i is Element of NAT : verum  }  by A20;
then A23: d2 c= FD by A2, ZFMISC_1:74;
A25: [Aq2,dq2] `1 = Aq2 ;
then  Aq2 in  {  ((S . i) `1) where i is Element of NAT : verum  }  by A21;
then A26: Aq2 c= FS by A1, ZFMISC_1:74;
 [Aq2,dq2] `2 = dq2 ;
then  dq2 in  {  ((S . i) `2) where i is Element of NAT : verum  }  by A21;
then A27: dq2 c= FD by A2, ZFMISC_1:74;
percases ( k <= l or l <= k ) ;
suppose k <= l ; ::_thesis: ex z1, z2, z3 being Element of FS st
( FD . (x,z1) = a & FD . (z2,z3) = a & FD . (z1,z2) = b & FD . (z3,y) = b )
then  A1 c= A2 by A12, A24, Th38, A20, A8;
then reconsider x9 = x, y9 = y as Element of A2 by A10, A16, A18, A20, A24;
A28: ( x9 in A2 & y9 in A2 ) ;
 A2 c= Aq2 by A24, A25, Th38, A20, A21, NAT_1:11;
then reconsider x99 = x9, y99 = y9 as Element of Aq2 by A28;
 dom d2 = [:A2,A2:] by FUNCT_2:def_1;
then FD . (x,y) = d2 . [x9,y9] by A23, GRFUNC_1:2
.= d2 . (x9,y9) ;
then consider z1, z2, z3 being Element of Aq2 such that
A29: dq2 . (x,z1) = a and
A30: dq2 . (z2,z3) = a and
A31: dq2 . (z1,z2) = b and
A32: dq2 . (z3,y) = b by A3, A19, Th37;
A33: z3 in Aq2 ;
 ( z1 in Aq2 & z2 in Aq2 ) ;
then reconsider z19 = z1, z29 = z2, z39 = z3 as Element of FS by A26, A33;
take  z19 ; ::_thesis: ex z2, z3 being Element of FS st
( FD . (x,z19) = a & FD . (z2,z3) = a & FD . (z19,z2) = b & FD . (z3,y) = b )
take  z29 ; ::_thesis: ex z3 being Element of FS st
( FD . (x,z19) = a & FD . (z29,z3) = a & FD . (z19,z29) = b & FD . (z3,y) = b )
take  z39 ; ::_thesis: ( FD . (x,z19) = a & FD . (z29,z39) = a & FD . (z19,z29) = b & FD . (z39,y) = b )
A34: dom dq2 = [:Aq2,Aq2:] by FUNCT_2:def_1;
hence FD . (x,z19) = dq2 . [x99,z1] by A27, GRFUNC_1:2
.= a by A29 ;
::_thesis: ( FD . (z29,z39) = a & FD . (z19,z29) = b & FD . (z39,y) = b )
thus FD . (z29,z39) = dq2 . [z2,z3] by A27, A34, GRFUNC_1:2
.= a by A30 ; ::_thesis: ( FD . (z19,z29) = b & FD . (z39,y) = b )
thus FD . (z19,z29) = dq2 . [z1,z2] by A27, A34, GRFUNC_1:2
.= b by A31 ; ::_thesis: FD . (z39,y) = b
thus FD . (z39,y) = dq2 . [z3,y99] by A27, A34, GRFUNC_1:2
.= b by A32 ; ::_thesis: verum
end;
suppose l <= k ; ::_thesis: ex z1, z2, z3 being Element of FS st
( FD . (x,z1) = a & FD . (z2,z3) = a & FD . (z1,z2) = b & FD . (z3,y) = b )
then  A2 c= A1 by A12, A24, Th38, A20, A8;
then reconsider x9 = x, y9 = y as Element of A1 by A4, A6, A8, A22, A12;
A35: ( x9 in A1 & y9 in A1 ) ;
 A1 c= Aq1 by A12, A13, Th38, A8, A9, NAT_1:11;
then reconsider x99 = x9, y99 = y9 as Element of Aq1 by A35;
 dom d1 = [:A1,A1:] by FUNCT_2:def_1;
then FD . (x,y) = d1 . [x9,y9] by A11, GRFUNC_1:2
.= d1 . (x9,y9) ;
then consider z1, z2, z3 being Element of Aq1 such that
A36: dq1 . (x,z1) = a and
A37: dq1 . (z2,z3) = a and
A38: dq1 . (z1,z2) = b and
A39: dq1 . (z3,y) = b by A3, A7, Th37;
A40: z3 in Aq1 ;
 ( z1 in Aq1 & z2 in Aq1 ) ;
then reconsider z19 = z1, z29 = z2, z39 = z3 as Element of FS by A14, A40;
take  z19 ; ::_thesis: ex z2, z3 being Element of FS st
( FD . (x,z19) = a & FD . (z2,z3) = a & FD . (z19,z2) = b & FD . (z3,y) = b )
take  z29 ; ::_thesis: ex z3 being Element of FS st
( FD . (x,z19) = a & FD . (z29,z3) = a & FD . (z19,z29) = b & FD . (z3,y) = b )
take  z39 ; ::_thesis: ( FD . (x,z19) = a & FD . (z29,z39) = a & FD . (z19,z29) = b & FD . (z39,y) = b )
A41: dom dq1 = [:Aq1,Aq1:] by FUNCT_2:def_1;
hence FD . (x,z19) = dq1 . [x99,z1] by A15, GRFUNC_1:2
.= a by A36 ;
::_thesis: ( FD . (z29,z39) = a & FD . (z19,z29) = b & FD . (z39,y) = b )
thus FD . (z29,z39) = dq1 . [z2,z3] by A15, A41, GRFUNC_1:2
.= a by A37 ; ::_thesis: ( FD . (z19,z29) = b & FD . (z39,y) = b )
thus FD . (z19,z29) = dq1 . [z1,z2] by A15, A41, GRFUNC_1:2
.= b by A38 ; ::_thesis: FD . (z39,y) = b
