latquasi.miz
begin
reserve L for non
empty
LattStr;
reserve v3,v101,v100,v102,v103,v2,v1,v0 for
Element of L;
definition
let L be non
empty
LattStr;
::
LATQUASI:def1
attr L is
satisfying_QLT1 means for v0,v2,v1 be
Element of L holds ((v0
"/\" (v1
"\/" v2))
"\/" (v0
"/\" v1))
= (v0
"/\" (v1
"\/" v2));
::
LATQUASI:def2
attr L is
satisfying_QLT2 means for v0,v2,v1 be
Element of L holds ((v0
"\/" (v1
"/\" v2))
"/\" (v0
"\/" v1))
= (v0
"\/" (v1
"/\" v2));
::
LATQUASI:def3
attr L is
QLT-distributive means for v1,v2,v0 be
Element of L holds (v0
"/\" (v1
"\/" (v0
"/\" v2)))
= (v0
"/\" (v1
"\/" v2));
end
registration
cluster
trivial ->
satisfying_QLT1
satisfying_QLT2
QLT-distributive for non
empty
LattStr;
coherence by
STRUCT_0:def 10;
end
registration
cluster
trivial ->
join-idempotent
meet-idempotent for non
empty
LattStr;
coherence
proof
let L be non
empty
LattStr;
assume
A1: L is
trivial;
then for x be
Element of L holds (x
"\/" x)
= x by
STRUCT_0:def 10;
hence L is
join-idempotent by
ROBBINS1:def 7;
for x be
Element of L holds (x
"/\" x)
= x by
A1,
STRUCT_0:def 10;
hence thesis by
SHEFFER1:def 9;
end;
end
registration
cluster
join-commutative
join-associative
join-idempotent
meet-commutative
meet-associative
meet-idempotent
satisfying_QLT1
satisfying_QLT2 for non
empty
LattStr;
existence
proof
take the
trivial
Lattice;
thus thesis;
end;
end
definition
let L be
join-commutative non
empty
LattStr;
:: original:
satisfying_QLT1
redefine
::
LATQUASI:def4
attr L is
satisfying_QLT1 means for v0,v1,v2 be
Element of L holds (v0
"/\" v1)
[= (v0
"/\" (v1
"\/" v2));
compatibility
proof
thus L is
satisfying_QLT1 implies for v0,v1,v2 be
Element of L holds (v0
"/\" v1)
[= (v0
"/\" (v1
"\/" v2))
proof
assume
A1: L is
satisfying_QLT1;
let v0,v1,v2 be
Element of L;
((v0
"/\" (v1
"\/" v2))
"\/" (v0
"/\" v1))
= (v0
"/\" (v1
"\/" v2)) by
A1;
hence thesis by
LATTICES:def 3;
end;
assume
B1: for v0,v1,v2 be
Element of L holds (v0
"/\" v1)
[= (v0
"/\" (v1
"\/" v2));
let v0,v2,v1 be
Element of L;
((v0
"/\" (v1
"\/" v2))
"\/" (v0
"/\" v1))
= (v0
"/\" (v1
"\/" v2)) by
B1,
LATTICES:def 3;
hence thesis;
end;
end
registration
cluster
{
0 , 1, 2} ->
real-membered;
coherence ;
end
registration
cluster ->
real for
Element of
{
0 , 1, 2};
coherence ;
end
T2: for x,y be
Element of
{
0 , 1, 2} holds (
min (x,y))
in
{
0 , 1, 2}
proof
let x,y be
Element of
{
0 , 1, 2};
(
min (x,y))
= x or (
min (x,y))
= y by
XXREAL_0: 15;
hence thesis;
end;
definition
let x,y be
Element of
{
0 , 1, 2};
::
LATQUASI:def5
func
OpEx2 (x,y) ->
Element of
{
0 , 1, 2} equals
:
OpEx2Def: 1 if x
= 1 or y
= 1,
(
min (x,y)) if x
<> 1 & y
<> 1;
coherence by
T2;
consistency ;
end
definition
::
LATQUASI:def6
func
QLT_Ex1 ->
BinOp of
{
0 , 1, 2} means
:
QLTEx1Def: for x,y be
Element of
{
0 , 1, 2} holds (x
= y implies (it
. (x,y))
= x) & (x
<> y implies (it
. (x,y))
=
0 );
existence
proof
deffunc
X() =
{
0 , 1, 2};
deffunc
F(
object,
object) = (
In ((
IFEQ ($1,$2,$1,
0 )),
X()));
ex f be
BinOp of
X() st for x be
Element of
X() holds for y be
Element of
X() holds (f
. (x,y))
=
F(x,y) from
BINOP_1:sch 4;
then
consider f be
BinOp of
X() such that
A1: for x be
Element of
X() holds for y be
Element of
X() holds (f
. (x,y))
=
F(x,y);
take f;
let x,y be
Element of
{
0 , 1, 2};
A4:
0
in
X() by
ENUMSET1:def 1;
hereby
assume
A2: x
= y;
thus (f
. (x,y))
=
F(x,y) by
A1
.= (
In (x,
X())) by
A2,
FUNCOP_1:def 8
.= x;
end;
assume
A3: x
<> y;
thus (f
. (x,y))
=
F(x,y) by
A1
.= (
In (
0 ,
X())) by
A3,
FUNCOP_1:def 8
.=
0 by
SUBSET_1:def 8,
A4;
end;
uniqueness
proof
let f1,f2 be
BinOp of
{
0 , 1, 2} such that
A1: for x,y be
Element of
{
0 , 1, 2} holds (x
= y implies (f1
. (x,y))
= x) & (x
<> y implies (f1
. (x,y))
=
0 ) and
A2: for x,y be
Element of
{
0 , 1, 2} holds (x
= y implies (f2
. (x,y))
= x) & (x
<> y implies (f2
. (x,y))
=
0 );
for x,y be
Element of
{
0 , 1, 2} holds (f1
. (x,y))
= (f2
. (x,y))
proof
let x,y be
Element of
{
0 , 1, 2};
per cases ;
suppose
A3: x
= y;
then (f1
. (x,y))
= x by
A1
.= (f2
. (x,y)) by
A2,
A3;
hence thesis;
end;
suppose
A3: x
<> y;
then (f1
. (x,y))
=
0 by
A1
.= (f2
. (x,y)) by
A2,
A3;
hence thesis;
end;
end;
hence thesis by
BINOP_1: 2;
end;
::
LATQUASI:def7
func
QLT_Ex2 ->
BinOp of
{
0 , 1, 2} means
:
QLTEx2Def: for x,y be
Element of
{
0 , 1, 2} holds (x
= 1 or y
= 1 implies (it
. (x,y))
= 1) & (x
<> 1 & y
<> 1 implies (it
. (x,y))
= (
min (x,y)));
existence
proof
deffunc
X() =
{
0 , 1, 2};
deffunc
F(
Element of
X(),
Element of
X()) = (
OpEx2 ($1,$2));
ex f be
BinOp of
X() st for x,y be
Element of
X() holds (f
. (x,y))
=
F(x,y) from
BINOP_1:sch 4;
then
consider f be
BinOp of
X() such that
A1: for x,y be
Element of
X() holds (f
. (x,y))
=
F(x,y);
take f;
let x,y be
Element of
{
0 , 1, 2};
hereby
assume
A2: x
= 1 or y
= 1;
(f
. (x,y))
= (
OpEx2 (x,y)) by
A1;
hence (f
. (x,y))
= 1 by
A2,
OpEx2Def;
end;
assume
A2: x
<> 1 & y
<> 1;
(f
. (x,y))
= (
OpEx2 (x,y)) by
A1;
hence thesis by
OpEx2Def,
A2;
end;
uniqueness
proof
let f1,f2 be
BinOp of
{
0 , 1, 2} such that
A1: for x,y be
Element of
{
0 , 1, 2} holds (x
= 1 or y
= 1 implies (f1
. (x,y))
= 1) & (x
<> 1 & y
<> 1 implies (f1
. (x,y))
= (
min (x,y))) and
A2: for x,y be
Element of
{
0 , 1, 2} holds (x
= 1 or y
= 1 implies (f2
. (x,y))
= 1) & (x
<> 1 & y
<> 1 implies (f2
. (x,y))
= (
min (x,y)));
for x,y be
Element of
{
0 , 1, 2} holds (f1
. (x,y))
= (f2
. (x,y))
proof
let x,y be
Element of
{
0 , 1, 2};
per cases ;
suppose
A3: x
= 1 or y
= 1;
then (f1
. (x,y))
= 1 by
A1
.= (f2
. (x,y)) by
A2,
A3;
hence thesis;
end;
suppose
A3: x
<> 1 & y
<> 1;
then (f1
. (x,y))
= (
min (x,y)) by
A1
.= (f2
. (x,y)) by
A2,
A3;
hence thesis;
end;
end;
hence thesis by
BINOP_1: 2;
end;
end
theorem ::
LATQUASI:1
WazneQLT7:
QLT_Ex1
<>
QLT_Ex2
proof
A0:
0
in
{
0 , 1, 2} & 1
in
{
0 , 1, 2} by
ENUMSET1:def 1;
then (
QLT_Ex1
. (
0 ,1))
=
0 by
QLTEx1Def;
hence thesis by
A0,
QLTEx2Def;
end;
definition
::
LATQUASI:def8
func
QLTLattice1 ->
strict non
empty
LattStr equals
LattStr (#
{
0 , 1, 2},
QLT_Ex1 ,
QLT_Ex1 #);
coherence ;
::
LATQUASI:def9
func
QLTLattice2 ->
strict non
empty
LattStr equals
LattStr (#
{
0 , 1, 2},
QLT_Ex1 ,
QLT_Ex2 #);
coherence ;
end
registration
cluster
QLT_Ex1 ->
commutative
associative
idempotent;
coherence
proof
set X =
{
0 , 1, 2};
XX:
0
in X by
ENUMSET1:def 1;
set f =
QLT_Ex1 ;
thus f is
commutative
proof
let a,b be
Element of X;
per cases ;
suppose a
= b;
hence thesis;
end;
suppose
A2: a
<> b;
then (f
. (a,b))
=
0 by
QLTEx1Def
.= (f
. (b,a)) by
A2,
QLTEx1Def;
hence thesis;
end;
end;
thus f is
associative
proof
let a,b,c be
Element of X;
per cases ;
suppose
A1: a
= b & b
= c;
then (f
. (a,b))
= a by
QLTEx1Def;
hence thesis by
A1;
end;
suppose
A1: a
<> b & b
= c;
then
A2: (f
. (a,b))
=
0 by
QLTEx1Def;
D2:
0
in X by
ENUMSET1:def 1;
(f
. ((f
. (a,b)),c))
= (f
. (
0 ,c)) by
A1,
QLTEx1Def
.=
0 by
QLTEx1Def,
D2;
hence thesis by
A2,
QLTEx1Def,
A1;
end;
suppose
A1: a
= b & b
<> c;
then
A2: (f
. (a,b))
= a by
QLTEx1Def;
A3: (f
. (b,c))
=
0 by
QLTEx1Def,
A1;
per cases ;
suppose a
=
0 ;
hence thesis by
A3,
A1,
A2;
end;
suppose a
<>
0 ;
hence thesis by
QLTEx1Def,
A3,
A1,
A2;
end;
end;
suppose
A1: a
<> b & b
<> c;
then
A3: (f
. (b,c))
=
0 by
QLTEx1Def;
per cases ;
suppose
B1: a
=
0 ;
(f
. ((f
. (a,b)),c))
= (f
. (
0 ,c)) by
A1,
QLTEx1Def
.=
0 by
B1,
QLTEx1Def;
hence thesis by
B1,
QLTEx1Def,
A3;
end;
suppose
b1: a
<>
0 ;
(f
. ((f
. (a,b)),c))
= (f
. (
0 ,c)) by
A1,
QLTEx1Def
.=
0 by
QLTEx1Def,
XX;
hence thesis by
b1,
QLTEx1Def,
A3;
end;
end;
end;
let a be
Element of X;
thus thesis by
QLTEx1Def;
end;
end
registration
cluster
QLT_Ex2 ->
commutative
associative
idempotent;
coherence
proof
set X =
{
0 , 1, 2};
set f =
QLT_Ex2 ;
thus f is
commutative
proof
let a,b be
Element of X;
per cases ;
suppose
A1: a
= 1 or b
= 1;
then (f
. (a,b))
= 1 by
QLTEx2Def
.= (f
. (b,a)) by
A1,
QLTEx2Def;
hence thesis;
end;
suppose
A2: a
<> 1 & b
<> 1;
then (f
. (a,b))
= (
min (a,b)) by
QLTEx2Def
.= (f
. (b,a)) by
A2,
QLTEx2Def;
hence thesis;
end;
end;
thus f is
associative
proof
let a,b,c be
Element of X;
per cases ;
suppose
A1: a
= 1 & (b
= 1 or c
= 1);
then
A2: (f
. (a,b))
= 1 by
QLTEx2Def;
A3: (f
. (b,c))
= 1 by
QLTEx2Def,
A1;
(f
. ((f
. (a,b)),c))
= 1 by
QLTEx2Def,
A2;
hence thesis by
QLTEx2Def,
A3;
end;
suppose
A1: (a
= 1 or b
= 1) & c
= 1;
then
A5: (f
. ((f
. (a,b)),c))
= 1 by
QLTEx2Def;
(f
. (b,c))
= 1 by
QLTEx2Def,
A1;
hence thesis by
A5,
QLTEx2Def;
end;
suppose
A1: a
<> 1 & b
= 1 & c
<> 1;
then (f
. (a,b))
= 1 by
QLTEx2Def;
then
A5: (f
. ((f
. (a,b)),c))
= 1 by
QLTEx2Def;
(f
. (b,c))
= 1 by
QLTEx2Def,
A1;
hence thesis by
A5,
QLTEx2Def;
end;
suppose
A1: a
= 1 & b
<> 1 & c
= 1;
then (f
. (b,c))
= 1 by
QLTEx2Def;
hence thesis by
QLTEx2Def,
A1;
end;
suppose
A1: a
= 1 & b
<> 1 & c
<> 1;
then
A2: (f
. (a,b))
= 1 by
QLTEx2Def;
A3: (f
. (b,c))
= (
min (b,c)) by
QLTEx2Def,
A1;
per cases by
XXREAL_0: 15;
suppose (
min (b,c))
= b;
hence (f
. ((f
. (a,b)),c))
= (f
. (a,(f
. (b,c)))) by
A2,
A3,
QLTEx2Def;
end;
suppose (
min (b,c))
= c;
hence (f
. ((f
. (a,b)),c))
= (f
. (a,(f
. (b,c)))) by
A1,
QLTEx2Def,
A2;
end;
end;
suppose
A1: a
<> 1 & b
= 1 & c
<> 1;
then
A2: (f
. (a,b))
= 1 by
QLTEx2Def;
A3: (f
. (b,c))
= 1 by
QLTEx2Def,
A1;
(f
. ((f
. (a,b)),c))
= 1 by
QLTEx2Def,
A2;
hence thesis by
QLTEx2Def,
A3;
end;
suppose
A1: a
<> 1 & b
<> 1 & c
= 1;
then
A2: (f
. (a,b))
= (
min (a,b)) by
QLTEx2Def;
A3: (f
. (b,c))
= 1 by
QLTEx2Def,
A1;
per cases by
XXREAL_0: 15;
suppose (
min (a,b))
= a;
(f
. (a,(f
. (b,c))))
= (f
. (a,1)) by
QLTEx2Def,
A1
.= 1 by
QLTEx2Def,
A1;
hence (f
. ((f
. (a,b)),c))
= (f
. (a,(f
. (b,c)))) by
QLTEx2Def,
A1;
end;
suppose (
min (a,b))
= b;
hence (f
. ((f
. (a,b)),c))
= (f
. (a,(f
. (b,c)))) by
QLTEx2Def,
A3,
A2;
end;
end;
suppose
A1: a
<> 1 & b
<> 1 & c
<> 1;
then
A2: (f
. (a,b))
= (
min (a,b)) by
QLTEx2Def;
A3: (f
. (b,c))
= (
min (b,c)) by
QLTEx2Def,
A1;
per cases by
XXREAL_0: 15;
suppose
C1: (
min (a,c))
= a;
per cases by
XXREAL_0: 15;
suppose (
min (a,b))
= a & (
min (b,c))
= b;
hence (f
. ((f
. (a,b)),c))
= (f
. (a,(f
. (b,c)))) by
A2,
A3,
C1,
QLTEx2Def,
A1;
end;
suppose (
min (a,b))
= a & (
min (b,c))
= c;
hence (f
. ((f
. (a,b)),c))
= (f
. (a,(f
. (b,c)))) by
QLTEx2Def,
A1,
A2;
end;
suppose (
min (a,b))
= b & (
min (b,c))
= b;
hence (f
. ((f
. (a,b)),c))
= (f
. (a,(f
. (b,c)))) by
A3,
A2;
end;
suppose
B1: (
min (a,b))
= b & (
min (b,c))
= c;
then a
<= c & b
<= a & c
<= b by
C1,
XXREAL_0:def 9;
then a
<= c & c
<= a by
XXREAL_0: 2;
then a
= c by
XXREAL_0: 1;
hence (f
. ((f
. (a,b)),c))
= (f
. (a,(f
. (b,c)))) by
B1,
A3;
end;
end;
suppose
C1: (
min (a,c))
= c;
per cases by
XXREAL_0: 15;
suppose
B1: (
min (a,b))
= a & (
min (b,c))
= b;
then c
<= a & a
<= b & b
<= c by
C1,
XXREAL_0:def 9;
then a
<= c & c
<= a by
XXREAL_0: 2;
then a
= c by
XXREAL_0: 1;
hence (f
. ((f
. (a,b)),c))
= (f
. (a,(f
. (b,c)))) by
A2,
B1;
end;
suppose (
min (a,b))
= a & (
min (b,c))
= c;
hence (f
. ((f
. (a,b)),c))
= (f
. (a,(f
. (b,c)))) by
QLTEx2Def,
A1,
A2;
end;
suppose (
min (a,b))
= b & (
min (b,c))
= b;
hence (f
. ((f
. (a,b)),c))
= (f
. (a,(f
. (b,c)))) by
A2,
A3;
end;
suppose (
min (a,b))
= b & (
min (b,c))
= c;
hence (f
. ((f
. (a,b)),c))
= (f
. (a,(f
. (b,c)))) by
C1,
QLTEx2Def,
A1,
A3,
A2;
end;
end;
end;
end;
let a be
Element of X;
per cases ;
suppose a
= 1;
hence thesis by
QLTEx2Def;
end;
suppose a
<> 1;
then (f
. (a,a))
= (
min (a,a)) by
QLTEx2Def
.= a;
hence thesis;
end;
end;
end
registration
cluster
QLTLattice1 ->
join-commutative
join-associative
join-idempotent;
coherence
proof
set L =
QLTLattice1 ;
set f = the
L_join of L;
Y1: for x,y be
Element of L holds (x
"\/" y)
= (y
"\/" x) by
BINOP_1:def 2;
S1: for x,y,z be
Element of L holds (x
"\/" (y
"\/" z))
= ((x
"\/" y)
"\/" z) by
BINOP_1:def 3;
for x be
Element of L holds (x
"\/" x)
= x by
BINOP_1:def 4;
hence thesis by
Y1,
LATTICES:def 4,
LATTICES:def 5,
S1,
ROBBINS1:def 7;
end;
end
registration
cluster
QLTLattice1 ->
meet-commutative
meet-associative
meet-idempotent;
coherence
proof
set L =
QLTLattice1 ;
set f = the
L_meet of L;
Y1: for x,y be
Element of L holds (x
"/\" y)
= (y
"/\" x) by
BINOP_1:def 2;
S1: for x,y,z be
Element of L holds (x
"/\" (y
"/\" z))
= ((x
"/\" y)
"/\" z) by
BINOP_1:def 3;
for x be
Element of L holds (x
"/\" x)
= x by
BINOP_1:def 4;
hence thesis by
Y1,
LATTICES:def 6,
LATTICES:def 7,
S1,
SHEFFER1:def 9;
end;
end
theorem ::
LATQUASI:2
Lemacik00: for v0,v1 be
Element of
QLTLattice1 st v1
=
0 holds (v0
"/\" v1)
= v1
proof
let v0,v1 be
Element of
QLTLattice1 ;
assume
A1: v1
=
0 ;
per cases ;
suppose v0
= v1;
hence thesis by
QLTEx1Def;
end;
suppose v0
<> v1;
hence thesis by
A1,
QLTEx1Def;
end;
end;
theorem ::
LATQUASI:3
Lemacik001: for v0,v1 be
Element of
QLTLattice1 st v1
=
0 holds (v0
"\/" v1)
= v1
proof
let v0,v1 be
Element of
QLTLattice1 ;
assume
A1: v1
=
0 ;
per cases ;
suppose v0
= v1;
hence thesis by
QLTEx1Def;
end;
suppose v0
<> v1;
hence thesis by
A1,
QLTEx1Def;
end;
end;
registration
cluster
QLTLattice1 ->
satisfying_QLT1;
coherence
proof
set L =
QLTLattice1 ;
for v0,v2,v1 be
Element of L holds ((v0
"/\" (v1
"\/" v2))
"\/" (v0
"/\" v1))
= (v0
"/\" (v1
"\/" v2))
proof
let v0,v2,v1 be
Element of L;
reconsider z =
0 as
Element of L by
ENUMSET1:def 1;
per cases ;
suppose v0
= v1 & v1
= v2;
then (v1
"\/" v2)
= v1 & (v0
"/\" v1)
= v0 by
QLTEx1Def;
hence thesis by
ROBBINS1:def 7;
end;
suppose v0
= v1 & v1
<> v2;
then
A1: (v1
"\/" v2)
=
0 & (v0
"/\" v1)
= v0 by
QLTEx1Def;
then ((v0
"/\" (v1
"\/" v2))
"\/" (v0
"/\" v1))
= (z
"\/" v0) by
Lemacik00
.= z by
Lemacik001
.= (v0
"/\" (v1
"\/" v2)) by
A1,
Lemacik00;
hence thesis;
end;
suppose v0
<> v1 & v1
<> v2;
then
A1: (v1
"\/" v2)
=
0 & (v0
"/\" v1)
=
0 by
QLTEx1Def;
then ((v0
"/\" (v1
"\/" v2))
"\/" (v0
"/\" v1))
= z by
Lemacik001
.= (v0
"/\" (v1
"\/" v2)) by
A1,
Lemacik00;
hence thesis;
end;
suppose v0
<> v1 & v1
= v2;
then (v1
"\/" v2)
= v1 & (v0
"/\" v1)
=
0 by
QLTEx1Def;
hence thesis by
Lemacik001;
end;
end;
hence thesis;
end;
end
registration
cluster
QLTLattice1 ->
satisfying_QLT2;
coherence
proof
set L =
QLTLattice1 ;
let v0,v2,v1 be
Element of L;
reconsider z =
0 as
Element of L by
ENUMSET1:def 1;
per cases ;
suppose v0
= v1 & v1
= v2;
then (v1
"/\" v2)
= v1 & (v0
"\/" v1)
= v0 by
QLTEx1Def;
hence thesis by
SHEFFER1:def 9;
end;
suppose v0
= v1 & v1
<> v2;
then
A1: (v1
"/\" v2)
=
0 & (v0
"\/" v1)
= v0 by
QLTEx1Def;
(z
"\/" v0)
= (v0
"\/" z);
then ((v0
"\/" (v1
"/\" v2))
"/\" (v0
"\/" v1))
= (z
"/\" v0) by
Lemacik00,
A1
.= z by
Lemacik00
.= (v0
"\/" (v1
"/\" v2)) by
A1,
Lemacik001;
hence thesis;
end;
suppose v0
<> v1 & v1
<> v2;
then
A1: (v1
"/\" v2)
=
0 & (v0
"\/" v1)
=
0 by
QLTEx1Def;
then ((v0
"\/" (v1
"/\" v2))
"/\" (v0
"\/" v1))
= z by
Lemacik00
.= (v0
"\/" (v1
"/\" v2)) by
A1,
Lemacik001;
hence thesis;
end;
suppose v0
<> v1 & v1
= v2;
then (v1
"/\" v2)
= v1 & (v0
"\/" v1)
=
0 by
QLTEx1Def;
hence thesis by
Lemacik00;
end;
end;
end
registration
cluster ->
real for
Element of
QLTLattice2 ;
coherence ;
end
registration
cluster
QLTLattice2 ->
join-commutative
join-associative
join-idempotent;
coherence
proof
set L =
QLTLattice2 ;
Y1: for x,y be
Element of L holds (x
"\/" y)
= (y
"\/" x) by
BINOP_1:def 2;
S1: for x,y,z be
Element of L holds (x
"\/" (y
"\/" z))
= ((x
"\/" y)
"\/" z) by
BINOP_1:def 3;
for x be
Element of L holds (x
"\/" x)
= x by
BINOP_1:def 4;
hence thesis by
Y1,
LATTICES:def 4,
LATTICES:def 5,
S1,
ROBBINS1:def 7;
end;
end
registration
cluster
QLTLattice2 ->
meet-commutative
meet-associative
meet-idempotent;
coherence
proof
set L =
QLTLattice2 ;
Y1: for x,y be
Element of L holds (x
"/\" y)
= (y
"/\" x) by
BINOP_1:def 2;
S1: for x,y,z be
Element of L holds (x
"/\" (y
"/\" z))
= ((x
"/\" y)
"/\" z) by
BINOP_1:def 3;
for x be
Element of L holds (x
"/\" x)
= x by
BINOP_1:def 4;
hence thesis by
Y1,
LATTICES:def 6,
LATTICES:def 7,
S1,
SHEFFER1:def 9;
end;
end
registration
cluster
QLTLattice2 ->
satisfying_QLT1;
coherence
proof
set L =
QLTLattice2 ;
for v0,v2,v1 be
Element of L holds ((v0
"/\" (v1
"\/" v2))
"\/" (v0
"/\" v1))
= (v0
"/\" (v1
"\/" v2))
proof
let v0,v2,v1 be
Element of L;
reconsider z =
0 as
Element of L by
ENUMSET1:def 1;
p2: z
<= v0 by
ENUMSET1:def 1;
reconsider o = 1 as
Element of L by
ENUMSET1:def 1;
reconsider dwa = 2 as
Element of L by
ENUMSET1:def 1;
per cases by
ENUMSET1:def 1;
suppose v0
= v1 & v1
= v2;
then (v1
"\/" v2)
= v1 & (v0
"/\" v1)
= v1 by
BINOP_1:def 4;
hence thesis by
ROBBINS1:def 7;
end;
suppose
Z1: v0
= v1 & v1
<> v2 & v0
= 1;
then (v1
"\/" v2)
=
0 & (v0
"/\" v1)
= 1 by
QLTEx1Def,
BINOP_1:def 4;
then ((v0
"/\" (v1
"\/" v2))
"\/" (v0
"/\" v1))
= (o
"\/" v0) by
QLTEx2Def,
Z1
.= o by
QLTEx1Def,
Z1
.= (v0
"/\" (v1
"\/" v2)) by
Z1,
QLTEx2Def;
hence thesis;
end;
suppose
A0: v0
= v1 & v1
<> v2 & v0
=
0 ;
then (v1
"\/" v2)
=
0 & (v0
"/\" v1)
= (
min (v0,v1)) by
QLTEx1Def,
QLTEx2Def;
hence thesis by
QLTEx1Def,
A0;
end;
suppose
A0: v0
= v1 & v1
<> v2 & v0
= 2;
then
A1: (v1
"\/" v2)
=
0 & (v0
"/\" v1)
= (
min (v0,v1)) & v0
<> 1 by
QLTEx1Def,
QLTEx2Def;
(v0
"/\" z)
= (
min (v0,z)) by
A0,
QLTEx2Def
.= z by
XXREAL_0:def 9,
p2;
hence thesis by
A1,
QLTEx1Def,
A0;
end;
suppose
A0: v0
<> v1 & v1
<> v2 & v0
= 1;
then
A1: (v1
"\/" v2)
=
0 & (v0
"/\" v1)
= 1 by
QLTEx1Def,
QLTEx2Def;
(v0
"/\" z)
= o by
A0,
QLTEx2Def;
hence thesis by
QLTEx1Def,
A1;
end;
suppose
A0: v0
<> v1 & v1
<> v2 & v0
= 2;
per cases ;
suppose v1
= 1;
then
A1: (v1
"\/" v2)
=
0 & (v0
"/\" v1)
= 1 by
QLTEx1Def,
QLTEx2Def,
A0;
(v0
"/\" z)
= (
min (v0,z)) by
A0,
QLTEx2Def
.= z by
XXREAL_0:def 9,
p2;
hence thesis by
QLTEx1Def,
A1;
end;
suppose
B: v1
<> 1;
then
A1: (v1
"\/" v2)
=
0 & (v0
"/\" v1)
= (
min (v0,v1)) by
QLTEx1Def,
QLTEx2Def,
A0;
v1
=
0 or v1
= 2 by
ENUMSET1:def 1,
B;
hence thesis by
QLTEx1Def,
A0,
A1;
end;
end;
suppose
A0: v0
<> v1 & v1
<> v2 & v0
=
0 ;
per cases ;
suppose v1
= 1;
then
A1: (v1
"\/" v2)
=
0 & (v0
"/\" v1)
= 1 by
QLTEx1Def,
QLTEx2Def,
A0;
(v0
"/\" z)
= (
min (v0,z)) by
A0,
QLTEx2Def
.= z by
XXREAL_0:def 9,
p2;
hence thesis by
A1,
QLTEx1Def;
end;
suppose v1
<> 1;
then
A1: (v1
"\/" v2)
=
0 & (v0
"/\" v1)
= (
min (v0,v1)) by
QLTEx1Def,
QLTEx2Def,
A0;
p2: v0
<= v1 by
A0,
ENUMSET1:def 1;
P4: (v0
"/\" z)
= (
min (v0,z)) by
A0,
QLTEx2Def
.= z by
A0;
((v0
"/\" (v1
"\/" v2))
"\/" (v0
"/\" v1))
= ((v0
"/\" z)
"\/" v0) by
p2,
A1,
XXREAL_0:def 9
.= (v0
"/\" (v1
"\/" v2)) by
A1,
P4,
QLTEx1Def,
A0;
hence thesis;
end;
end;
suppose v0
<> v1 & v1
= v2 & v0
= 1;
then (v1
"\/" v2)
= v1 & (v0
"/\" v1)
= 1 by
QLTEx1Def,
QLTEx2Def;
hence thesis by
QLTEx1Def;
end;
suppose
A0: v0
<> v1 & v1
= v2 & v0
=
0 ;
per cases ;
suppose v1
= 1;
then (v1
"\/" v2)
= v1 & (v0
"/\" v1)
= 1 by
QLTEx1Def,
QLTEx2Def,
A0;
hence thesis by
ROBBINS1:def 7;
end;
suppose v1
<> 1;
then (v1
"\/" v2)
= v1 & (v0
"/\" v1)
= (
min (v0,v1)) by
QLTEx1Def,
QLTEx2Def,
A0;
hence thesis by
ROBBINS1:def 7;
end;
end;
suppose
A0: v0
<> v1 & v1
= v2 & v0
= 2;
per cases ;
suppose v1
= 1;
then (v1
"\/" v2)
= v1 & (v0
"/\" v1)
= 1 by
QLTEx1Def,
QLTEx2Def,
A0;
hence thesis by
ROBBINS1:def 7;
end;
suppose v1
<> 1;
then (v1
"\/" v2)
= v1 & (v0
"/\" v1)
= (
min (v0,v1)) by
QLTEx1Def,
QLTEx2Def,
A0;
hence thesis by
ROBBINS1:def 7;
end;
end;
end;
hence thesis;
end;
end
registration
cluster
QLTLattice2 ->
satisfying_QLT2;
coherence
proof
set L =
QLTLattice2 ;
for v0,v2,v1 be
Element of L holds ((v0
"\/" (v1
"/\" v2))
"/\" (v0
"\/" v1))
= (v0
"\/" (v1
"/\" v2))
proof
let v0,v2,v1 be
Element of L;
reconsider z =
0 as
Element of L by
ENUMSET1:def 1;
reconsider o = 1 as
Element of L by
ENUMSET1:def 1;
reconsider dwa = 2 as
Element of L by
ENUMSET1:def 1;
per cases by
ENUMSET1:def 1;
suppose v0
= v1 & v1
= v2;
then (v1