thus FD . (z39,y) = dq1 . [z3,y99] by A15, A41, GRFUNC_1:2
.= b by A39 ; ::_thesis: verum
end;
end;
end;

theorem Th43: :: LATTICE5:43
for L being lower-bounded LATTICE
for S being ExtensionSeq of the carrier of L, BasicDF L
for FS being non empty set
for FD being distance_function of FS,L
for f being Homomorphism of L,(EqRelLATT FS)
for x, y being Element of FS
for e1, e2 being Equivalence_Relation of FS
for x, y being set st f = alpha FD & FS = union  {  ((S . i) `1) where i is Element of NAT : verum  }  & FD = union  {  ((S . i) `2) where i is Element of NAT : verum  }  & e1 in the carrier of (Image f) & e2 in the carrier of (Image f) & [x,y] in e1 "\/" e2 holds
ex F being non empty FinSequence of FS st
( len F = 3 + 2 & x,y are_joint_by F,e1,e2 )
proof
let L be lower-bounded LATTICE; ::_thesis: for S being ExtensionSeq of the carrier of L, BasicDF L
for FS being non empty set
for FD being distance_function of FS,L
for f being Homomorphism of L,(EqRelLATT FS)
for x, y being Element of FS
for e1, e2 being Equivalence_Relation of FS
for x, y being set st f = alpha FD & FS = union  {  ((S . i) `1) where i is Element of NAT : verum  }  & FD = union  {  ((S . i) `2) where i is Element of NAT : verum  }  & e1 in the carrier of (Image f) & e2 in the carrier of (Image f) & [x,y] in e1 "\/" e2 holds
ex F being non empty FinSequence of FS st
( len F = 3 + 2 & x,y are_joint_by F,e1,e2 )
let S be ExtensionSeq of the carrier of L, BasicDF L; ::_thesis: for FS being non empty set
for FD being distance_function of FS,L
for f being Homomorphism of L,(EqRelLATT FS)
for x, y being Element of FS
for e1, e2 being Equivalence_Relation of FS
for x, y being set st f = alpha FD & FS = union  {  ((S . i) `1) where i is Element of NAT : verum  }  & FD = union  {  ((S . i) `2) where i is Element of NAT : verum  }  & e1 in the carrier of (Image f) & e2 in the carrier of (Image f) & [x,y] in e1 "\/" e2 holds
ex F being non empty FinSequence of FS st
( len F = 3 + 2 & x,y are_joint_by F,e1,e2 )
let FS be non empty set ; ::_thesis: for FD being distance_function of FS,L
for f being Homomorphism of L,(EqRelLATT FS)
for x, y being Element of FS
for e1, e2 being Equivalence_Relation of FS
for x, y being set st f = alpha FD & FS = union  {  ((S . i) `1) where i is Element of NAT : verum  }  & FD = union  {  ((S . i) `2) where i is Element of NAT : verum  }  & e1 in the carrier of (Image f) & e2 in the carrier of (Image f) & [x,y] in e1 "\/" e2 holds
ex F being non empty FinSequence of FS st
( len F = 3 + 2 & x,y are_joint_by F,e1,e2 )
let FD be distance_function of FS,L; ::_thesis: for f being Homomorphism of L,(EqRelLATT FS)
for x, y being Element of FS
for e1, e2 being Equivalence_Relation of FS
for x, y being set st f = alpha FD & FS = union  {  ((S . i) `1) where i is Element of NAT : verum  }  & FD = union  {  ((S . i) `2) where i is Element of NAT : verum  }  & e1 in the carrier of (Image f) & e2 in the carrier of (Image f) & [x,y] in e1 "\/" e2 holds
ex F being non empty FinSequence of FS st
( len F = 3 + 2 & x,y are_joint_by F,e1,e2 )
let f be Homomorphism of L,(EqRelLATT FS); ::_thesis: for x, y being Element of FS
for e1, e2 being Equivalence_Relation of FS
for x, y being set st f = alpha FD & FS = union  {  ((S . i) `1) where i is Element of NAT : verum  }  & FD = union  {  ((S . i) `2) where i is Element of NAT : verum  }  & e1 in the carrier of (Image f) & e2 in the carrier of (Image f) & [x,y] in e1 "\/" e2 holds
ex F being non empty FinSequence of FS st
( len F = 3 + 2 & x,y are_joint_by F,e1,e2 )
let x, y be Element of FS; ::_thesis: for e1, e2 being Equivalence_Relation of FS