"/\" v2)
= v1 & (v0
"\/" v1)
= v1 by
BINOP_1:def 4;
hence thesis by
SHEFFER1:def 9;
end;
suppose
Z1: v0
= v1 & v1
<> v2 & v0
= 1;
then (v1
"/\" v2)
= 1 & (v0
"\/" v1)
= v1 by
QLTEx1Def,
QLTEx2Def;
hence thesis by
SHEFFER1:def 9,
Z1;
end;
suppose
A0: v0
= v1 & v1
<> v2 & v0
=
0 ;
per cases ;
suppose v2
= 1;
then
A1: (v1
"/\" v2)
= 1 & (v0
"\/" v1)
= v0 by
QLTEx1Def,
QLTEx2Def,
A0;
P4: (v0
"/\" z)
= (
min (v0,z)) by
A0,
QLTEx2Def
.= z by
A0;
((v0
"\/" (v1
"/\" v2))
"/\" (v0
"\/" v1))
= z by
P4,
QLTEx1Def,
A0,
A1;
hence thesis by
QLTEx1Def,
A0,
A1;
end;
suppose v2
<> 1;
then
A1: (v1
"/\" v2)
= (
min (v1,v2)) & (v0
"\/" v1)
= v0 by
QLTEx1Def,
QLTEx2Def,
A0;
P3: v1
<= v2 by
A0,
ENUMSET1:def 1;
(v1
"/\" v2)
= v1 by
XXREAL_0:def 9,
P3,
A1;
hence thesis by
SHEFFER1:def 9;
end;
end;
suppose
A0: v0
= v1 & v1
<> v2 & v0
= 2;
P4: (v0
"/\" z)
= (
min (v0,z)) by
A0,
QLTEx2Def
.= z by
XXREAL_0:def 9,
A0;
per cases ;
suppose v2
= 1;
then
A1: (v1
"/\" v2)
= 1 & (v0
"\/" v1)
= v0 by
QLTEx1Def,
QLTEx2Def,
A0;
then ((v0
"\/" (v1
"/\" v2))
"/\" (v0
"\/" v1))
= z by
P4,
QLTEx1Def,
A0;
hence thesis by
QLTEx1Def,
A0,
A1;
end;
suppose
B: v2
<> 1;
then
A1: (v1
"/\" v2)
= (
min (v1,v2)) & (v0
"\/" v1)
= v0 by
QLTEx1Def,
QLTEx2Def,
A0;
p4: v2
=
0 or v2
= 2 by
ENUMSET1:def 1,
B;
then
P4A: (v1
"/\" v2)
= v2 by
XXREAL_0:def 9,
A1,
A0;
per cases by
ENUMSET1:def 1,
B;
suppose v2
=
0 ;
((v0
"\/" (v1
"/\" v2))
"/\" (v0
"\/" v1))
= z by
QLTEx1Def,
A0,
A1,
p4,
P4;
hence thesis by
QLTEx1Def,
A0,
P4A;
end;
suppose v2
= 2;
hence thesis by
A0;
end;
end;
end;
suppose
A0: v0
<> v1 & v1
<> v2 & v0
= 1;
per cases ;
suppose v2
= 1;
then
A1: (v1
"/\" v2)
= 1 & (v0
"\/" v1)
=
0 by
QLTEx1Def,
QLTEx2Def,
A0;
then ((v0
"\/" (v1
"/\" v2))
"/\" (v0
"\/" v1))
= (v0
"/\" z) by
A0,
QLTEx1Def
.= 1 by
QLTEx2Def,
A0;
hence thesis by
A0,
QLTEx1Def,
A1;
end;
suppose
B: v2
<> 1;
then
A1: (v1
"/\" v2)
= (
min (v1,v2)) & (v0
"\/" v1)
=
0 by
QLTEx1Def,
QLTEx2Def,
A0;
per cases by
XXREAL_0: 15;
suppose
C1: (
min (v1,v2))
= v1;
((v0
"\/" (v1
"/\" v2))
"/\" (v0
"\/" v1))
= ((v0
"\/" v1)
"/\" (v0
"\/" v1)) by
B,
C1,
QLTEx2Def,
A0
.= (v0
"\/" v1) by
SHEFFER1:def 9;
hence thesis by
B,
C1,
QLTEx2Def,
A0;
end;
suppose
D1: (
min (v1,v2))
= v2;
then ((v0
"\/" (v1
"/\" v2))
"/\" (v0
"\/" v1))
= (z
"/\" z) by
A0,
B,
QLTEx1Def,
A1
.= z by
BINOP_1:def 4;
hence thesis by
A0,
B,
QLTEx1Def,
A1,
D1;
end;
end;
end;
suppose
A0: v0
<> v1 & v1
<> v2 & v0
= 2;
per cases ;
suppose v2
= 1 or v1
= 1;
then
A1: (v1
"/\" v2)
= 1 & (v0
"\/" v1)
=
0 by
QLTEx1Def,
QLTEx2Def,
A0;
then ((v0
"\/" (v1
"/\" v2))
"/\" (v0
"\/" v1))
= (z
"/\" z) by
A0,
QLTEx1Def
.= z by
BINOP_1:def 4;
hence thesis by
A0,
QLTEx1Def,
A1;
end;
suppose
B: v2
<> 1 & v1
<> 1;
then
A1: (v1
"/\" v2)
= (
min (v1,v2)) & (v0
"\/" v1)
=
0 by
QLTEx1Def,
QLTEx2Def,
A0;
per cases by
XXREAL_0: 15;
suppose (
min (v1,v2))
= v1;
hence thesis by
SHEFFER1:def 9,
A1;
end;
suppose
D1: (
min (v1,v2))
= v2;
(v1
= 1 & v2
<> 1) or (v1
=
0 & v2
<>
0 ) by
A0,
ENUMSET1:def 1;
then (v1
= 1 & (v2
=
0 or v2
= 2)) or (v1
=
0 & (v2
= 1 or v2
= 2)) by
ENUMSET1:def 1;
hence thesis by
XXREAL_0:def 9,
D1,
B;
end;
end;
end;
suppose
A0: v0
<> v1 & v1
<> v2 & v0
=
0 ;
per cases ;
suppose v2
= 1 or v1
= 1;
then
A1: (v1
"/\" v2)
= 1 & (v0
"\/" v1)
=
0 by
QLTEx1Def,
QLTEx2Def,
A0;
(v0
"\/" o)
= z by
QLTEx1Def,
A0;
hence thesis by
BINOP_1:def 4,
A1;
end;
suppose v2
<> 1 & v1
<> 1;
then
A1: (v1
"/\" v2)
= (
min (v1,v2)) & (v0
"\/" v1)
=
0 by
QLTEx1Def,
QLTEx2Def,
A0;
per cases by
XXREAL_0: 15;
suppose (
min (v1,v2))
= v1;
hence thesis by
SHEFFER1:def 9,
A1;
end;
suppose
D1: (
min (v1,v2))
= v2;
per cases ;
suppose
D2: v2
= v0;
(v0
"/\" z)
= (
min (v0,z)) by
QLTEx2Def,
A0
.= v0 by
A0;
then ((v0
"\/" (v1
"/\" v2))
"/\" (v0
"\/" v1))
= v0 by
A1,
D2,
D1,
QLTEx1Def;
hence thesis by
D2,
QLTEx1Def,
A1,
D1;
end;
suppose
D2: v2
<> v0;
then ((v0
"\/" (v1
"/\" v2))
"/\" (v0
"\/" v1))
= (z
"/\" z) by
QLTEx1Def,
A1,
D1
.= z by
BINOP_1:def 4;
hence thesis by
D2,
QLTEx1Def,
A1,
D1;
end;
end;
end;
end;
suppose v0
<> v1 & v1
= v2;
then (v1
"/\" v2)
= v1 & (v0
"\/" v1)
=
0 by
QLTEx1Def,
BINOP_1:def 4;
hence thesis by
SHEFFER1:def 9;
end;
end;
hence thesis;
end;
end
registration
cluster
QLTLattice2 -> non
join-absorbing;
coherence
proof
set L =
QLTLattice2 ;
reconsider z =
0 as
Element of L by
ENUMSET1:def 1;
reconsider o = 1 as
Element of L by
ENUMSET1:def 1;
reconsider dwa = 2 as
Element of L by
ENUMSET1:def 1;
(dwa
"/\" (dwa
"\/" o))
= (dwa
"/\" z) by
QLTEx1Def
.= (
min (dwa,z)) by
QLTEx2Def
.= z by
XXREAL_0:def 9;
hence thesis by
LATTICES:def 9;
end;
cluster
QLTLattice2 -> non
meet-absorbing;
coherence
proof
set L =
QLTLattice2 ;
reconsider z =
0 as
Element of L by
ENUMSET1:def 1;
reconsider o = 1 as
Element of L by
ENUMSET1:def 1;
reconsider dwa = 2 as
Element of L by
ENUMSET1:def 1;
((dwa
"/\" o)
"\/" dwa)
= (o
"\/" dwa) by
QLTEx2Def
.= z by
QLTEx1Def;
hence thesis by
LATTICES:def 8;
end;
end
registration
cluster
QLTLattice1 -> non
join-absorbing;
coherence
proof
set L =
QLTLattice1 ;
reconsider z =
0 as
Element of L by
ENUMSET1:def 1;
reconsider o = 1 as
Element of L by
ENUMSET1:def 1;
reconsider dwa = 2 as
Element of L by
ENUMSET1:def 1;
(dwa
"/\" (dwa
"\/" o))
= (dwa
"/\" z) by
QLTEx1Def
.= z by
QLTEx1Def;
hence thesis by
LATTICES:def 9;
end;
cluster
QLTLattice1 -> non
meet-absorbing;
coherence
proof
set L =
QLTLattice1 ;
reconsider z =
0 as
Element of L by
ENUMSET1:def 1;
reconsider o = 1 as
Element of L by
ENUMSET1:def 1;
reconsider dwa = 2 as
Element of L by
ENUMSET1:def 1;
((dwa
"/\" o)
"\/" dwa)
= (z
"\/" dwa) by
QLTEx1Def
.= z by
QLTEx1Def;
hence thesis by
LATTICES:def 8;
end;
end
definition
mode
QuasiLattice is
join-commutative
join-associative
meet-commutative
meet-associative
join-idempotent
meet-idempotent
satisfying_QLT1
satisfying_QLT2 non
empty
LattStr;
end
begin
theorem ::
LATQUASI:4
Lemma1: (for v1, v0 holds (v0
"/\" v1)
= (v1
"/\" v0)) & (for v0, v2, v1 holds ((v0
"/\" (v1
"\/" v2))
"\/" (v0
"/\" v1))
= (v0
"/\" (v1
"\/" v2))) & (for v0 holds (v0
"\/" v0)
= v0) & (for v2, v1, v0 holds ((v0
"\/" v1)
"\/" v2)
= (v0
"\/" (v1
"\/" v2))) & (for v1, v0 holds (v0
"\/" v1)
= (v1
"\/" v0)) & (for v0, v2, v1 holds ((v0
"\/" (v1
"/\" v2))
"/\" (v0
"\/" v1))
= (v0
"\/" (v1
"/\" v2))) & (for v1, v2, v0 holds (v0
"/\" (v1
"\/" (v0
"/\" v2)))
= (v0
"/\" (v1
"\/" v2))) implies for v1, v2, v3 holds ((v1
"/\" v2)
"\/" (v1
"/\" v3))
= (v1
"/\" (v2
"\/" v3))
proof
assume
A1: for v1, v0 holds (v0
"/\" v1)
= (v1
"/\" v0);
assume
A2: for v0, v2, v1 holds ((v0
"/\" (v1
"\/" v2))
"\/" (v0
"/\" v1))
= (v0
"/\" (v1
"\/" v2));
assume
A3: for v0 holds (v0
"\/" v0)
= v0;
assume
A4: for v2, v1, v0 holds ((v0
"\/" v1)
"\/" v2)
= (v0
"\/" (v1
"\/" v2));
assume
A5: for v1, v0 holds (v0
"\/" v1)
= (v1
"\/" v0);
assume
A6: for v0, v2, v1 holds ((v0
"\/" (v1
"/\" v2))
"/\" (v0
"\/" v1))
= (v0
"\/" (v1
"/\" v2));
A8: for v0, v2, v1 holds ((v0
"\/" v1)
"/\" (v0
"\/" (v1
"/\" v2)))
= (v0
"\/" (v1
"/\" v2))
proof
let v0, v2, v1;
((v0
"\/" (v1
"/\" v2))
"/\" (v0
"\/" v1))
= ((v0
"\/" v1)
"/\" (v0
"\/" (v1
"/\" v2))) by
A1;
hence thesis by
A6;
end;
assume
A9: for v1, v2, v0 holds (v0
"/\" (v1
"\/" (v0
"/\" v2)))
= (v0
"/\" (v1
"\/" v2));
A12: for v0, v2, v1 holds ((v0
"/\" v1)
"\/" (v0
"/\" (v1
"\/" v2)))
= (v0
"/\" (v1
"\/" v2))
proof
let v0, v2, v1;
((v0
"/\" (v1
"\/" v2))
"\/" (v0
"/\" v1))
= ((v0
"/\" v1)
"\/" (v0
"/\" (v1
"\/" v2))) by
A5;
hence thesis by
A2;
end;
A15: for v102, v100 holds ((v100
"\/" v102)
"\/" v102)
= (v100
"\/" v102)
proof
let v102, v100;
(v102
"\/" v102)
= v102 by
A3;
hence thesis by
A4;
end;
A18: for v1, v0 holds (v1
"\/" (v0
"\/" v1))
= (v0
"\/" v1)
proof
let v1, v0;
((v0
"\/" v1)
"\/" v1)
= (v1
"\/" (v0
"\/" v1)) by
A5;
hence thesis by
A15;
end;
A21: for v2, v0, v1 holds ((v1
"\/" v0)
"\/" v2)
= (v0
"\/" (v1
"\/" v2))
proof
let v2, v0, v1;
(v0
"\/" v1)
= (v1
"\/" v0) by
A5;
hence thesis by
A4;
end;
A24: for v0, v2, v1 holds (v0
"\/" (v1
"\/" v2))
= (v1
"\/" (v0
"\/" v2))
proof
let v0, v2, v1;
((v0
"\/" v1)
"\/" v2)
= (v0
"\/" (v1
"\/" v2)) by
A4;
hence thesis by
A21;
end;
A27: for v102, v101 holds (v101
"/\" (v101
"\/" (v101
"/\" v102)))
= (v101
"\/" (v101
"/\" v102))
proof
let v102, v101;
(v101
"\/" v101)
= v101 by
A3;
hence thesis by
A8;
end;
A30: for v1, v0 holds (v0
"/\" (v0
"\/" v1))
= (v0
"\/" (v0
"/\" v1))
proof
let v1, v0;
(v0
"/\" (v0
"\/" (v0
"/\" v1)))
= (v0
"/\" (v0
"\/" v1)) by
A9;
hence thesis by
A27;
end;
A33: for v1, v0, v2 holds (v0
"/\" (v1
"\/" (v2
"/\" v0)))
= (v0
"/\" (v1
"\/" v2))
proof
let v1, v0, v2;
(v0
"/\" v2)
= (v2
"/\" v0) by
A1;
hence thesis by
A9;
end;
A35: for v0, v1, v2 holds ((v0
"/\" v1)
"\/" (v0
"/\" (v2
"\/" v1)))
= (v0
"/\" (v1
"\/" v2))
proof
let v0, v1, v2;
(v1
"\/" v2)
= (v2
"\/" v1) by
A5;
hence thesis by
A12;
end;
A37: for v0, v1 holds (v0
"\/" (v1
"/\" v0))
= (v0
"/\" (v0
"\/" v1))
proof
let v0, v1;
(v0
"/\" v1)
= (v1
"/\" v0) by
A1;
hence thesis by
A30;
end;
A40: for v1, v2, v101 holds ((v1
"\/" (v101
"/\" v2))
"\/" (v101
"/\" (v1
"\/" v2)))
= ((v1
"\/" (v101
"/\" v2))
"/\" ((v1
"\/" (v101
"/\" v2))
"\/" v101))
proof
let v1, v2, v101;
(v101
"/\" (v1
"\/" (v101
"/\" v2)))
= (v101
"/\" (v1
"\/" v2)) by
A9;
hence thesis by
A37;
end;
A43: for v1, v2, v0 holds ((v1
"/\" (v0
"\/" v2))
"\/" (v0
"\/" (v1
"/\" v2)))
= ((v0
"\/" (v1
"/\" v2))
"/\" ((v0
"\/" (v1
"/\" v2))
"\/" v1))
proof
let v1, v2, v0;
((v0
"\/" (v1
"/\" v2))
"\/" (v1
"/\" (v0
"\/" v2)))
= ((v1
"/\" (v0
"\/" v2))
"\/" (v0
"\/" (v1
"/\" v2))) by
A5;
hence thesis by
A40;
end;
A46: for v0, v2, v1 holds (v1
"\/" ((v0
"/\" (v1
"\/" v2))
"\/" (v0
"/\" v2)))
= ((v1
"\/" (v0
"/\" v2))
"/\" ((v1
"\/" (v0
"/\" v2))
"\/" v0))
proof
let v0, v2, v1;
((v0
"/\" (v1
"\/" v2))
"\/" (v1
"\/" (v0
"/\" v2)))
= (v1
"\/" ((v0
"/\" (v1
"\/" v2))
"\/" (v0
"/\" v2))) by
A24;
hence thesis by
A43;
end;
A49: for v1, v2, v0 holds (v0
"\/" ((v1
"/\" v2)
"\/" (v1
"/\" (v0
"\/" v2))))
= ((v0
"\/" (v1
"/\" v2))
"/\" ((v0
"\/" (v1
"/\" v2))
"\/" v1))
proof
let v1, v2, v0;
((v1
"/\" (v0
"\/" v2))
"\/" (v1
"/\" v2))
= ((v1
"/\" v2)
"\/" (v1
"/\" (v0
"\/" v2))) by
A5;
hence thesis by
A46;
end;
A51: for v1, v0, v2 holds (v0
"\/" (v1
"/\" (v2
"\/" v0)))
= ((v0
"\/" (v1
"/\" v2))
"/\" ((v0
"\/" (v1
"/\" v2))
"\/" v1))
proof
let v1, v0, v2;
((v1
"/\" v2)
"\/" (v1
"/\" (v0
"\/" v2)))
= (v1
"/\" (v2
"\/" v0)) by
A35;
hence thesis by
A49;
end;
A53: for v1, v0, v2 holds (v0
"\/" (v1
"/\" (v2
"\/" v0)))
= ((v0
"\/" (v1
"/\" v2))
"/\" (v1
"\/" (v0
"\/" (v1
"/\" v2))))
proof
let v1, v0, v2;
((v0
"\/" (v1
"/\" v2))
"\/" v1)
= (v1
"\/" (v0
"\/" (v1
"/\" v2))) by
A5;
hence thesis by
A51;
end;
A56: for v101, v0, v2, v1 holds ((v0
"\/" (v1
"/\" v2))
"/\" (v101
"\/" (v0
"\/" (v1
"/\" v2))))
= ((v0
"\/" (v1
"/\" v2))
"/\" (v101
"\/" (v0
"\/" v1)))
proof
let v101, v0, v2, v1;
((v0
"\/" v1)
"/\" (v0
"\/" (v1
"/\" v2)))
= (v0
"\/" (v1
"/\" v2)) by
A8;
hence thesis by
A33;
end;
A59: for v1, v0, v2 holds (v0
"\/" (v1
"/\" (v2
"\/" v0)))
= ((v0
"\/" (v1
"/\" v2))
"/\" (v1
"\/" (v0
"\/" v1)))
proof
let v1, v0, v2;
((v0
"\/" (v1
"/\" v2))
"/\" (v1
"\/" (v0
"\/" (v1
"/\" v2))))
= ((v0
"\/" (v1
"/\" v2))
"/\" (v1
"\/" (v0
"\/" v1))) by
A56;
hence thesis by
A53;
end;
A61: for v1, v0, v2 holds (v0
"\/" (v1
"/\" (v2
"\/" v0)))
= ((v0
"\/" (v1
"/\" v2))
"/\" (v0
"\/" v1))
proof
let v1, v0, v2;
(v1
"\/" (v0
"\/" v1))
= (v0
"\/" v1) by
A18;
hence thesis by
A59;
end;
A63: for v1, v0, v2 holds (v0
"\/" (v1
"/\" (v2
"\/" v0)))
= ((v0
"\/" v1)
"/\" (v0
"\/" (v1
"/\" v2)))
proof
let v1, v0, v2;
((v0
"\/" (v1
"/\" v2))
"/\" (v0
"\/" v1))
= ((v0
"\/" v1)
"/\" (v0
"\/" (v1
"/\" v2))) by
A1;
hence thesis by
A61;
end;
A65: for v1, v0, v2 holds (v0
"\/" (v1
"/\" (v2
"\/" v0)))
= (v0
"\/" (v1
"/\" v2))
proof
let v1, v0, v2;
((v0
"\/" v1)
"/\" (v0
"\/" (v1
"/\" v2)))
= (v0
"\/" (v1
"/\" v2)) by
A8;
hence thesis by
A63;
end;
A68: for v101, v2, v102 holds ((v101
"/\" v2)
"\/" (v101
"/\" (v102
"\/" v2)))
= ((v101
"/\" v2)
"\/" (v101
"/\" v102))
proof
let v101, v2, v102;
(v101
"/\" (v102
"\/" (v101
"/\" v2)))
= (v101
"/\" (v102
"\/" v2)) by
A9;
hence thesis by
A65;
end;
for v0, v2, v1 holds (v0
"/\" (v1
"\/" v2))
= ((v0
"/\" v1)
"\/" (v0
"/\" v2))
proof
let v0, v2, v1;
((v0
"/\" v1)
"\/" (v0
"/\" (v2
"\/" v1)))
= (v0
"/\" (v1
"\/" v2)) by
A35;
hence thesis by
A68;
end;
hence thesis;
end;
theorem ::
LATQUASI:5
Cluster1: L is
meet-commutative
satisfying_QLT1
join-idempotent
join-associative
join-commutative
satisfying_QLT2
QLT-distributive implies L is
distributive
proof
assume that
A1: L is
meet-commutative and
A2: L is
satisfying_QLT1 and
A3: L is
join-idempotent and
A4: L is
join-associative and
A5: L is
join-commutative and
A6: L is
satisfying_QLT2 and
A7: L is
QLT-distributive;
S: for v1, v0 holds (v0
"/\" v1)
= (v1
"/\" v0) by
LATTICES:def 6,
A1;
S2: for v0 holds (v0
"\/" v0)
= v0 by
A3,
ROBBINS1:def 7;
S3: for v2, v1, v0 holds ((v0
"\/" v1)
"\/" v2)
= (v0
"\/" (v1
"\/" v2)) by
A4,
LATTICES:def 5;
S4: for v1, v0 holds (v0
"\/" v1)
= (v1
"\/" v0) by
A5,
LATTICES:def 4;
let v1,v2,v3 be
Element of L;
thus thesis by
Lemma1,
S,
A2,
A6,
S2,
S3,
S4,
A7;
end;
registration
cluster
meet-commutative
satisfying_QLT1
join-idempotent
join-associative
join-commutative
satisfying_QLT2
QLT-distributive ->
distributive for non
empty
LattStr;
coherence by
Cluster1;
end
begin
theorem ::
LATQUASI:6
ThQLT2: (for v0 holds (v0
"/\" v0)
= v0) & (for v2, v1, v0 holds ((v0
"/\" v1)
"/\" v2)
= (v0
"/\" (v1
"/\" v2))) & (for v1, v0 holds (v0
"/\" v1)
= (v1
"/\" v0)) & (for v0 holds (v0
"\/" v0)
= v0) & (for v2, v1, v0 holds ((v0
"\/" v1)
"\/" v2)
= (v0
"\/" (v1
"\/" v2))) & (for v0, v2, v1 holds ((v0
"\/" (v1
"/\" v2))
"/\" (v0
"\/" v1))
= (v0
"\/" (v1
"/\" v2))) & (for v0, v2, v1 holds (v0
"/\" (v1
"\/" v2))
= ((v0
"/\" v1)
"\/" (v0
"/\" v2))) implies for v1, v2, v3 holds (v1
"\/" (v2
"/\" v3))
= ((v1
"\/" v2)
"/\" (v1
"\/" v3))
proof
assume
A2: for v0 holds (v0
"/\" v0)
= v0;
assume
A3: for v2, v1, v0 holds ((v0
"/\" v1)
"/\" v2)
= (v0
"/\" (v1
"/\" v2));
assume
A4: for v1, v0 holds (v0
"/\" v1)
= (v1
"/\" v0);
assume
A5: for v0 holds (v0
"\/" v0)
= v0;
assume
A6: for v2, v1, v0 holds ((v0
"\/" v1)
"\/" v2)
= (v0
"\/" (v1
"\/" v2));
assume
A7: for v0, v2, v1 holds ((v0
"\/" (v1
"/\" v2))
"/\" (v0
"\/" v1))
= (v0
"\/" (v1
"/\" v2));
A9: for v0, v2, v1 holds ((v0
"\/" v1)
"/\" (v0
"\/" (v1
"/\" v2)))
= (v0
"\/" (v1
"/\" v2))
proof
let v0, v2, v1;
((v0
"\/" (v1
"/\" v2))
"/\" (v0
"\/" v1))
= ((v0
"\/" v1)
"/\" (v0
"\/" (v1
"/\" v2))) by
A4;
hence thesis by
A7;
end;
assume
A10: for v0, v2, v1 holds (v0
"/\" (v1
"\/" v2))
= ((v0
"/\" v1)
"\/" (v0
"/\" v2));
A15: for v102, v101 holds (v101
"/\" v102)
= (v101
"/\" (v101
"/\" v102))
proof
let v102, v101;
(v101
"/\" v101)
= v101 by
A2;
hence thesis by
A3;
end;
A20: for v102, v101 holds (v101
"\/" v102)
= (v101
"\/" (v101
"\/" v102))
proof
let v102, v101;
(v101
"\/" v101)
= v101 by
A5;
hence thesis by
A6;
end;
A24: for v2, v0, v1 holds ((v1
"/\" v0)
"\/" (v0
"/\" v2))
= (v0
"/\" (v1
"\/" v2))
proof
let v2, v0, v1;
(v0
"/\" v1)
= (v1
"/\" v0) by
A4;
hence thesis by
A10;
end;
A28: for v102, v0, v2, v1 holds ((v0
"/\" (v1
"\/" v2))
"\/" v102)
= ((v0
"/\" v1)
"\/" ((v0
"/\" v2)
"\/" v102))
proof
let v102, v0, v2, v1;
((v0
"/\" v1)
"\/" (v0
"/\" v2))
= (v0
"/\" (v1
"\/" v2)) by
A10;
hence thesis by
A6;
end;
A32: for v102, v1, v100 holds ((v100
"/\" v1)
"\/" (v100
"/\" v102))
= (v100
"/\" ((v100
"/\" v1)
"\/" v102))
proof
let v102, v1, v100;
(v100
"/\" (v100
"/\" v1))
= (v100
"/\" v1) by
A15;
hence thesis by
A10;
end;
A35: for v0, v2, v1 holds (v0
"/\" (v1
"\/" v2))
= (v0
"/\" ((v0
"/\" v1)
"\/" v2))
proof
let v0, v2, v1;
((v0
"/\" v1)
"\/" (v0
"/\" v2))
= (v0
"/\" (v1
"\/" v2)) by
A10;
hence thesis by
A32;
end;
A39: for v1, v101, v100 holds ((v100
"/\" v101)
"\/" (v100
"/\" v1))
= (v100
"/\" (v101
"\/" (v100
"/\" v1)))
proof
let v1, v101, v100;
(v100
"/\" (v100
"/\" v1))
= (v100
"/\" v1) by
A15;
hence thesis by
A10;
end;
A42: for v0, v2, v1 holds (v0
"/\" (v1
"\/" v2))
= (v0
"/\" (v1
"\/" (v0
"/\" v2)))
proof
let v0, v2, v1;
((v0
"/\" v1)
"\/" (v0
"/\" v2))
= (v0
"/\" (v1
"\/" v2)) by
A10;
hence thesis by
A39;
end;
A46: for v102, v1, v100 holds ((v100
"\/" v1)
"/\" (v100
"\/" ((v100
"\/" v1)
"/\" v102)))
= (v100
"\/" ((v100
"\/" v1)
"/\" v102))
proof
let v102, v1, v100;
(v100
"\/" (v100
"\/" v1))
= (v100
"\/" v1) by
A20;
hence thesis by
A9;
end;
A49: for v2, v1, v0 holds ((v0
"\/" v1)
"/\" (v0
"\/" v2))
= (v0
"\/" ((v0
"\/" v1)
"/\" v2))
proof
let v2, v1, v0;
((v0
"\/" v1)
"/\" (v0
"\/" ((v0
"\/" v1)
"/\" v2)))
= ((v0
"\/" v1)
"/\" (v0
"\/" v2)) by
A42;
hence thesis by
A46;
end;
A53: for v2, v1, v101 holds ((v101
"\/" v1)
"/\" (((v101
"\/" v1)
"/\" v101)
"\/" (v1
"/\" v2)))
= (v101
"\/" (v1
"/\" v2))
proof
let v2, v1, v101;
((v101
"\/" v1)
"/\" (v101
"\/" (v1
"/\" v2)))
= (v101
"\/" (v1
"/\" v2)) by
A9;
hence thesis by
A35;
end;
A56: for v2, v1, v0 holds ((v0
"\/" v1)
"/\" ((v0
"/\" (v0
"\/" v1))
"\/" (v1
"/\" v2)))
= (v0
"\/" (v1
"/\" v2))
proof
let v2, v1, v0;
((v0
"\/" v1)
"/\" v0)
= (v0
"/\" (v0
"\/" v1)) by
A4;
hence thesis by
A53;
end;
A58: for v2, v1, v0 holds ((v0
"\/" v1)
"/\" ((v0
"/\" v0)
"\/" ((v0
"/\" v1)
"\/" (v1
"/\" v2))))
= (v0
"\/" (v1
"/\" v2))
proof
let v2, v1, v0;
((v0
"/\" (v0
"\/" v1))
"\/" (v1
"/\" v2))
= ((v0
"/\" v0)
"\/" ((v0
"/\" v1)
"\/" (v1
"/\" v2))) by
A28;
hence thesis by
A56;
end;
A60: for v2, v1, v0 holds ((v0
"\/" v1)
"/\" (v0
"\/" ((v0
"/\" v1)
"\/" (v1
"/\" v2))))
= (v0
"\/" (v1
"/\" v2))
proof
let v2, v1, v0;
(v0
"/\" v0)
= v0 by
A2;
hence thesis by
A58;
end;
A62: for v1, v2, v0 holds ((v0
"\/" v1)
"/\" (v0
"\/" (v1
"/\" (v0
"\/" v2))))
= (v0
"\/" (v1
"/\" v2))
proof
let v1, v2, v0;
((v0
"/\" v1)
"\/" (v1
"/\" v2))
= (v1
"/\" (v0
"\/" v2)) by
A24;
hence thesis by
A60;
end;
A64: for v1, v2, v0 holds (v0
"\/" (v1
"/\" (v0
"\/" v2)))
= (v0
"\/" (v1
"/\" v2))
proof
let v1, v2, v0;
((v0
"\/" v1)
"/\" (v0
"\/" (v1
"/\" (v0
"\/" v2))))
= (v0
"\/" (v1
"/\" (v0
"\/" v2))) by
A9;
hence thesis by
A62;
end;
A66: for v1, v2, v0 holds (v0
"\/" ((v0
"\/" v2)
"/\" v1))
= (v0
"\/" (v1
"/\" v2))
proof
let v1, v2, v0;
(v1
"/\" (v0
"\/" v2))
= ((v0
"\/" v2)
"/\" v1) by
A4;
hence thesis by
A64;
end;
A69: for v2, v1, v0 holds ((v0
"\/" v1)
"/\" (v0
"\/" v2))
= (v0
"\/" (v2
"/\" v1))
proof
let v2, v1, v0;
(v0
"\/" ((v0
"\/" v1)
"/\" v2))
= ((v0
"\/" v1)
"/\" (v0
"\/" v2)) by
A49;
hence thesis by
A66;
end;
let v1, v2, v3;
(v1
"\/" (v2
"/\" v3))
= ((v1
"\/" v3)
"/\" (v1
"\/" v2)) by
A69;
hence thesis by
A4;
end;
theorem ::
LATQUASI:7
Cluster2: L is
meet-idempotent
meet-associative
meet-commutative
join-idempotent
join-associative
satisfying_QLT2
distributive implies L is
distributive'
proof
assume L is
meet-idempotent
meet-associative
meet-commutative
join-idempotent
join-associative
satisfying_QLT2
distributive;
then (for v0 holds (v0
"/\" v0)
= v0) & (for v2, v1, v0 holds ((v0
"/\" v1)
"/\" v2)
= (v0
"/\" (v1
"/\" v2))) & (for v1, v0 holds (v0
"/\" v1)
= (v1
"/\" v0)) & (for v0 holds (v0
"\/" v0)
= v0) & (for v2, v1, v0 holds ((v0
"\/" v1)
"\/" v2)
= (v0
"\/" (v1
"\/" v2))) & (for v0, v2, v1 holds ((v0
"\/" (v1
"/\" v2))
"/\" (v0
"\/" v1))
= (v0
"\/" (v1
"/\" v2))) & (for v0, v2, v1 holds (v0
"/\" (v1
"\/" v2))
= ((v0
"/\" v1)
"\/" (v0
"/\" v2))) by
LATTICES:def 5,
LATTICES:def 6,
LATTICES:def 7,
LATTICES:def 11,
SHEFFER1:def 9,
ROBBINS1:def 7;
then for v1, v2, v3 holds (v1
"\/" (v2
"/\" v3))
= ((v1
"\/" v2)
"/\" (v1
"\/" v3)) by
ThQLT2;
hence thesis by
SHEFFER1:def 5;
end;
registration
cluster
meet-idempotent
meet-associative
meet-commutative
join-idempotent