for x, y being set st f = alpha FD & FS = union  {  ((S . i) `1) where i is Element of NAT : verum  }  & FD = union  {  ((S . i) `2) where i is Element of NAT : verum  }  & e1 in the carrier of (Image f) & e2 in the carrier of (Image f) & [x,y] in e1 "\/" e2 holds
ex F being non empty FinSequence of FS st
( len F = 3 + 2 & x,y are_joint_by F,e1,e2 )
let e1, e2 be Equivalence_Relation of FS; ::_thesis: for x, y being set st f = alpha FD & FS = union  {  ((S . i) `1) where i is Element of NAT : verum  }  & FD = union  {  ((S . i) `2) where i is Element of NAT : verum  }  & e1 in the carrier of (Image f) & e2 in the carrier of (Image f) & [x,y] in e1 "\/" e2 holds
ex F being non empty FinSequence of FS st
( len F = 3 + 2 & x,y are_joint_by F,e1,e2 )
let x, y be set ; ::_thesis: ( f = alpha FD & FS = union  {  ((S . i) `1) where i is Element of NAT : verum  }  & FD = union  {  ((S . i) `2) where i is Element of NAT : verum  }  & e1 in the carrier of (Image f) & e2 in the carrier of (Image f) & [x,y] in e1 "\/" e2 implies ex F being non empty FinSequence of FS st
( len F = 3 + 2 & x,y are_joint_by F,e1,e2 ) )
assume that
A1: f = alpha FD and
A2: ( FS = union  {  ((S . i) `1) where i is Element of NAT : verum  }  & FD = union  {  ((S . i) `2) where i is Element of NAT : verum  }  ) and
A3: e1 in the carrier of (Image f) and
A4: e2 in the carrier of (Image f) and
A5: [x,y] in e1 "\/" e2 ; ::_thesis: ex F being non empty FinSequence of FS st
( len F = 3 + 2 & x,y are_joint_by F,e1,e2 )
A6: 4 in Seg 5 ;
 field (e1 "\/" e2) = FS by ORDERS_1:12;
then reconsider u = x, v = y as Element of FS by A5, RELAT_1:15;
A7: 1 in Seg 5 ;
 Image f = subrelstr (rng f) by YELLOW_2:def_2;
then A8: the carrier of (Image f) = rng f by YELLOW_0:def_15;
then consider a being set  such that
A9: a in dom f and
A10: e1 = f . a by A3, FUNCT_1:def_3;
consider b being set  such that
A11: b in dom f and
A12: e2 = f . b by A4, A8, FUNCT_1:def_3;
reconsider a = a, b = b as Element of L by A9, A11;
reconsider a = a, b = b as Element of L ;
consider e being Equivalence_Relation of FS such that
A13: e = f . (a "\/" b) and
A14: for u, v being Element of FS holds
( [u,v] in e iff FD . (u,v) <= a "\/" b ) by A1, Def8;
consider e19 being Equivalence_Relation of FS such that
A15: e19 = f . a and
A16: for u, v being Element of FS holds
( [u,v] in e19 iff FD . (u,v) <= a ) by A1, Def8;
consider e29 being Equivalence_Relation of FS such that
A17: e29 = f . b and
A18: for u, v being Element of FS holds
( [u,v] in e29 iff FD . (u,v) <= b ) by A1, Def8;
A19: 3 in Seg 5 ;
e = (f . a) "\/" (f . b) by A13, WAYBEL_6:2
.= e1 "\/" e2 by A10, A12, Th10 ;
then  FD . (u,v) <= a "\/" b by A5, A14;
then consider z1, z2, z3 being Element of FS such that
A20: FD . (u,z1) = a and
A21: FD . (z2,z3) = a and
A22: FD . (z1,z2) = b and
A23: FD . (z3,v) = b by A2, Th42;
defpred S1[ set , set ] means ( ( $1 = 1 implies $2 = u ) & ( $1 = 2 implies $2 = z1 ) & ( $1 = 3 implies $2 = z2 ) & ( $1 = 4 implies $2 = z3 ) & ( $1 = 5 implies $2 = v ) );
A24: for m being Nat st m in Seg 5 holds
ex w being set st S1[m,w]
proof
let m be Nat; ::_thesis: ( m in Seg 5 implies ex w being set st S1[m,w] )
assume A25: m in Seg 5 ; ::_thesis: ex w being set st S1[m,w]
percases ( m = 1 or m = 2 or m = 3 or m = 4 or m = 5 ) by A25, Lm3;
supposeA26: m = 1 ; ::_thesis: ex w being set st S1[m,w]
take  x ; ::_thesis: S1[m,x]
thus  S1[m,x] by A26; ::_thesis: verum
end;
supposeA27: m = 2 ; ::_thesis: ex w being set st S1[m,w]
take  z1 ; ::_thesis: S1[m,z1]
thus  S1[m,z1] by A27; ::_thesis: verum
end;
supposeA28: m = 3 ; ::_thesis: ex w being set st S1[m,w]
take  z2 ; ::_thesis: S1[m,z2]
thus  S1[m,z2] by A28; ::_thesis: verum