join-associative
satisfying_QLT2
distributive ->
distributive' for non
empty
LattStr;
coherence by
Cluster2;
end
begin
definition
let L;
::
LATQUASI:def10
attr L is
QLT-selfdistributive means for v2, v1, v0 holds ((((v0
"/\" v1)
"\/" v2)
"/\" v1)
"\/" (v2
"/\" v0))
= ((((v0
"\/" v1)
"/\" v2)
"\/" v1)
"/\" (v2
"\/" v0));
end
theorem ::
LATQUASI:8
ThQLT3: (for v0 holds (v0
"/\" v0)
= v0) & (for v2, v1, v0 holds ((v0
"/\" v1)
"/\" v2)
= (v0
"/\" (v1
"/\" v2))) & (for v1, v0 holds (v0
"/\" v1)
= (v1
"/\" v0)) & (for v0, v2, v1 holds ((v0
"/\" (v1
"\/" v2))
"\/" (v0
"/\" v1))
= (v0
"/\" (v1
"\/" v2))) & (for v0 holds (v0
"\/" v0)
= v0) & (for v2, v1, v0 holds ((v0
"\/" v1)
"\/" v2)
= (v0
"\/" (v1
"\/" v2))) & (for v1, v0 holds (v0
"\/" v1)
= (v1
"\/" v0)) & (for v0, v2, v1 holds ((v0
"\/" (v1
"/\" v2))
"/\" (v0
"\/" v1))
= (v0
"\/" (v1
"/\" v2))) & (for v2, v1, v0 holds ((((v0
"/\" v1)
"\/" v2)
"/\" v1)
"\/" (v2
"/\" v0))
= ((((v0
"\/" v1)
"/\" v2)
"\/" v1)
"/\" (v2
"\/" v0))) implies for v1, v2, v3 holds (v1
"\/" (v2
"/\" v3))
= ((v1
"\/" v2)
"/\" (v1
"\/" v3))
proof
assume
A2: for v0 holds (v0
"/\" v0)
= v0;
assume
A3: for v2, v1, v0 holds ((v0
"/\" v1)
"/\" v2)
= (v0
"/\" (v1
"/\" v2));
assume
A4: for v1, v0 holds (v0
"/\" v1)
= (v1
"/\" v0);
assume
A5: for v0, v2, v1 holds ((v0
"/\" (v1
"\/" v2))
"\/" (v0
"/\" v1))
= (v0
"/\" (v1
"\/" v2));
assume
A6: for v0 holds (v0
"\/" v0)
= v0;
assume
A7: for v2, v1, v0 holds ((v0
"\/" v1)
"\/" v2)
= (v0
"\/" (v1
"\/" v2));
assume
A8: for v1, v0 holds (v0
"\/" v1)
= (v1
"\/" v0);
assume
A9: for v0, v2, v1 holds ((v0
"\/" (v1
"/\" v2))
"/\" (v0
"\/" v1))
= (v0
"\/" (v1
"/\" v2));
A11: for v0, v2, v1 holds ((v0
"\/" v1)
"/\" (v0
"\/" (v1
"/\" v2)))
= (v0
"\/" (v1
"/\" v2))
proof
let v0, v2, v1;
((v0
"\/" (v1
"/\" v2))
"/\" (v0
"\/" v1))
= ((v0
"\/" v1)
"/\" (v0
"\/" (v1
"/\" v2))) by
A4;
hence thesis by
A9;
end;
assume
A12: for v2, v1, v0 holds ((((v0
"/\" v1)
"\/" v2)
"/\" v1)
"\/" (v2
"/\" v0))
= ((((v0
"\/" v1)
"/\" v2)
"\/" v1)
"/\" (v2
"\/" v0));
A14: for v2, v1, v0 holds ((v1
"/\" ((v0
"/\" v1)
"\/" v2))
"\/" (v2
"/\" v0))
= ((((v0
"\/" v1)
"/\" v2)
"\/" v1)
"/\" (v2
"\/" v0))
proof
let v2, v1, v0;
(((v0
"/\" v1)
"\/" v2)
"/\" v1)
= (v1
"/\" ((v0
"/\" v1)
"\/" v2)) by
A4;
hence thesis by
A12;
end;
A17: for v2, v0, v1 holds ((v2
"/\" v1)
"\/" (v0
"/\" ((v1
"/\" v0)
"\/" v2)))
= ((((v1
"\/" v0)
"/\" v2)
"\/" v0)
"/\" (v2
"\/" v1))
proof
let v2, v0, v1;
((v0
"/\" ((v1
"/\" v0)
"\/" v2))
"\/" (v2
"/\" v1))
= ((v2
"/\" v1)
"\/" (v0
"/\" ((v1
"/\" v0)
"\/" v2))) by
A8;
hence thesis by
A14;
end;
A20: for v0, v2, v1 holds ((v0
"/\" v1)
"\/" (v2
"/\" ((v1
"/\" v2)
"\/" v0)))
= ((v2
"\/" ((v1
"\/" v2)
"/\" v0))
"/\" (v0
"\/" v1))
proof
let v0, v2, v1;
(((v1
"\/" v2)
"/\" v0)
"\/" v2)
= (v2
"\/" ((v1
"\/" v2)
"/\" v0)) by
A8;
hence thesis by
A17;
end;
A23: for v0, v2, v1 holds ((v0
"/\" v1)
"\/" (v0
"/\" (v1
"\/" v2)))
= (v0
"/\" (v1
"\/" v2))
proof
let v0, v2, v1;
((v0
"/\" (v1
"\/" v2))
"\/" (v0
"/\" v1))
= ((v0
"/\" v1)
"\/" (v0
"/\" (v1
"\/" v2))) by
A8;
hence thesis by
A5;
end;
A26: for v102, v101 holds (v101
"/\" v102)
= (v101
"/\" (v101
"/\" v102))
proof
let v102, v101;
(v101
"/\" v101)
= v101 by
A2;
hence thesis by
A3;
end;
A31: for v102, v100 holds ((v100
"/\" v102)
"/\" v102)
= (v100
"/\" v102)
proof
let v102, v100;
(v102
"/\" v102)
= v102 by
A2;
hence thesis by
A3;
end;
A34: for v1, v0 holds (v1
"/\" (v0
"/\" v1))
= (v0
"/\" v1)
proof
let v1, v0;
((v0
"/\" v1)
"/\" v1)
= (v1
"/\" (v0
"/\" v1)) by
A4;
hence thesis by
A31;
end;
A37: for v2, v0, v1 holds ((v1
"/\" v0)
"/\" v2)
= (v0
"/\" (v1
"/\" v2))
proof
let v2, v0, v1;
(v0
"/\" v1)
= (v1
"/\" v0) by
A4;
hence thesis by
A3;
end;
A40: for v0, v2, v1 holds (v0
"/\" (v1
"/\" v2))
= (v1
"/\" (v0
"/\" v2))
proof
let v0, v2, v1;
((v0
"/\" v1)
"/\" v2)
= (v0
"/\" (v1
"/\" v2)) by
A3;
hence thesis by
A37;
end;
A43: for v102, v101 holds (v101
"\/" v102)
= (v101
"\/" (v101
"\/" v102))
proof
let v102, v101;
(v101
"\/" v101)
= v101 by
A6;
hence thesis by
A7;
end;
A48: for v102, v100 holds ((v100
"\/" v102)
"\/" v102)
= (v100
"\/" v102)
proof
let v102, v100;
(v102
"\/" v102)
= v102 by
A6;
hence thesis by
A7;
end;
A51: for v1, v0 holds (v1
"\/" (v0
"\/" v1))
= (v0
"\/" v1)
proof
let v1, v0;
((v0
"\/" v1)
"\/" v1)
= (v1
"\/" (v0
"\/" v1)) by
A8;
hence thesis by
A48;
end;
A54: for v2, v0, v1 holds ((v1
"\/" v0)
"\/" v2)
= (v0
"\/" (v1
"\/" v2))
proof
let v2, v0, v1;
(v0
"\/" v1)
= (v1
"\/" v0) by
A8;
hence thesis by
A7;
end;
A57: for v0, v2, v1 holds (v0
"\/" (v1
"\/" v2))
= (v1
"\/" (v0
"\/" v2))
proof
let v0, v2, v1;
((v0
"\/" v1)
"\/" v2)
= (v0
"\/" (v1
"\/" v2)) by
A7;
hence thesis by
A54;
end;
A60: for v102, v0, v2, v1 holds ((v0
"\/" (v1
"/\" v2))
"/\" v102)
= ((v0
"\/" v1)
"/\" ((v0
"\/" (v1
"/\" v2))
"/\" v102))
proof
let v102, v0, v2, v1;
((v0
"\/" v1)
"/\" (v0
"\/" (v1
"/\" v2)))
= (v0
"\/" (v1
"/\" v2)) by
A11;
hence thesis by
A3;
end;
A65: for v100, v0, v102, v1 holds ((v100
"\/" (v0
"/\" v1))
"/\" (v100
"\/" (v0
"/\" (v1
"/\" v102))))
= (v100
"\/" ((v0
"/\" v1)
"/\" v102))
proof
let v100, v0, v102, v1;
((v0
"/\" v1)
"/\" v102)
= (v0
"/\" (v1
"/\" v102)) by
A3;
hence thesis by
A11;
end;
A68: for v0, v1, v3, v2 holds ((v0
"\/" (v1
"/\" v2))
"/\" (v0
"\/" (v1
"/\" (v2
"/\" v3))))
= (v0
"\/" (v1
"/\" (v2
"/\" v3)))
proof
let v0, v1, v3, v2;
((v1
"/\" v2)
"/\" v3)
= (v1
"/\" (v2
"/\" v3)) by
A3;
hence thesis by
A65;
end;
A71: for v102, v101 holds (v101
"/\" (v101
"\/" (v101
"/\" v102)))
= (v101
"\/" (v101
"/\" v102))
proof
let v102, v101;
(v101
"\/" v101)
= v101 by
A6;
hence thesis by
A11;
end;
A75: for v102, v101 holds (((v101
"/\" v102)
"\/" v101)
"/\" (v101
"/\" v102))
= ((v101
"/\" v102)
"\/" (v101
"/\" v102))
proof
let v102, v101;
((v101
"/\" v102)
"\/" (v101
"/\" v102))
= (v101
"/\" v102) by
A6;
hence thesis by
A11;
end;
A78: for v1, v0 holds ((v0
"\/" (v0
"/\" v1))
"/\" (v0
"/\" v1))
= ((v0
"/\" v1)
"\/" (v0
"/\" v1))
proof
let v1, v0;
((v0
"/\" v1)
"\/" v0)
= (v0
"\/" (v0
"/\" v1)) by
A8;
hence thesis by
A75;
end;
A80: for v1, v0 holds ((v0
"/\" v1)
"/\" (v0
"\/" (v0
"/\" v1)))
= ((v0
"/\" v1)
"\/" (v0
"/\" v1))
proof
let v1, v0;
((v0
"\/" (v0
"/\" v1))
"/\" (v0
"/\" v1))
= ((v0
"/\" v1)
"/\" (v0
"\/" (v0
"/\" v1))) by
A4;
hence thesis by
A78;
end;
A82: for v1, v0 holds (v0
"/\" (v1
"/\" (v0
"\/" (v0
"/\" v1))))
= ((v0
"/\" v1)
"\/" (v0
"/\" v1))
proof
let v1, v0;
((v0
"/\" v1)
"/\" (v0
"\/" (v0
"/\" v1)))
= (v0
"/\" (v1
"/\" (v0
"\/" (v0
"/\" v1)))) by
A3;
hence thesis by
A80;
end;
A84: for v1, v0 holds (v0
"/\" (v1
"/\" (v0
"\/" (v0
"/\" v1))))
= (v0
"/\" v1)
proof
let v1, v0;
((v0
"/\" v1)
"\/" (v0
"/\" v1))
= (v0
"/\" v1) by
A6;
hence thesis by
A82;
end;
A86: for v0, v2, v1 holds ((v1
"\/" v0)
"/\" (v0
"\/" (v1
"/\" v2)))
= (v0
"\/" (v1
"/\" v2))
proof
let v0, v2, v1;
(v0
"\/" v1)
= (v1
"\/" v0) by
A8;
hence thesis by
A11;
end;
A90: for v102, v101 holds (v101
"\/" (v102
"/\" ((v101
"/\" v102)
"\/" v101)))
= ((v102
"\/" ((v101
"\/" v102)
"/\" v101))
"/\" (v101
"\/" v101))
proof
let v102, v101;
(v101
"/\" v101)
= v101 by
A2;
hence thesis by
A20;
end;
A93: for v1, v0 holds (v0
"\/" (v1
"/\" (v0
"\/" (v0
"/\" v1))))
= ((v1
"\/" ((v0
"\/" v1)
"/\" v0))
"/\" (v0
"\/" v0))
proof
let v1, v0;
((v0
"/\" v1)
"\/" v0)
= (v0
"\/" (v0
"/\" v1)) by
A8;
hence thesis by
A90;
end;
A95: for v1, v0 holds (v0
"\/" (v1
"/\" (v0
"\/" (v0
"/\" v1))))
= ((v1
"\/" (v0
"/\" (v0
"\/" v1)))
"/\" (v0
"\/" v0))
proof
let v1, v0;
((v0
"\/" v1)
"/\" v0)
= (v0
"/\" (v0
"\/" v1)) by
A4;
hence thesis by
A93;
end;
A97: for v1, v0 holds (v0
"\/" (v1
"/\" (v0
"\/" (v0
"/\" v1))))
= ((v1
"\/" (v0
"/\" (v0
"\/" v1)))
"/\" v0)
proof
let v1, v0;
(v0
"\/" v0)
= v0 by
A6;
hence thesis by
A95;
end;
A99: for v1, v0 holds (v0
"\/" (v1
"/\" (v0
"\/" (v0
"/\" v1))))
= (v0
"/\" (v1
"\/" (v0
"/\" (v0
"\/" v1))))
proof
let v1, v0;
((v1
"\/" (v0
"/\" (v0
"\/" v1)))
"/\" v0)
= (v0
"/\" (v1
"\/" (v0
"/\" (v0
"\/" v1)))) by
A4;
hence thesis by
A97;
end;
A101: for v0, v2, v1 holds ((v1
"/\" v0)
"\/" (v2
"/\" ((v1
"/\" v2)
"\/" v0)))
= ((v2
"\/" ((v1
"\/" v2)
"/\" v0))
"/\" (v0
"\/" v1))
proof
let v0, v2, v1;
(v0
"/\" v1)
= (v1
"/\" v0) by
A4;
hence thesis by
A20;
end;
A104: for v0, v1, v2 holds ((v0
"/\" v1)
"\/" (v2
"/\" ((v2
"/\" v1)
"\/" v0)))
= ((v2
"\/" ((v1
"\/" v2)
"/\" v0))
"/\" (v0
"\/" v1))
proof
let v0, v1, v2;
(v1
"/\" v2)
= (v2
"/\" v1) by
A4;
hence thesis by
A20;
end;
A107: for v101, v1, v0 holds (((v0
"/\" v1)
"\/" v101)
"/\" ((v101
"\/" ((v1
"\/" v101)
"/\" v0))
"/\" (v0
"\/" v1)))
= ((v0
"/\" v1)
"\/" (v101
"/\" ((v1
"/\" v101)
"\/" v0)))
proof
let v101, v1, v0;
((v0
"/\" v1)
"\/" (v101
"/\" ((v1
"/\" v101)
"\/" v0)))
= ((v101
"\/" ((v1
"\/" v101)
"/\" v0))
"/\" (v0
"\/" v1)) by
A20;
hence thesis by
A11;
end;
A110: for v2, v1, v0 holds (((v0
"/\" v1)
"\/" v2)
"/\" ((v2
"\/" ((v1
"\/" v2)
"/\" v0))
"/\" (v0
"\/" v1)))
= ((v2
"\/" ((v1
"\/" v2)
"/\" v0))
"/\" (v0
"\/" v1))
proof
let v2, v1, v0;
((v0
"/\" v1)
"\/" (v2
"/\" ((v1
"/\" v2)
"\/" v0)))
= ((v2
"\/" ((v1
"\/" v2)
"/\" v0))
"/\" (v0
"\/" v1)) by
A20;
hence thesis by
A107;
end;
A113: for v102, v101 holds (v101
"\/" (v101
"/\" (v101
"\/" v102)))
= (v101
"/\" (v101
"\/" v102))
proof
let v102, v101;
(v101
"/\" v101)
= v101 by
A2;
hence thesis by
A23;
end;
A117: for v102, v101 holds (((v101
"\/" v102)
"/\" v101)
"\/" (v101
"\/" v102))
= ((v101
"\/" v102)
"/\" (v101
"\/" v102))
proof
let v102, v101;
((v101
"\/" v102)
"/\" (v101
"\/" v102))
= (v101
"\/" v102) by
A2;
hence thesis by
A23;
end;
A120: for v1, v0 holds ((v0
"/\" (v0
"\/" v1))
"\/" (v0
"\/" v1))
= ((v0
"\/" v1)
"/\" (v0
"\/" v1))
proof
let v1, v0;
((v0
"\/" v1)
"/\" v0)
= (v0
"/\" (v0
"\/" v1)) by
A4;
hence thesis by
A117;
end;
A122: for v1, v0 holds ((v0
"\/" v1)
"\/" (v0
"/\" (v0
"\/" v1)))
= ((v0
"\/" v1)
"/\" (v0
"\/" v1))
proof
let v1, v0;
((v0
"/\" (v0
"\/" v1))
"\/" (v0
"\/" v1))
= ((v0
"\/" v1)
"\/" (v0
"/\" (v0
"\/" v1))) by
A8;
hence thesis by
A120;
end;
A124: for v1, v0 holds (v0
"\/" (v1
"\/" (v0
"/\" (v0
"\/" v1))))
= ((v0
"\/" v1)
"/\" (v0
"\/" v1))
proof
let v1, v0;
((v0
"\/" v1)
"\/" (v0
"/\" (v0
"\/" v1)))
= (v0
"\/" (v1
"\/" (v0
"/\" (v0
"\/" v1)))) by
A7;
hence thesis by
A122;
end;
A126: for v1, v0 holds (v0
"\/" (v1
"\/" (v0
"/\" (v0
"\/" v1))))
= (v0
"\/" v1)
proof
let v1, v0;
((v0
"\/" v1)
"/\" (v0
"\/" v1))
= (v0
"\/" v1) by
A2;
hence thesis by
A124;
end;
A128: for v0, v2, v1 holds ((v1
"/\" v0)
"\/" (v0
"/\" (v1
"\/" v2)))
= (v0
"/\" (v1
"\/" v2))
proof
let v0, v2, v1;
(v0
"/\" v1)
= (v1
"/\" v0) by
A4;
hence thesis by
A23;
end;
A131: for v0, v2, v1 holds ((v0
"/\" v1)
"\/" ((v1
"\/" v2)
"/\" v0))
= (v0
"/\" (v1
"\/" v2))
proof
let v0, v2, v1;
(v0
"/\" (v1
"\/" v2))
= ((v1
"\/" v2)
"/\" v0) by
A4;
hence thesis by
A23;
end;
A133: for v0, v1, v2 holds ((v0
"/\" v1)
"\/" (v0
"/\" (v2
"\/" v1)))
= (v0
"/\" (v1
"\/" v2))
proof
let v0, v1, v2;
(v1
"\/" v2)
= (v2
"\/" v1) by
A8;
hence thesis by
A23;
end;
A136: for v101, v2, v1 holds (((v101
"/\" v1)
"\/" v101)
"/\" (v101
"/\" (v1
"\/" v2)))
= ((v101
"/\" v1)
"\/" (v101
"/\" (v1
"\/" v2)))
proof
let v101, v2, v1;
((v101
"/\" v1)
"\/" (v101
"/\" (v1
"\/" v2)))
= (v101
"/\" (v1
"\/" v2)) by
A23;
hence thesis by
A11;
end;
A139: for v0, v2, v1 holds ((v0
"\/" (v0
"/\" v1))
"/\" (v0
"/\" (v1
"\/" v2)))
= ((v0
"/\" v1)
"\/" (v0
"/\" (v1
"\/" v2)))
proof
let v0, v2, v1;
((v0
"/\" v1)
"\/" v0)
= (v0
"\/" (v0
"/\" v1)) by
A8;
hence thesis by
A136;
end;
A141: for v2, v1, v0 holds (v0
"/\" ((v0
"\/" (v0
"/\" v1))
"/\" (v1
"\/" v2)))
= ((v0
"/\" v1)
"\/" (v0
"/\" (v1
"\/" v2)))
proof
let v2, v1, v0;
((v0
"\/" (v0
"/\" v1))
"/\" (v0
"/\" (v1
"\/" v2)))
= (v0
"/\" ((v0
"\/" (v0
"/\" v1))
"/\" (v1
"\/" v2))) by
A40;
hence thesis by
A139;
end;
A143: for v2, v1, v0 holds (v0
"/\" ((v0
"\/" (v0
"/\" v1))
"/\" (v1
"\/" v2)))
= (v0
"/\" (v1
"\/" v2))
proof
let v2, v1, v0;
((v0
"/\" v1)
"\/" (v0
"/\" (v1
"\/" v2)))
= (v0
"/\" (v1
"\/" v2)) by
A23;
hence thesis by
A141;
end;
A146: for v101, v2, v1 holds (((v101
"\/" v1)
"/\" v101)
"\/" (v101
"\/" (v1
"/\" v2)))
= ((v101
"\/" v1)
"/\" (v101
"\/" (v1
"/\" v2)))
proof
let v101, v2, v1;
((v101
"\/" v1)
"/\" (v101
"\/" (v1
"/\" v2)))
= (v101
"\/" (v1
"/\" v2)) by
A11;
hence thesis by
A23;
end;
A149: for v0, v2, v1 holds ((v0
"/\" (v0
"\/" v1))
"\/" (v0
"\/" (v1
"/\" v2)))
= ((v0
"\/" v1)
"/\" (v0
"\/" (v1
"/\" v2)))
proof
let v0, v2, v1;
((v0
"\/" v1)
"/\" v0)
= (v0
"/\" (v0
"\/" v1)) by
A4;
hence thesis by
A146;
end;
A151: for v2, v1, v0 holds (v0
"\/" ((v0
"/\" (v0
"\/" v1))
"\/" (v1
"/\" v2)))
= ((v0
"\/" v1)
"/\" (v0
"\/" (v1
"/\" v2)))
proof
let v2, v1, v0;
((v0
"/\" (v0
"\/" v1))
"\/" (v0
"\/" (v1
"/\" v2)))
= (v0
"\/" ((v0
"/\" (v0
"\/" v1))
"\/" (v1
"/\" v2))) by
A57;
hence thesis by
A149;
end;
A153: for v2, v1, v0 holds (v0
"\/" ((v0
"/\" (v0
"\/" v1))
"\/" (v1
"/\" v2)))
= (v0
"\/" (v1
"/\" v2))
proof
let v2, v1, v0;
((v0
"\/" v1)
"/\" (v0
"\/" (v1
"/\" v2)))
= (v0
"\/" (v1
"/\" v2)) by
A11;
hence thesis by
A151;
end;
A156: for v1, v101, v100 holds ((v100
"/\" v101)
"/\" (v101
"/\" v1))
= (v100
"/\" (v101
"/\" v1))
proof
let v1, v101, v100;
(v101
"/\" (v101
"/\" v1))
= (v101
"/\" v1) by
A26;
hence thesis by
A3;
end;
A159: for v2, v1, v0 holds (v1
"/\" ((v0
"/\" v1)
"/\" v2))
= (v0
"/\" (v1
"/\" v2))
proof
let v2, v1, v0;
((v0
"/\" v1)
"/\" (v1
"/\" v2))
= (v1
"/\" ((v0
"/\" v1)
"/\" v2)) by
A40;
hence thesis by
A156;
end;
A162: for v1, v2, v0 holds (v0
"/\" (v1
"/\" (v0
"/\" v2)))
= (v1
"/\" (v0
"/\" v2))
proof
let v1, v2, v0;
((v1
"/\" v0)
"/\" v2)
= (v1
"/\" (v0
"/\" v2)) by
A3;
hence thesis by
A159;
end;
A165: for v102, v1, v100 holds ((v100
"/\" v1)
"\/" (v102
"/\" (((v100
"/\" v1)
"/\" v102)
"\/" v100)))
= ((v102
"\/" (((v100
"/\" v1)
"\/" v102)
"/\" v100))
"/\" (v100
"\/" (v100
"/\" v1)))
proof
let v102, v1, v100;
(v100
"/\" (v100
"/\" v1))
= (v100
"/\" v1) by
A26;
hence thesis by
A20;
end;
A168: for v0, v2, v1 holds ((v0
"/\" v1)
"\/" (v2
"/\" ((v0
"/\" (v1
"/\" v2))
"\/" v0)))
= ((v2
"\/" (((v0
"/\" v1)
"\/" v2)
"/\" v0))
"/\" (v0
"\/" (v0
"/\" v1)))
proof
let v0, v2, v1;
((v0
"/\" v1)
"/\" v2)
= (v0
"/\" (v1
"/\" v2)) by
A3;
hence thesis by
A165;
end;
A170: for v0, v2, v1 holds ((v0
"/\" v1)
"\/" (v2
"/\" (v0
"\/" (v0
"/\" (v1
"/\" v2)))))
= ((v2
"\/" (((v0
"/\" v1)
"\/" v2)
"/\" v0))
"/\" (v0
"\/" (v0
"/\" v1)))
proof
let v0, v2, v1;
((v0
"/\" (v1
"/\" v2))
"\/" v0)
= (v0
"\/" (v0
"/\" (v1
"/\" v2))) by
A8;
hence thesis by
A168;
end;
A172: for v0, v2, v1 holds ((v0
"/\" v1)
"\/" (v2
"/\" (v0
"\/" (v0
"/\" (v1
"/\" v2)))))
= ((v2
"\/" (v0
"/\" ((v0
"/\" v1)
"\/" v2)))
"/\" (v0
"\/" (v0
"/\" v1)))
proof
let v0, v2, v1;
(((v0
"/\" v1)
"\/" v2)
"/\" v0)
= (v0
"/\" ((v0
"/\" v1)
"\/" v2)) by
A4;
hence thesis by
A170;
end;
A174: for v0, v2, v1 holds ((v0
"/\" v1)
"\/" (v2
"/\" (v0
"\/" (v0
"/\" (v1
"/\" v2)))))
= ((v0
"\/" (v0
"/\" v1))
"/\" (v2
"\/" (v0
"/\" ((v0
"/\" v1)
"\/" v2))))
proof
let v0, v2, v1;
((v2
"\/" (v0
"/\" ((v0
"/\" v1)
"\/" v2)))
"/\" (v0
"\/" (v0
"/\" v1)))
= ((v0
"\/" (v0
"/\" v1))
"/\" (v2
"\/" (v0
"/\" ((v0
"/\" v1)
"\/" v2)))) by
A4;
hence thesis by
A172;
end;
A177: for v0, v100, v1 holds (v100
"/\" (v0
"/\" (v1
"/\" v100)))
= ((v0
"/\" v1)
"/\" v100)
proof
let v0, v100, v1;
((v0
"/\" v1)
"/\" v100)
= (v0
"/\" (v1
"/\" v100)) by
A3;
hence thesis by
A34;
end;
A180: for v1, v0, v2 holds (v0
"/\" (v1
"/\" (v2
"/\" v0)))
= (v1
"/\" (v2
"/\" v0))
proof
let v1, v0, v2;
((v1
"/\" v2)
"/\" v0)
= (v1
"/\" (v2
"/\" v0)) by
A3;
hence thesis by
A177;
end;
A183: for v100, v101, v1 holds ((v100
"\/" v101)
"/\" (v100
"\/" (v1
"/\" v101)))
= (v100
"\/" (v101
"/\" (v1
"/\" v101)))
proof
let v100, v101, v1;
(v101
"/\" (v1
"/\" v101))
= (v1
"/\" v101) by
A34;
hence thesis by
A11;
end;
A186: for v0, v1, v2 holds ((v0
"\/" v1)
"/\" (v0
"\/" (v2
"/\" v1)))
= (v0
"\/" (v2
"/\" v1))
proof
let v0, v1, v2;
(v1
"/\" (v2
"/\" v1))
= (v2
"/\" v1) by
A34;
hence thesis by
A183;
end;
A189: for v1, v101, v100 holds ((v100
"\/" v101)
"\/" (v101
"\/" v1))
= (v100
"\/" (v101
"\/" v1))
proof
let v1, v101, v100;
(v101
"\/" (v101
"\/" v1))
= (v101
"\/" v1) by
A43;
hence thesis by
A7;
end;
A192: for v2, v1, v0 holds (v1
"\/" ((v0
"\/" v1)
"\/" v2))
= (v0
"\/" (v1
"\/" v2))
proof
let v2, v1, v0;
((v0
"\/" v1)
"\/" (v1
"\/" v2))
= (v1
"\/" ((v0
"\/" v1)
"\/" v2)) by
A57;
hence thesis by
A189;
end;
A195: for v1, v2, v0 holds (v0
"\/" (v1
"\/" (v0
"\/" v2)))
= (v1
"\/" (v0
"\/" v2))
proof
let v1, v2, v0;
((v1
"\/" v0)
"\/" v2)
= (v1
"\/" (v0
"\/" v2)) by
A7;
hence thesis by
A192;
end;
A198: for v102, v1, v100 holds ((v100
"\/" v1)
"/\" (v100
"\/" ((v100
"\/" v1)
"/\" v102)))
= (v100
"\/" ((v100
"\/" v1)
"/\" v102))
proof
let v102, v1, v100;
(v100
"\/" (v100
"\/" v1))
= (v100
"\/" v1) by
A43;
hence thesis by
A11;
end;
A202: for v0, v100, v1 holds (v100
"\/" (v0
"\/" (v1
"\/" v100)))
= ((v0
"\/" v1)
"\/" v100)
proof
let v0, v100, v1;
((v0
"\/" v1)
"\/" v100)
= (v0
"\/" (v1
"\/" v100)) by
A7;
hence thesis by
A51;
end;
A205: for v1, v0, v2 holds (v0
"\/" (v1
"\/" (v2
"\/" v0)))
= (v1
"\/" (v2
"\/" v0))
proof
let v1, v0, v2;
((v1
"\/" v2)
"\/" v0)
= (v1
"\/" (v2
"\/" v0)) by
A7;
hence thesis by
A202;
end;
A208: for v102, v100, v1 holds ((v1
"\/" v100)
"/\" (v100
"\/" ((v1
"\/" v100)
"/\" v102)))
= (v100
"\/" ((v1
"\/" v100)
"/\" v102))
proof
let v102, v100, v1;
(v100
"\/" (v1
"\/" v100))
= (v1
"\/" v100) by
A51;
hence thesis by
A11;
end;
A212: for v102, v1, v100 holds ((v100
"\/" (v100
"/\" v1))
"/\" v102)
= (v100
"/\" ((v100
"\/" (v100
"/\" v1))
"/\" v102))
proof
let v102, v1, v100;
(v100
"/\" (v100
"\/" (v100
"/\" v1)))
= (v100
"\/" (v100
"/\" v1)) by
A71;
hence thesis by
A3;
end;
A217: for v1, v0 holds ((v0
"/\" v1)
"/\" ((v0
"/\" v1)
"\/" ((v0
"/\" v1)
"/\" ((v1
"/\" (v0
"/\" v1))
"\/" v0))))
= (((v0
"/\" v1)
"\/" ((v1
"\/" (v0
"/\" v1))
"/\" v0))
"/\" (v0
"\/" v1))
proof
let v1, v0;
((v0
"/\" v1)
"\/" ((v0
"/\" v1)
"/\" ((v1
"/\" (v0
"/\" v1))
"\/" v0)))
= (((v0
"/\" v1)
"\/" ((v1
"\/" (v0
"/\" v1))
"/\" v0))
"/\" (v0
"\/" v1)) by
A20;
hence thesis by
A71;
end;
A219: for v1, v0 holds ((v0
"/\" v1)
"/\" ((v0
"/\" v1)
"\/" ((v0
"/\" v1)
"/\" ((v0
"/\" v1)
"\/" v0))))
= (((v0
"/\" v1)
"\/" ((v1
"\/" (v0
"/\" v1))
"/\" v0))
"/\" (v0
"\/" v1))
proof
let v1, v0;
(v1
"/\" (v0
"/\" v1))
= (v0
"/\" v1) by
A34;
hence thesis by
A217;
end;
A221: for v1, v0 holds ((v0
"/\" v1)
"/\" ((v0
"/\" v1)
"\/" ((v0
"/\" v1)
"/\" (v0
"\/" (v0
"/\" v1)))))
= (((v0
"/\" v1)
"\/" ((v1
"\/" (v0
"/\" v1))
"/\" v0))
"/\" (v0
"\/" v1))
proof
let v1, v0;
((v0
"/\" v1)
"\/" v0)
= (v0
"\/" (v0
"/\" v1)) by
A8;
hence thesis by
A219;
end;
A223: for v1, v0 holds ((v0
"/\" v1)
"/\" ((v0
"/\" v1)
"\/" (v0
"/\" (v1
"/\" (v0
"\/" (v0
"/\" v1))))))
= (((v0
"/\" v1)
"\/" ((v1
"\/" (v0
"/\" v1))
"/\" v0))
"/\" (v0
"\/" v1))
proof
let v1, v0;
((v0
"/\" v1)
"/\" (v0
"\/" (v0
"/\" v1)))
= (v0
"/\" (v1
"/\" (v0
"\/" (v0
"/\" v1)))) by
A3;
hence thesis by
A221;
end;
A225: for v1, v0 holds ((v0
"/\" v1)
"/\" ((v0
"/\" v1)
"\/" (v0
"/\" v1)))
= (((v0
"/\" v1)
"\/" ((v1
"\/" (v0
"/\" v1))
"/\" v0))
"/\" (v0
"\/" v1))
proof
let v1, v0;
(v0
"/\" (v1
"/\" (v0
"\/" (v0
"/\" v1))))
= (v0
"/\" v1) by
A84;
hence thesis by
A223;
end;
A227: for v1, v0 holds ((v0
"/\" v1)
"/\" (v0
"/\" v1))
= (((v0
"/\" v1)
"\/" ((v1
"\/" (v0
"/\" v1))
"/\" v0))
"/\" (v0
"\/" v1))
proof
let v1, v0;
((v0
"/\" v1)
"\/" (v0
"/\" v1))
= (v0
"/\" v1) by
A6;
hence thesis by
A225;
end;
A229: for v1, v0 holds (v0
"/\" v1)
= (((v0
"/\" v1)
"\/" ((v1
"\/" (v0
"/\" v1))
"/\" v0))
"/\" (v0
"\/" v1))
proof
let v1, v0;
((v0
"/\" v1)
"/\" (v0
"/\" v1))
= (v0
"/\" v1) by
A2;