end;
supposeA29: m = 4 ; ::_thesis: ex w being set st S1[m,w]
take  z3 ; ::_thesis: S1[m,z3]
thus  S1[m,z3] by A29; ::_thesis: verum
end;
supposeA30: m = 5 ; ::_thesis: ex w being set st S1[m,w]
take  y ; ::_thesis: S1[m,y]
thus  S1[m,y] by A30; ::_thesis: verum
end;
end;
end;
 ex p being FinSequence st
( dom p = Seg 5 & ( for k being Nat st k in Seg 5 holds
S1[k,p . k] ) ) from FINSEQ_1:sch_1(A24);
then consider h being FinSequence such that
A31: dom h = Seg 5 and
A32: for m being Nat st m in Seg 5 holds
( ( m = 1 implies h . m = u ) & ( m = 2 implies h . m = z1 ) & ( m = 3 implies h . m = z2 ) & ( m = 4 implies h . m = z3 ) & ( m = 5 implies h . m = v ) ) ;
A33: len h = 5 by A31, FINSEQ_1:def_3;
A34: 5 in Seg 5 ;
A35: 2 in Seg 5 ;
 rng h c= FS
proof
let w be set ; :: according to TARSKI:def_3 ::_thesis: ( not w in rng h or w in FS )
assume  w in rng h ; ::_thesis: w in FS
then consider j being set  such that
A36: j in dom h and
A37: w = h . j by FUNCT_1:def_3;
percases ( j = 1 or j = 2 or j = 3 or j = 4 or j = 5 ) by A31, A36, Lm3;
suppose j = 1 ; ::_thesis: w in FS
then  h . j = u by A32, A7;
hence  w in FS by A37; ::_thesis: verum
end;
suppose j = 2 ; ::_thesis: w in FS
then  h . j = z1 by A32, A35;
hence  w in FS by A37; ::_thesis: verum
end;
suppose j = 3 ; ::_thesis: w in FS
then  h . j = z2 by A32, A19;
hence  w in FS by A37; ::_thesis: verum
end;
suppose j = 4 ; ::_thesis: w in FS
then  h . j = z3 by A32, A6;
hence  w in FS by A37; ::_thesis: verum
end;
suppose j = 5 ; ::_thesis: w in FS
then  h . j = v by A32, A34;
hence  w in FS by A37; ::_thesis: verum
end;
end;
end;
then reconsider h = h as FinSequence of FS by FINSEQ_1:def_4;
reconsider h = h as non empty FinSequence of FS by A31;
A38: h . 1 = x by A32, A7;
A39: for j being Element of NAT st 1 <= j & j < len h holds
( ( j is odd implies [(h . j),(h . (j + 1))] in e1 ) & ( j is even implies [(h . j),(h . (j + 1))] in e2 ) )
proof
let j be Element of NAT ; ::_thesis: ( 1 <= j & j < len h implies ( ( j is odd implies [(h . j),(h . (j + 1))] in e1 ) & ( j is even implies [(h . j),(h . (j + 1))] in e2 ) ) )
assume A40: ( 1 <= j & j < len h ) ; ::_thesis: ( ( j is odd implies [(h . j),(h . (j + 1))] in e1 ) & ( j is even implies [(h . j),(h . (j + 1))] in e2 ) )
percases ( j = 1 or j = 3 or j = 2 or j = 4 ) by A33, A40, Lm2;
supposeA41: j = 1 ; ::_thesis: ( ( j is odd implies [(h . j),(h . (j + 1))] in e1 ) & ( j is even implies [(h . j),(h . (j + 1))] in e2 ) )
 [u,z1] in e19 by A16, A20;
then  [(h . 1),z1] in e19 by A32, A7;
hence  ( ( j is odd implies [(h . j),(h . (j + 1))] in e1 ) & ( j is even implies [(h . j),(h . (j + 1))] in e2 ) ) by A10, A15, A32, A35, A41; ::_thesis: verum
end;
supposeA42: j = 3 ; ::_thesis: ( ( j is odd implies [(h . j),(h . (j + 1))] in e1 ) & ( j is even implies [(h . j),(h . (j + 1))] in e2 ) )
 [z2,z3] in e19 by A16, A21;
then A43: [(h . 3),z3] in e19 by A32, A19;
 (2 * 1) + 1 = j by A42;
hence  ( ( j is odd implies [(h . j),(h . (j + 1))] in e1 ) & ( j is even implies [(h . j),(h . (j + 1))] in e2 ) ) by A10, A15, A32, A6, A43; ::_thesis: verum
end;
supposeA44: j = 2 ; ::_thesis: ( ( j is odd implies [(h . j),(h . (j + 1))] in e1 ) & ( j is even implies [(h . j),(h . (j + 1))] in e2 ) )
 [z1,z2] in e29 by A18, A22;
then A45: [(h . 2),z2] in e29 by A32, A35;
 2 * 1 = j by A44;
hence  ( ( j is odd implies [(h . j),(h . (j + 1))] in e1 ) & ( j is even implies [(h . j),(h . (j + 1))] in e2 ) ) by A12, A17, A32, A19, A45; ::_thesis: verum
end;
supposeA46: j = 4 ; ::_thesis: ( ( j is odd implies [(h . j),(h . (j + 1))] in e1 ) & ( j is even implies [(h . j),(h . (j + 1))] in e2 ) )
 [z3,v] in e29 by A18, A23;
then A47: [(h . 4),v] in e29 by A32, A6;
 2 * 2 = j by A46;
hence  ( ( j is odd implies [(h . j),(h . (j + 1))] in e1 ) & ( j is even implies [(h . j),(h . (j + 1))] in e2 ) ) by A12, A17, A32, A34, A47; ::_thesis: verum
end;
end;
end;
take  h ; ::_thesis: ( len h = 3 + 2 & x,y are_joint_by h,e1,e2 )
thus  len h = 3 + 2 by A31, FINSEQ_1:def_3; ::_thesis: x,y are_joint_by h,e1,e2
h . (len h) = h . 5 by A31, FINSEQ_1:def_3
.= y by A32, A34 ;
hence  x,y are_joint_by h,e1,e2 by A38, A39, Def3; ::_thesis: verum
end;

theorem :: LATTICE5:44
for L being lower-bounded LATTICE ex A being non empty set ex f being Homomorphism of L,(EqRelLATT A) st
( f is one-to-one & type_of (Image f) <= 3 )
proof
let L be lower-bounded LATTICE; ::_thesis: ex A being non empty set ex f being Homomorphism of L,(EqRelLATT A) st
( f is one-to-one & type_of (Image f) <= 3 )
set A = the carrier of L;
set D = BasicDF L;
set S = the ExtensionSeq of the carrier of L, BasicDF L;
set FS = union  {  (( the ExtensionSeq of the carrier of L, BasicDF L . i) `1) where i is Element of NAT : verum  }  ;
A1: ( the ExtensionSeq of the carrier of L, BasicDF L . 0) `1 in  {  (( the ExtensionSeq of the carrier of L, BasicDF L . i) `1) where i is Element of NAT : verum  }  ;
A2: the ExtensionSeq of the carrier of L, BasicDF L . 0 = [ the carrier of L,(BasicDF L)] by Def20;
 [ the carrier of L,(BasicDF L)] `1 = the carrier of L ;
then  the carrier of L c= union  {  (( the ExtensionSeq of the carrier of L, BasicDF L . i) `1) where i is Element of NAT : verum  }  by A1, A2, ZFMISC_1:74;
then reconsider FS = union  {  (( the ExtensionSeq of the carrier of L, BasicDF L . i) `1) where i is Element of NAT : verum  }  as non empty set ;
reconsider FD = union  {  (( the ExtensionSeq of the carrier of L, BasicDF L . i) `2) where i is Element of NAT : verum  }  as distance_function of FS,L by Th41;
 alpha FD is join-preserving
proof
set f = alpha FD;
let a, b be Element of L; :: according to WAYBEL_0:def_35 ::_thesis: alpha FD preserves_sup_of {a,b}
A3: ex_sup_of (alpha FD) .: {a,b}, EqRelLATT FS by YELLOW_0:17;
consider e3 being Equivalence_Relation of FS such that
A4: e3 = (alpha FD) . (a "\/" b) and
A5: for x, y being Element of FS holds
( [x,y] in e3 iff FD . (x,y) <= a "\/" b ) by Def8;
consider e2 being Equivalence_Relation of FS such that
A6: e2 = (alpha FD) . b and
A7: for x, y being Element of FS holds
( [x,y] in e2 iff FD . (x,y) <= b ) by Def8;
consider e1 being Equivalence_Relation of FS such that
A8: e1 = (alpha FD) . a and
A9: for x, y being Element of FS holds
( [x,y] in e1 iff FD . (x,y) <= a ) by Def8;
A10: field e2 = FS by ORDERS_1:12;
now__::_thesis:_for_x,_y_being_set_st_[x,y]_in_e2_holds_
[x,y]_in_e3
let x, y be set ; ::_thesis: ( [x,y] in e2 implies [x,y] in e3 )
A11: b <= b "\/" a by YELLOW_0:22;
assume A12: [x,y] in e2 ; ::_thesis: [x,y] in e3
then reconsider x9 = x, y9 = y as Element of FS by A10, RELAT_1:15;
 FD . (x9,y9) <= b by A7, A12;
then  FD . (x9,y9) <= b "\/" a by A11, ORDERS_2:3;
hence  [x,y] in e3 by A5; ::_thesis: verum
end;
then A13: e2 c= e3 by RELAT_1:def_3;
A14: field e3 = FS by ORDERS_1:12;
 for u, v being set st [u,v] in e3 holds
[u,v] in e1 "\/" e2
proof
let u, v be set ; ::_thesis: ( [u,v] in e3 implies [u,v] in e1 "\/" e2 )
A15: 3 in Seg 5 ;
assume A16: [u,v] in e3 ; ::_thesis: [u,v] in e1 "\/" e2
then reconsider x = u, y = v as Element of FS by A14, RELAT_1:15;
 FD . (x,y) <= a "\/" b by A5, A16;
then consider z1, z2, z3 being Element of FS such that
A17: FD . (x,z1) = a and
A18: FD . (z2,z3) = a and
A19: FD . (z1,z2) = b and
A20: FD . (z3,y) = b by Th42;
A21: u in FS by A14, A16, RELAT_1:15;