hence thesis by
A227;
end;
A231: for v1, v0 holds (v0
"/\" v1)
= (((v0
"/\" v1)
"\/" (v0
"/\" (v1
"\/" (v0
"/\" v1))))
"/\" (v0
"\/" v1))
proof
let v1, v0;
((v1
"\/" (v0
"/\" v1))
"/\" v0)
= (v0
"/\" (v1
"\/" (v0
"/\" v1))) by
A4;
hence thesis by
A229;
end;
A233: for v1, v0 holds (v0
"/\" v1)
= ((v0
"/\" (v1
"\/" (v0
"/\" v1)))
"/\" (v0
"\/" v1))
proof
let v1, v0;
((v0
"/\" v1)
"\/" (v0
"/\" (v1
"\/" (v0
"/\" v1))))
= (v0
"/\" (v1
"\/" (v0
"/\" v1))) by
A23;
hence thesis by
A231;
end;
A235: for v1, v0 holds (v0
"/\" v1)
= (v0
"/\" ((v1
"\/" (v0
"/\" v1))
"/\" (v0
"\/" v1)))
proof
let v1, v0;
((v0
"/\" (v1
"\/" (v0
"/\" v1)))
"/\" (v0
"\/" v1))
= (v0
"/\" ((v1
"\/" (v0
"/\" v1))
"/\" (v0
"\/" v1))) by
A3;
hence thesis by
A233;
end;
A239: for v100, v1, v101 holds (v100
"/\" (v101
"\/" (v101
"/\" v1)))
= (v101
"/\" (v100
"/\" (v101
"\/" (v101
"/\" v1))))
proof
let v100, v1, v101;
(v101
"/\" (v101
"\/" (v101
"/\" v1)))
= (v101
"\/" (v101
"/\" v1)) by
A71;
hence thesis by
A40;
end;
A244: for v2, v1, v0 holds ((v0
"\/" (v0
"/\" v1))
"/\" (v1
"\/" v2))
= (v0
"/\" (v1
"\/" v2))
proof
let v2, v1, v0;
(v0
"/\" ((v0
"\/" (v0
"/\" v1))
"/\" (v1
"\/" v2)))
= ((v0
"\/" (v0
"/\" v1))
"/\" (v1
"\/" v2)) by
A212;
hence thesis by
A143;
end;
A246: for v1, v0 holds (v1
"/\" (v0
"\/" (v0
"/\" v1)))
= (v0
"/\" v1)
proof
let v1, v0;
(v0
"/\" (v1
"/\" (v0
"\/" (v0
"/\" v1))))
= (v1
"/\" (v0
"\/" (v0
"/\" v1))) by
A239;
hence thesis by
A84;
end;
A249: for v1, v0 holds (v0
"\/" (v0
"/\" v1))
= (v0
"/\" (v1
"\/" (v0
"/\" (v0
"\/" v1))))
proof
let v1, v0;
(v1
"/\" (v0
"\/" (v0
"/\" v1)))
= (v0
"/\" v1) by
A246;
hence thesis by
A99;
end;
A251: for v1, v0 holds (v0
"/\" (v1
"\/" (v0
"/\" v1)))
= (v1
"/\" v0)
proof
let v1, v0;
(v1
"/\" v0)
= (v0
"/\" v1) by
A4;
hence thesis by
A246;
end;
A254: for v1, v0 holds (((v1
"/\" (v0
"/\" v1))
"\/" v0)
"/\" (((v0
"/\" v1)
"\/" ((v1
"\/" (v0
"/\" v1))
"/\" v0))
"/\" (v0
"\/" v1)))
= ((v0
"/\" v1)
"/\" ((v1
"/\" (v0
"/\" v1))
"\/" v0))
proof
let v1, v0;
((v0
"/\" v1)
"\/" ((v0
"/\" v1)
"/\" ((v1
"/\" (v0
"/\" v1))
"\/" v0)))
= (((v0
"/\" v1)
"\/" ((v1
"\/" (v0
"/\" v1))
"/\" v0))
"/\" (v0
"\/" v1)) by
A20;
hence thesis by
A246;
end;
A257: for v0, v1 holds (((v1
"/\" v0)
"\/" v1)
"/\" (((v1
"/\" v0)
"\/" ((v0
"\/" (v1
"/\" v0))
"/\" v1))
"/\" (v1
"\/" v0)))
= ((v1
"/\" v0)
"/\" ((v0
"/\" (v1
"/\" v0))
"\/" v1))
proof
let v0, v1;
(v0
"/\" (v1
"/\" v0))
= (v1
"/\" v0) by
A34;
hence thesis by
A254;
end;
A260: for v1, v0 holds ((v0
"\/" (v0
"/\" v1))
"/\" (((v0
"/\" v1)
"\/" ((v1
"\/" (v0
"/\" v1))
"/\" v0))
"/\" (v0
"\/" v1)))
= ((v0
"/\" v1)
"/\" ((v1
"/\" (v0
"/\" v1))
"\/" v0))
proof
let v1, v0;
((v0
"/\" v1)
"\/" v0)
= (v0
"\/" (v0
"/\" v1)) by
A8;
hence thesis by
A257;
end;
A262: for v1, v0 holds ((v0
"\/" (v0
"/\" v1))
"/\" (((v0
"/\" v1)
"\/" (v0
"/\" (v1
"\/" (v0
"/\" v1))))
"/\" (v0
"\/" v1)))
= ((v0
"/\" v1)
"/\" ((v1
"/\" (v0
"/\" v1))
"\/" v0))
proof
let v1, v0;
((v1
"\/" (v0
"/\" v1))
"/\" v0)
= (v0
"/\" (v1
"\/" (v0
"/\" v1))) by
A4;
hence thesis by
A260;
end;
A264: for v1, v0 holds ((v0
"\/" (v0
"/\" v1))
"/\" (((v0
"/\" v1)
"\/" (v1
"/\" v0))
"/\" (v0
"\/" v1)))
= ((v0
"/\" v1)
"/\" ((v1
"/\" (v0
"/\" v1))
"\/" v0))
proof
let v1, v0;
(v0
"/\" (v1
"\/" (v0
"/\" v1)))
= (v1
"/\" v0) by
A251;
hence thesis by
A262;
end;
A266: for v1, v0 holds ((v0
"\/" (v0
"/\" v1))
"/\" ((v0
"\/" v1)
"/\" ((v0
"/\" v1)
"\/" (v1
"/\" v0))))
= ((v0
"/\" v1)
"/\" ((v1
"/\" (v0
"/\" v1))
"\/" v0))
proof
let v1, v0;
(((v0
"/\" v1)
"\/" (v1
"/\" v0))
"/\" (v0
"\/" v1))
= ((v0
"\/" v1)
"/\" ((v0
"/\" v1)
"\/" (v1
"/\" v0))) by
A4;
hence thesis by
A264;
end;
A268: for v1, v0 holds ((v0
"\/" v1)
"/\" ((v0
"\/" (v0
"/\" v1))
"/\" ((v0
"/\" v1)
"\/" (v1
"/\" v0))))
= ((v0
"/\" v1)
"/\" ((v1
"/\" (v0
"/\" v1))
"\/" v0))
proof
let v1, v0;
((v0
"\/" (v0
"/\" v1))
"/\" ((v0
"\/" v1)
"/\" ((v0
"/\" v1)
"\/" (v1
"/\" v0))))
= ((v0
"\/" v1)
"/\" ((v0
"\/" (v0
"/\" v1))
"/\" ((v0
"/\" v1)
"\/" (v1
"/\" v0)))) by
A40;
hence thesis by
A266;
end;
A270: for v1, v0 holds ((v0
"\/" v1)
"/\" ((v0
"\/" (v0
"/\" v1))
"/\" ((v0
"/\" v1)
"\/" (v1
"/\" v0))))
= ((v0
"/\" v1)
"/\" ((v0
"/\" v1)
"\/" v0))
proof
let v1, v0;
(v1
"/\" (v0
"/\" v1))
= (v0
"/\" v1) by
A34;
hence thesis by
A268;
end;
A272: for v1, v0 holds ((v0
"\/" v1)
"/\" ((v0
"\/" (v0
"/\" v1))
"/\" ((v0
"/\" v1)
"\/" (v1
"/\" v0))))
= ((v0
"/\" v1)
"/\" (v0
"\/" (v0
"/\" v1)))
proof
let v1, v0;
((v0
"/\" v1)
"\/" v0)
= (v0
"\/" (v0
"/\" v1)) by
A8;
hence thesis by
A270;
end;
A274: for v1, v0 holds ((v0
"\/" v1)
"/\" ((v0
"\/" (v0
"/\" v1))
"/\" ((v0
"/\" v1)
"\/" (v1
"/\" v0))))
= (v0
"/\" (v1
"/\" (v0
"\/" (v0
"/\" v1))))
proof
let v1, v0;
((v0
"/\" v1)
"/\" (v0
"\/" (v0
"/\" v1)))
= (v0
"/\" (v1
"/\" (v0
"\/" (v0
"/\" v1)))) by
A3;
hence thesis by
A272;
end;
A276: for v1, v0 holds ((v0
"\/" v1)
"/\" ((v0
"\/" (v0
"/\" v1))
"/\" ((v0
"/\" v1)
"\/" (v1
"/\" v0))))
= (v0
"/\" (v0
"/\" v1))
proof
let v1, v0;
(v1
"/\" (v0
"\/" (v0
"/\" v1)))
= (v0
"/\" v1) by
A246;
hence thesis by
A274;
end;
A278: for v1, v0 holds ((v0
"\/" v1)
"/\" ((v0
"\/" (v0
"/\" v1))
"/\" ((v0
"/\" v1)
"\/" (v1
"/\" v0))))
= (v0
"/\" v1)
proof
let v1, v0;
(v0
"/\" (v0
"/\" v1))
= (v0
"/\" v1) by
A26;
hence thesis by
A276;
end;
A281: for v101, v100 holds ((v100
"/\" v101)
"\/" (v101
"/\" v100))
= (v100
"/\" (v101
"\/" (v101
"/\" v100)))
proof
let v101, v100;
(v100
"/\" (v101
"\/" (v101
"/\" v100)))
= (v101
"/\" v100) by
A246;
hence thesis by
A23;
end;
A284: for v1, v0 holds ((v0
"/\" v1)
"\/" (v1
"/\" v0))
= (v1
"/\" v0)
proof
let v1, v0;
(v0
"/\" (v1
"\/" (v1
"/\" v0)))
= (v1
"/\" v0) by
A246;
hence thesis by
A281;
end;
A286: for v1, v0 holds ((v0
"\/" v1)
"/\" ((v0
"\/" (v0
"/\" v1))
"/\" (v1
"/\" v0)))
= (v0
"/\" v1)
proof
let v1, v0;
((v0
"/\" v1)
"\/" (v1
"/\" v0))
= (v1
"/\" v0) by
A284;
hence thesis by
A278;
end;
A288: for v1, v0 holds ((v0
"\/" v1)
"/\" ((v1
"/\" v0)
"/\" (v0
"\/" (v0
"/\" v1))))
= (v0
"/\" v1)
proof
let v1, v0;
((v0
"\/" (v0
"/\" v1))
"/\" (v1
"/\" v0))
= ((v1
"/\" v0)
"/\" (v0
"\/" (v0
"/\" v1))) by
A4;
hence thesis by
A286;
end;
A290: for v1, v0 holds ((v0
"\/" v1)
"/\" (v1
"/\" (v0
"/\" (v0
"\/" (v0
"/\" v1)))))
= (v0
"/\" v1)
proof
let v1, v0;
((v1
"/\" v0)
"/\" (v0
"\/" (v0
"/\" v1)))
= (v1
"/\" (v0
"/\" (v0
"\/" (v0
"/\" v1)))) by
A3;
hence thesis by
A288;
end;
A292: for v1, v0 holds ((v0
"\/" v1)
"/\" (v1
"/\" (v0
"\/" (v0
"/\" v1))))
= (v0
"/\" v1)
proof
let v1, v0;
(v0
"/\" (v0
"\/" (v0
"/\" v1)))
= (v0
"\/" (v0
"/\" v1)) by
A71;
hence thesis by
A290;
end;
A294: for v1, v0 holds ((v0
"\/" v1)
"/\" (v0
"/\" v1))
= (v0
"/\" v1)
proof
let v1, v0;
(v1
"/\" (v0
"\/" (v0
"/\" v1)))
= (v0
"/\" v1) by
A246;
hence thesis by
A292;
end;
A296: for v1, v0 holds ((v0
"/\" v1)
"/\" (v0
"\/" v1))
= (v0
"/\" v1)
proof
let v1, v0;
((v0
"\/" v1)
"/\" (v0
"/\" v1))
= ((v0
"/\" v1)
"/\" (v0
"\/" v1)) by
A4;
hence thesis by
A294;
end;
A298: for v1, v0 holds (v0
"/\" (v1
"/\" (v0
"\/" v1)))
= (v0
"/\" v1)
proof
let v1, v0;
((v0
"/\" v1)
"/\" (v0
"\/" v1))
= (v0
"/\" (v1
"/\" (v0
"\/" v1))) by
A3;
hence thesis by
A296;
end;
A301: for v102, v100, v1 holds ((v1
"/\" v100)
"/\" v102)
= (v100
"/\" ((v1
"\/" (v100
"/\" v1))
"/\" v102))
proof
let v102, v100, v1;
(v100
"/\" (v1
"\/" (v100
"/\" v1)))
= (v1
"/\" v100) by
A251;
hence thesis by
A3;
end;
A304: for v0, v2, v1 holds (v0
"/\" (v1
"/\" v2))
= (v1
"/\" ((v0
"\/" (v1
"/\" v0))
"/\" v2))
proof
let v0, v2, v1;
((v0
"/\" v1)
"/\" v2)
= (v0
"/\" (v1
"/\" v2)) by
A3;
hence thesis by
A301;
end;
A308: for v1, v0 holds (v1
"/\" (v0
"/\" (v0
"\/" v1)))
= (v0
"/\" v1)
proof
let v1, v0;
(v0
"/\" ((v1
"\/" (v0
"/\" v1))
"/\" (v0
"\/" v1)))
= (v1
"/\" (v0
"/\" (v0
"\/" v1))) by
A304;
hence thesis by
A235;
end;
A312: for v103, v100, v101, v1 holds ((v100
"\/" v101)
"/\" ((v100
"\/" (v1
"/\" v101))
"/\" v103))
= ((v100
"\/" (v101
"/\" (v1
"/\" v101)))
"/\" v103)
proof
let v103, v100, v101, v1;
(v101
"/\" (v1
"/\" v101))
= (v1
"/\" v101) by
A34;
hence thesis by
A60;
end;
A315: for v3, v0, v1, v2 holds ((v0
"\/" v1)
"/\" ((v0
"\/" (v2
"/\" v1))
"/\" v3))
= ((v0
"\/" (v2
"/\" v1))
"/\" v3)
proof
let v3, v0, v1, v2;
(v1
"/\" (v2
"/\" v1))
= (v2
"/\" v1) by
A34;
hence thesis by
A312;
end;
A318: for v102, v100, v1 holds ((v1
"/\" v100)
"/\" v102)
= (v100
"/\" ((v1
"/\" (v1
"\/" v100))
"/\" v102))
proof
let v102, v100, v1;
(v100
"/\" (v1
"/\" (v1
"\/" v100)))
= (v1
"/\" v100) by
A308;
hence thesis by
A3;
end;
A321: for v0, v2, v1 holds (v0
"/\" (v1
"/\" v2))
= (v1
"/\" ((v0
"/\" (v0
"\/" v1))
"/\" v2))
proof
let v0, v2, v1;
((v0
"/\" v1)
"/\" v2)
= (v0
"/\" (v1
"/\" v2)) by
A3;
hence thesis by
A318;
end;
A323: for v0, v2, v1 holds (v0
"/\" (v1
"/\" v2))
= (v1
"/\" (v0
"/\" ((v0
"\/" v1)
"/\" v2)))
proof
let v0, v2, v1;
((v0
"/\" (v0
"\/" v1))
"/\" v2)
= (v0
"/\" ((v0
"\/" v1)
"/\" v2)) by
A3;
hence thesis by
A321;
end;
A328: for v0, v100, v1 holds (v100
"/\" ((v0
"\/" v1)
"/\" (v0
"\/" (v1
"\/" v100))))
= ((v0
"\/" v1)
"/\" v100)
proof
let v0, v100, v1;
((v0
"\/" v1)
"\/" v100)
= (v0
"\/" (v1
"\/" v100)) by
A7;
hence thesis by
A308;
end;
A332: for v102, v1, v100 holds ((v100
"/\" (v100
"\/" v1))
"\/" v102)
= (v100
"\/" ((v100
"/\" (v100
"\/" v1))
"\/" v102))
proof
let v102, v1, v100;
(v100
"\/" (v100
"/\" (v100
"\/" v1)))
= (v100
"/\" (v100
"\/" v1)) by
A113;
hence thesis by
A7;
end;
A337: for v100, v1, v101 holds (v100
"\/" (v101
"/\" (v101
"\/" v1)))
= (v101
"\/" (v100
"\/" (v101
"/\" (v101
"\/" v1))))
proof
let v100, v1, v101;
(v101
"\/" (v101
"/\" (v101
"\/" v1)))
= (v101
"/\" (v101
"\/" v1)) by
A113;
hence thesis by
A57;
end;
A342: for v2, v1, v0 holds ((v0
"/\" (v0
"\/" v1))
"\/" (v1
"/\" v2))
= (v0
"\/" (v1
"/\" v2))
proof
let v2, v1, v0;
(v0
"\/" ((v0
"/\" (v0
"\/" v1))
"\/" (v1
"/\" v2)))
= ((v0
"/\" (v0
"\/" v1))
"\/" (v1