defpred S1[ set , set ] means ( ( $1 = 1 implies $2 = x ) & ( $1 = 2 implies $2 = z1 ) & ( $1 = 3 implies $2 = z2 ) & ( $1 = 4 implies $2 = z3 ) & ( $1 = 5 implies $2 = y ) );
A22: for m being Nat st m in Seg 5 holds
ex w being set st S1[m,w]
proof
let m be Nat; ::_thesis: ( m in Seg 5 implies ex w being set st S1[m,w] )
assume A23: m in Seg 5 ; ::_thesis: ex w being set st S1[m,w]
percases ( m = 1 or m = 2 or m = 3 or m = 4 or m = 5 ) by A23, Lm3;
supposeA24: m = 1 ; ::_thesis: ex w being set st S1[m,w]
take  x ; ::_thesis: S1[m,x]
thus  S1[m,x] by A24; ::_thesis: verum
end;
supposeA25: m = 2 ; ::_thesis: ex w being set st S1[m,w]
take  z1 ; ::_thesis: S1[m,z1]
thus  S1[m,z1] by A25; ::_thesis: verum
end;
supposeA26: m = 3 ; ::_thesis: ex w being set st S1[m,w]
take  z2 ; ::_thesis: S1[m,z2]
thus  S1[m,z2] by A26; ::_thesis: verum
end;
supposeA27: m = 4 ; ::_thesis: ex w being set st S1[m,w]
take  z3 ; ::_thesis: S1[m,z3]
thus  S1[m,z3] by A27; ::_thesis: verum
end;
supposeA28: m = 5 ; ::_thesis: ex w being set st S1[m,w]
take  y ; ::_thesis: S1[m,y]
thus  S1[m,y] by A28; ::_thesis: verum
end;
end;
end;
 ex p being FinSequence st
( dom p = Seg 5 & ( for k being Nat st k in Seg 5 holds
S1[k,p . k] ) ) from FINSEQ_1:sch_1(A22);
then consider h being FinSequence such that
A29: dom h = Seg 5 and
A30: for m being Nat st m in Seg 5 holds
( ( m = 1 implies h . m = x ) & ( m = 2 implies h . m = z1 ) & ( m = 3 implies h . m = z2 ) & ( m = 4 implies h . m = z3 ) & ( m = 5 implies h . m = y ) ) ;
A31: len h = 5 by A29, FINSEQ_1:def_3;
A32: 5 in Seg 5 ;
A33: 4 in Seg 5 ;
A34: 1 in Seg 5 ;
then A35: u = h . 1 by A30;
A36: 2 in Seg 5 ;
A37: for j being Element of NAT st 1 <= j & j < len h holds
[(h . j),(h . (j + 1))] in e1 \/ e2
proof
let j be Element of NAT ; ::_thesis: ( 1 <= j & j < len h implies [(h . j),(h . (j + 1))] in e1 \/ e2 )
assume A38: ( 1 <= j & j < len h ) ; ::_thesis: [(h . j),(h . (j + 1))] in e1 \/ e2
percases ( j = 1 or j = 3 or j = 2 or j = 4 ) by A31, A38, Lm2;
supposeA39: j = 1 ; ::_thesis: [(h . j),(h . (j + 1))] in e1 \/ e2
 [x,z1] in e1 by A9, A17;
then  [(h . 1),z1] in e1 by A30, A34;
then  [(h . 1),(h . 2)] in e1 by A30, A36;
hence  [(h . j),(h . (j + 1))] in e1 \/ e2 by A39, XBOOLE_0:def_3; ::_thesis: verum
end;
supposeA40: j = 3 ; ::_thesis: [(h . j),(h . (j + 1))] in e1 \/ e2
 [z2,z3] in e1 by A9, A18;
then  [(h . 3),z3] in e1 by A30, A15;
then  [(h . 3),(h . 4)] in e1 by A30, A33;
hence  [(h . j),(h . (j + 1))] in e1 \/ e2 by A40, XBOOLE_0:def_3; ::_thesis: verum
end;
supposeA41: j = 2 ; ::_thesis: [(h . j),(h . (j + 1))] in e1 \/ e2
 [z1,z2] in e2 by A7, A19;
then  [(h . 2),z2] in e2 by A30, A36;
then  [(h . 2),(h . 3)] in e2 by A30, A15;
hence  [(h . j),(h . (j + 1))] in e1 \/ e2 by A41, XBOOLE_0:def_3; ::_thesis: verum
end;
supposeA42: j = 4 ; ::_thesis: [(h . j),(h . (j + 1))] in e1 \/ e2
 [z3,y] in e2 by A7, A20;
then  [(h . 4),y] in e2 by A30, A33;
then  [(h . 4),(h . 5)] in e2 by A30, A32;
hence  [(h . j),(h . (j + 1))] in e1 \/ e2 by A42, XBOOLE_0:def_3; ::_thesis: verum
end;
end;
end;
v = h . 5 by A30, A32
.= h . (len h) by A29, FINSEQ_1:def_3 ;
hence  [u,v] in e1 "\/" e2 by A21, A31, A35, A37, EQREL_1:28; ::_thesis: verum
end;
then A43: e3 c= e1 "\/" e2 by RELAT_1:def_3;
A44: field e1 = FS by ORDERS_1:12;
now__::_thesis:_for_x,_y_being_set_st_[x,y]_in_e1_holds_
[x,y]_in_e3
let x, y be set ; ::_thesis: ( [x,y] in e1 implies [x,y] in e3 )
A45: a <= a "\/" b by YELLOW_0:22;
assume A46: [x,y] in e1 ; ::_thesis: [x,y] in e3
then reconsider x9 = x, y9 = y as Element of FS by A44, RELAT_1:15;
 FD . (x9,y9) <= a by A9, A46;
then  FD . (x9,y9) <= a "\/" b by A45, ORDERS_2:3;
hence  [x,y] in e3 by A5; ::_thesis: verum
end;
then  e1 c= e3 by RELAT_1:def_3;
then  e1 \/ e2 c= e3 by A13, XBOOLE_1:8;
then A47: e1 "\/" e2 c= e3 by EQREL_1:def_2;
 dom (alpha FD) = the carrier of L by FUNCT_2:def_1;
then  sup ((alpha FD) .: {a,b}) = sup {((alpha FD) . a),((alpha FD) . b)} by FUNCT_1:60
.= ((alpha FD) . a) "\/" ((alpha FD) . b) by YELLOW_0:41
.= e1 "\/" e2 by A8, A6, Th10
.= (alpha FD) . (a "\/" b) by A4, A47, A43, XBOOLE_0:def_10
.= (alpha FD) . (sup {a,b}) by YELLOW_0:41 ;
hence  alpha FD preserves_sup_of {a,b} by A3, WAYBEL_0:def_31; ::_thesis: verum
end;
then reconsider f = alpha FD as Homomorphism of L,(EqRelLATT FS) by Th14;
A48: dom f = the carrier of L by FUNCT_2:def_1;
A49: Image f = subrelstr (rng f) by YELLOW_2:def_2;
A50: ex e being Equivalence_Relation of FS st
( e in the carrier of (Image f) & e <> id FS )
proof
A51: { the carrier of L} <> {{ the carrier of L}}
proof
assume  { the carrier of L} = {{ the carrier of L}} ; ::_thesis: contradiction
then  { the carrier of L} in { the carrier of L} by TARSKI:def_1;
hence  contradiction ; ::_thesis: verum
end;
consider A9 being non empty set , d9 being distance_function of A9,L, Aq9 being non empty set , dq9 being distance_function of Aq9,L such that
A52: Aq9,dq9 is_extension_of A9,d9 and
A53: the ExtensionSeq of the carrier of L, BasicDF L . 0 = [A9,d9] and
A54: the ExtensionSeq of the carrier of L, BasicDF L . (0 + 1) = [Aq9,dq9] by Def20;
 ( A9 = the carrier of L & d9 = BasicDF L ) by A2, A53, XTUPLE_0:1;
then consider q being QuadrSeq of BasicDF L such that
A55: Aq9 = NextSet (BasicDF L) and
A56: dq9 = NextDelta q by A52, Def19;
 ConsecutiveSet ( the carrier of L,{}) = the carrier of L by Th21;
then reconsider Q = Quadr (q,{}) as Element of [: the carrier of L, the carrier of L, the carrier of L, the carrier of L:] ;
A57: ( the ExtensionSeq of the carrier of L, BasicDF L . 1) `2 in  {  (( the ExtensionSeq of the carrier of L, BasicDF L . i) `2) where i is Element of NAT : verum  }  ;
 succ {} c= DistEsti (BasicDF L) by Lm4;
then  {} in DistEsti (BasicDF L) by ORDINAL1:21;
then A58: {} in dom q by Th25;
then  q . {} in rng q by FUNCT_1:def_3;
then  q . {} in  {  [u,v,a9,b9] where u, v is Element of the carrier of L, a9, b9 is Element of L : (BasicDF L) . (u,v) <= a9 "\/" b9  }  by Def13;
then consider u, v being Element of the carrier of L, a, b being Element of L such that
A59: q . {} = [u,v,a,b] and
 (BasicDF L) . (u,v) <= a "\/" b ;
consider e being Equivalence_Relation of FS such that
A60: e = f . b and
A61: for x, y being Element of FS holds
( [x,y] in e iff FD . (x,y) <= b ) by Def8;
A62: Quadr (q,{}) = [u,v,a,b] by A58, A59, Def14;
 [Aq9,dq9] `2 = NextDelta q by A56;
then A63: NextDelta q c= FD by A57, A54, ZFMISC_1:74;
A64: {{ the carrier of L}} in {{ the carrier of L},{{ the carrier of L}},{{{ the carrier of L}}}} by ENUMSET1:def_1;
then A65: {{ the carrier of L}} in the carrier of L \/ {{ the carrier of L},{{ the carrier of L}},{{{ the carrier of L}}}} by XBOOLE_0:def_3;
take  e ; ::_thesis: ( e in the carrier of (Image f) & e <> id FS )
 e in rng f by A48, A60, FUNCT_1:def_3;
hence  e in the carrier of (Image f) by A49, YELLOW_0:def_15; ::_thesis: e <> id FS
A66: ( the ExtensionSeq of the carrier of L, BasicDF L . 1) `1 in  {  (( the ExtensionSeq of the carrier of L, BasicDF L . i) `1) where i is Element of NAT : verum  }  ;
 [Aq9,dq9] `1 = NextSet (BasicDF L) by A55;
then A67: NextSet (BasicDF L) c= FS by A66, A54, ZFMISC_1:74;
new_set the carrier of L = new_set (ConsecutiveSet ( the carrier of L,{})) by Th21
.= ConsecutiveSet ( the carrier of L,(succ {})) by Th22 ;
then  new_set the carrier of L c= NextSet (BasicDF L) by Lm4, Th29;
then A68: new_set the carrier of L c= FS by A67, XBOOLE_1:1;
A69: {{ the carrier of L}} in new_set the carrier of L by A64, XBOOLE_0:def_3;
A70: { the carrier of L} in {{ the carrier of L},{{ the carrier of L}},{{{ the carrier of L}}}} by ENUMSET1:def_1;
then  { the carrier of L} in the carrier of L \/ {{ the carrier of L},{{ the carrier of L}},{{{ the carrier of L}}}} by XBOOLE_0:def_3;
then reconsider W = { the carrier of L}, V = {{ the carrier of L}} as Element of FS by A68, A69;
A71: ( ConsecutiveSet ( the carrier of L,{}) = the carrier of L & ConsecutiveDelta (q,{}) = BasicDF L ) by Th21, Th26;
ConsecutiveDelta (q,(succ {})) = new_bi_fun ((BiFun ((ConsecutiveDelta (q,{})),(ConsecutiveSet ( the carrier of L,{})),L)),(Quadr (q,{}))) by Th27
.= new_bi_fun ((BasicDF L),Q) by A71, Def15 ;
then  new_bi_fun ((BasicDF L),Q) c= NextDelta q by Lm4, Th32;
then A72: new_bi_fun ((BasicDF L),Q) c= FD by A63, XBOOLE_1:1;
 ( dom (new_bi_fun ((BasicDF L),Q)) = [:(new_set the carrier of L),(new_set the carrier of L):] & { the carrier of L} in new_set the carrier of L ) by A70, FUNCT_2:def_1, XBOOLE_0:def_3;
then  [{ the carrier of L},{{ the carrier of L}}] in dom (new_bi_fun ((BasicDF L),Q)) by A65, ZFMISC_1:87;
then FD . (W,V) = (new_bi_fun ((BasicDF L),Q)) . ({ the carrier of L},{{ the carrier of L}}) by A72, GRFUNC_1:2
.= Q `4_4 by Def10
.= b by A62, MCART_1:def_11 ;
then  [{ the carrier of L},{{ the carrier of L}}] in e by A61;
hence  e <> id FS by A51, RELAT_1:def_10; ::_thesis: verum
end;
take  FS ; ::_thesis: ex f being Homomorphism of L,(EqRelLATT FS) st
( f is one-to-one & type_of (Image f) <= 3 )
take  f ; ::_thesis: ( f is one-to-one & type_of (Image f) <= 3 )
 BasicDF L is onto by Th40;
then A73: rng (BasicDF L) = the carrier of L by FUNCT_2:def_3;
 for w being set st w in the carrier of L holds
ex z being set st
( z in [:FS,FS:] & w = FD . z )
proof
let w be set ; ::_thesis: ( w in the carrier of L implies ex z being set st
( z in [:FS,FS:] & w = FD . z ) )
A74: ( the ExtensionSeq of the carrier of L, BasicDF L . 0) `1 in  {  (( the ExtensionSeq of the carrier of L, BasicDF L . i) `1) where i is Element of NAT : verum  }  ;
A75: ( the ExtensionSeq of the carrier of L, BasicDF L . 0) `2 in  {  (( the ExtensionSeq of the carrier of L, BasicDF L . i) `2) where i is Element of NAT : verum  }  ;
A76: the ExtensionSeq of the carrier of L, BasicDF L . 0 = [ the carrier of L,(BasicDF L)] by Def20;
 BasicDF L = [ the carrier of L,(BasicDF L)] `2 ;
then A77: BasicDF L c= FD by A75, A76, ZFMISC_1:74;
assume  w in the carrier of L ; ::_thesis: ex z being set st
( z in [:FS,FS:] & w = FD . z )
then consider z being set  such that
A78: z in [: the carrier of L, the carrier of L:] and
A79: (BasicDF L) . z = w by A73, FUNCT_2:11;
take  z ; ::_thesis: ( z in [:FS,FS:] & w = FD . z )
 the carrier of L = [ the carrier of L,(BasicDF L)] `1 ;
then  the carrier of L c= FS by A74, A76, ZFMISC_1:74;
then  [: the carrier of L, the carrier of L:] c= [:FS,FS:] by ZFMISC_1:96;
hence  z in [:FS,FS:] by A78; ::_thesis: w = FD . z
 z in dom (BasicDF L) by A78, FUNCT_2:def_1;
hence  w = FD . z by A79, A77, GRFUNC_1:2; ::_thesis: verum
end;
then  rng FD = the carrier of L by FUNCT_2:10;
then  FD is onto by FUNCT_2:def_3;
hence  f is one-to-one by Th15; ::_thesis: type_of (Image f) <= 3
 for e1, e2 being Equivalence_Relation of FS
for x, y being set st e1 in the carrier of (Image f) & e2 in the carrier of (Image f) & [x,y] in e1 "\/" e2 holds
ex F being non empty FinSequence of FS st
( len F = 3 + 2 & x,y are_joint_by F,e1,e2 ) by Th43;
hence  type_of (Image f) <= 3 by A50, Th13; ::_thesis: verum
end;