"/\" v2)) by
A332;
hence thesis by
A153;
end;
A344: for v1, v0 holds (v1
"\/" (v0
"/\" (v0
"\/" v1)))
= (v0
"\/" v1)
proof
let v1, v0;
(v0
"\/" (v1
"\/" (v0
"/\" (v0
"\/" v1))))
= (v1
"\/" (v0
"/\" (v0
"\/" v1))) by
A337;
hence thesis by
A126;
end;
A347: for v1, v0 holds (v0
"\/" (v0
"/\" v1))
= (v0
"/\" (v0
"\/" v1))
proof
let v1, v0;
(v1
"\/" (v0
"/\" (v0
"\/" v1)))
= (v0
"\/" v1) by
A344;
hence thesis by
A249;
end;
A349: for v2, v1, v0 holds ((v0
"/\" (v0
"\/" v1))
"/\" (v1
"\/" v2))
= (v0
"/\" (v1
"\/" v2))
proof
let v2, v1, v0;
(v0
"\/" (v0
"/\" v1))
= (v0
"/\" (v0
"\/" v1)) by
A347;
hence thesis by
A244;
end;
A351: for v2, v1, v0 holds (v0
"/\" ((v0
"\/" v1)
"/\" (v1
"\/" v2)))
= (v0
"/\" (v1
"\/" v2))
proof
let v2, v1, v0;
((v0
"/\" (v0
"\/" v1))
"/\" (v1
"\/" v2))
= (v0
"/\" ((v0
"\/" v1)
"/\" (v1
"\/" v2))) by
A3;
hence thesis by
A349;
end;
A353: for v0, v2, v1 holds ((v0
"/\" v1)
"\/" (v2
"/\" (v0
"/\" (v0
"\/" (v1
"/\" v2)))))
= ((v0
"\/" (v0
"/\" v1))
"/\" (v2
"\/" (v0
"/\" ((v0
"/\" v1)
"\/" v2))))
proof
let v0, v2, v1;
(v0
"\/" (v0
"/\" (v1
"/\" v2)))
= (v0
"/\" (v0
"\/" (v1
"/\" v2))) by
A347;
hence thesis by
A174;
end;
A355: for v0, v2, v1 holds ((v0
"/\" v1)
"\/" (v2
"/\" (v0
"/\" (v0
"\/" (v1
"/\" v2)))))
= ((v0
"/\" (v0
"\/" v1))
"/\" (v2
"\/" (v0
"/\" ((v0
"/\" v1)
"\/" v2))))
proof
let v0, v2, v1;
(v0
"\/" (v0
"/\" v1))
= (v0
"/\" (v0
"\/" v1)) by
A347;
hence thesis by
A353;
end;
A357: for v0, v2, v1 holds ((v0
"/\" v1)
"\/" (v2
"/\" (v0
"/\" (v0
"\/" (v1
"/\" v2)))))
= (v0
"/\" ((v0
"\/" v1)
"/\" (v2
"\/" (v0
"/\" ((v0
"/\" v1)
"\/" v2)))))
proof
let v0, v2, v1;
((v0
"/\" (v0
"\/" v1))
"/\" (v2
"\/" (v0
"/\" ((v0
"/\" v1)
"\/" v2))))
= (v0
"/\" ((v0
"\/" v1)
"/\" (v2
"\/" (v0
"/\" ((v0
"/\" v1)
"\/" v2))))) by
A3;
hence thesis by
A355;
end;
A360: for v101, v100 holds ((v100
"\/" v101)
"/\" (v101
"\/" v100))
= (v100
"\/" (v101
"/\" (v101
"\/" v100)))
proof
let v101, v100;
(v100
"\/" (v101
"/\" (v101
"\/" v100)))
= (v101
"\/" v100) by
A344;
hence thesis by
A11;
end;
A363: for v1, v0 holds ((v0
"\/" v1)
"/\" (v1
"\/" v0))
= (v1
"\/" v0)
proof
let v1, v0;
(v0
"\/" (v1
"/\" (v1
"\/" v0)))
= (v1
"\/" v0) by
A344;
hence thesis by
A360;
end;
A366: for v0, v2, v1 holds ((v0
"/\" (v1
"\/" v2))
"\/" ((v0
"/\" v1)
"/\" (v0
"/\" (v1
"\/" v2))))
= ((v0
"/\" v1)
"\/" (v0
"/\" (v1
"\/" v2)))
proof
let v0, v2, v1;
((v0
"/\" v1)
"\/" (v0
"/\" (v1
"\/" v2)))
= (v0
"/\" (v1
"\/" v2)) by
A23;
hence thesis by
A344;
end;
A368: for v0, v2, v1 holds ((v0
"/\" (v1
"\/" v2))
"\/" (v0
"/\" ((v0
"/\" v1)
"/\" (v1
"\/" v2))))
= ((v0
"/\" v1)
"\/" (v0
"/\" (v1
"\/" v2)))
proof
let v0, v2, v1;
((v0
"/\" v1)
"/\" (v0
"/\" (v1
"\/" v2)))
= (v0
"/\" ((v0
"/\" v1)
"/\" (v1
"\/" v2))) by
A40;
hence thesis by
A366;
end;
A370: for v0, v2, v1 holds ((v0
"/\" (v1
"\/" v2))
"\/" (v0
"/\" (v0
"/\" (v1
"/\" (v1
"\/" v2)))))
= ((v0
"/\" v1)
"\/" (v0
"/\" (v1
"\/" v2)))
proof
let v0, v2, v1;
((v0
"/\" v1)
"/\" (v1
"\/" v2))
= (v0
"/\" (v1
"/\" (v1
"\/" v2))) by
A3;
hence thesis by
A368;
end;
A372: for v0, v2, v1 holds ((v0
"/\" (v1
"\/" v2))
"\/" (v0
"/\" (v1
"/\" (v1
"\/" v2))))
= ((v0
"/\" v1)
"\/" (v0
"/\" (v1
"\/" v2)))
proof
let v0, v2, v1;
(v0
"/\" (v0
"/\" (v1
"/\" (v1
"\/" v2))))
= (v0
"/\" (v1
"/\" (v1
"\/" v2))) by
A26;
hence thesis by
A370;
end;
A374: for v0, v2, v1 holds ((v0
"/\" (v1
"\/" v2))
"\/" (v0
"/\" (v1
"/\" (v1
"\/" v2))))
= (v0
"/\" (v1
"\/" v2))
proof
let v0, v2, v1;
((v0
"/\" v1)
"\/" (v0
"/\" (v1
"\/" v2)))
= (v0
"/\" (v1
"\/" v2)) by
A23;
hence thesis by
A372;
end;
A377: for v101, v1 holds ((v1
"\/" v101)
"\/" (v101
"/\" (v1
"\/" v101)))
= (v101
"\/" (v1
"\/" v101))
proof
let v101, v1;
(v101
"\/" (v1
"\/" v101))
= (v1
"\/" v101) by
A51;
hence thesis by
A344;
end;
A380: for v1, v0 holds (v0
"\/" (v1
"\/" (v1
"/\" (v0
"\/" v1))))
= (v1
"\/" (v0
"\/" v1))
proof
let v1, v0;
((v0
"\/" v1)
"\/" (v1
"/\" (v0
"\/" v1)))
= (v0
"\/" (v1
"\/" (v1
"/\" (v0
"\/" v1)))) by
A7;
hence thesis by
A377;
end;
A382: for v1, v0 holds (v0
"\/" (v1
"/\" (v1
"\/" (v0
"\/" v1))))
= (v1
"\/" (v0
"\/" v1))
proof
let v1, v0;
(v1
"\/" (v1
"/\" (v0
"\/" v1)))
= (v1
"/\" (v1
"\/" (v0
"\/" v1))) by
A347;
hence thesis by
A380;
end;
A384: for v1, v0 holds (v0
"\/" (v1
"/\" (v0
"\/" v1)))
= (v1
"\/" (v0
"\/" v1))
proof
let v1, v0;
(v1
"\/" (v0
"\/" v1))
= (v0
"\/" v1) by
A51;
hence thesis by
A382;
end;
A386: for v1, v0 holds (v0
"\/" (v1
"/\" (v0
"\/" v1)))
= (v0
"\/" v1)
proof
let v1, v0;
(v1
"\/" (v0
"\/" v1))
= (v0
"\/" v1) by
A51;
hence thesis by
A384;
end;
A389: for v103, v100, v101 holds ((v100
"\/" v101)
"/\" ((v101
"\/" v100)
"/\" v103))
= ((v100
"\/" (v101
"/\" (v101
"\/" v100)))
"/\" v103)
proof
let v103, v100, v101;
(v100
"\/" (v101
"/\" (v101
"\/" v100)))
= (v101
"\/" v100) by
A344;
hence thesis by
A60;
end;
A392: for v2, v0, v1 holds ((v0
"\/" v1)
"/\" ((v1
"\/" v0)
"/\" v2))
= ((v1
"\/" v0)
"/\" v2)
proof
let v2, v0, v1;
(v0
"\/" (v1
"/\" (v1
"\/" v0)))
= (v1
"\/" v0) by
A344;
hence thesis by
A389;
end;
A395: for v0, v101, v1 holds ((v0
"/\" v1)
"\/" (v0
"/\" (v1
"/\" v101)))
= ((v0
"/\" v1)
"/\" ((v0
"/\" v1)
"\/" v101))
proof
let v0, v101, v1;
((v0
"/\" v1)
"/\" v101)
= (v0
"/\" (v1
"/\" v101)) by
A3;
hence thesis by
A347;
end;
A398: for v0, v2, v1 holds ((v0
"/\" v1)
"\/" (v0
"/\" (v1
"/\" v2)))
= (v0
"/\" (v1
"/\" ((v0
"/\" v1)
"\/" v2)))
proof
let v0, v2, v1;
((v0
"/\" v1)
"/\" ((v0
"/\" v1)
"\/" v2))
= (v0
"/\" (v1
"/\" ((v0
"/\" v1)
"\/" v2))) by
A3;
hence thesis by
A395;
end;
A400: for v0, v1 holds (v0
"\/" (v1
"/\" v0))
= (v0
"/\" (v0
"\/" v1))
proof
let v0, v1;
(v0
"/\" v1)
= (v1
"/\" v0) by
A4;
hence thesis by
A347;
end;
A403: for v102, v1, v100 holds ((v100
"/\" (v100
"\/" v1))
"\/" v102)
= (v100
"\/" ((v100
"/\" v1)
"\/" v102))
proof
let v102, v1, v100;
(v100
"\/" (v100
"/\" v1))
= (v100
"/\" (v100
"\/" v1)) by
A347;
hence thesis by
A7;
end;
A407: for v100, v1, v101 holds ((v100
"/\" v101)
"\/" (v100
"/\" (v101
"/\" (v101
"\/" v1))))
= (v100
"/\" (v101
"\/" (v101
"/\" v1)))
proof
let v100, v1, v101;
(v101
"\/" (v101
"/\" v1))
= (v101
"/\" (v101
"\/" v1)) by
A347;
hence thesis by
A23;
end;
A410: for v2, v1, v0 holds (v0
"/\" (v1
"/\" ((v0
"/\" v1)
"\/" (v1
"\/" v2))))
= (v0
"/\" (v1
"\/" (v1
"/\" v2)))
proof
let v2, v1, v0;
((v0
"/\" v1)
"\/" (v0
"/\" (v1
"/\" (v1
"\/" v2))))
= (v0
"/\" (v1
"/\" ((v0
"/\" v1)
"\/" (v1
"\/" v2)))) by
A398;
hence thesis by
A407;
end;
A412: for v2, v1, v0 holds (v0
"/\" (v1
"/\" (v1
"\/" ((v0
"/\" v1)
"\/" v2))))
= (v0
"/\" (v1
"\/" (v1
"/\" v2)))
proof
let v2, v1, v0;
((v0
"/\" v1)
"\/" (v1
"\/" v2))
= (v1
"\/" ((v0
"/\" v1)
"\/" v2)) by
A57;
hence thesis by
A410;
end;
A414: for v2, v1, v0 holds (v0
"/\" (v1
"/\" (v1
"\/" ((v0
"/\" v1)
"\/" v2))))
= (v0
"/\" (v1
"/\" (v1
"\/" v2)))
proof
let v2, v1, v0;
(v1
"\/" (v1
"/\" v2))
= (v1
"/\" (v1
"\/" v2)) by
A347;
hence thesis by
A412;
end;
A417: for v1, v2, v100 holds (v100
"\/" (v1
"/\" (v100
"/\" v2)))
= (v100
"/\" (v100
"\/" (v1
"/\" v2)))
proof
let v1, v2, v100;
(v100
"/\" (v1
"/\" v2))
= (v1
"/\" (v100
"/\" v2)) by
A40;
hence thesis by
A347;
end;
A421: for v100, v1, v101 holds (v100
"\/" (v101
"/\" (v101
"\/" v1)))
= (v101
"\/" (v100
"\/" (v101
"/\" v1)))
proof
let v100, v1, v101;
(v101
"\/" (v101
"/\" v1))
= (v101
"/\" (v101
"\/" v1)) by
A347;
hence thesis by
A57;
end;
A426: for v2, v1, v0 holds (v0
"\/" ((v0
"/\" v1)
"\/" (v1
"/\" v2)))
= (v0
"\/" (v1
"/\" v2))
proof
let v2, v1, v0;
((v0
"/\" (v0
"\/" v1))
"\/" (v1
"/\" v2))
= (v0
"\/" ((v0
"/\" v1)
"\/" (v1
"/\" v2))) by
A403;
hence thesis by
A342;
end;
A429: for v100, v102, v101 holds ((v100
"\/" v101)
"/\" ((v101
"/\" v102)
"\/" v100))
= ((v100
"\/" (v101
"/\" v102))
"/\" ((v101
"/\" v102)
"\/" v100))
proof
let v100, v102, v101;
((v100
"\/" (v101
"/\" v102))
"/\" ((v101
"/\" v102)
"\/" v100))
= ((v101
"/\" v102)
"\/" v100) by
A363;
hence thesis by
A60;
end;
A432: for v0, v2, v1 holds ((v0
"\/" v1)
"/\" ((v1
"/\" v2)
"\/" v0))
= ((v1
"/\" v2)
"\/" v0)
proof
let v0, v2, v1;
((v0
"\/" (v1
"/\" v2))
"/\" ((v1
"/\" v2)
"\/" v0))
= ((v1
"/\" v2)
"\/" v0) by
A363;
hence thesis by
A429;
end;
A435: for v100, v101, v1, v0 holds ((v100
"\/" (v101
"/\" (v0
"\/" v1)))
"/\" (v100
"\/" (v101
"/\" (v1
"\/" v0))))
= (v100
"\/" (v101
"/\" ((v0
"\/" v1)
"/\" (v1
"\/" v0))))
proof
let v100, v101, v1, v0;
((v0
"\/" v1)
"/\" (v1
"\/" v0))
= (v1
"\/" v0) by
A363;
hence thesis by
A68;
end;
A438: for v0, v1, v3, v2 holds ((v0
"\/" (v1
"/\" (v2
"\/" v3)))
"/\" (v0
"\/" (v1
"/\" (v3
"\/" v2))))
= (v0
"\/" (v1
"/\" (v3
"\/" v2)))
proof
let v0, v1, v3, v2;
((v2
"\/" v3)
"/\" (v3
"\/" v2))
= (v3
"\/" v2) by
A363;
hence thesis by
A435;
end;
A441: for v0, v100, v1 holds (v100
"\/" (v0
"/\" (v1
"/\" v100)))
= (v100
"/\" (v100
"\/" (v0
"/\" v1)))
proof
let v0, v100, v1;
((v0
"/\" v1)
"/\" v100)
= (v0
"/\" (v1
"/\" v100)) by
A3;
hence thesis by
A400;
end;
A445: for v1, v2, v101 holds ((v1
"/\" v2)
"\/" (v1
"/\" (v101
"/\" v2)))
= ((v1
"/\" v2)
"/\" ((v1
"/\" v2)
"\/" v101))
proof
let v1, v2, v101;
(v101
"/\" (v1
"/\" v2))
= (v1
"/\" (v101
"/\" v2)) by
A40;
hence thesis by
A400;
end;
A448: for v0, v1, v2 holds ((v0
"/\" v1)
"\/" (v0
"/\" (v2
"/\" v1)))
= (v0
"/\" (v1
"/\" ((v0
"/\" v1)
"\/" v2)))
proof
let v0, v1, v2;
((v0
"/\" v1)
"/\" ((v0
"/\" v1)
"\/" v2))
= (v0
"/\" (v1
"/\" ((v0
"/\" v1)
"\/" v2))) by
A3;
hence thesis by
A445;
end;
A451: for v100, v1, v101 holds (v100
"\/" (v101
"/\" (v101
"\/" v1)))
= (v101
"\/" (v100
"\/" (v1
"/\" v101)))
proof
let v100, v1, v101;
(v101
"\/" (v1
"/\" v101))
= (v101
"/\" (v101
"\/" v1)) by
A400;
hence thesis by
A57;
end;
A456: for v0, v2, v1 holds (v0
"/\" ((v1
"\/" v2)
"/\" ((v0
"/\" (v1
"\/" v2))
"\/" v1)))
= (v0
"/\" (v1
"\/" v2))
proof
let v0, v2, v1;
((v0
"/\" (v1
"\/" v2))
"\/" (v0
"/\" (v1
"/\" (v1
"\/" v2))))
= (v0
"/\" ((v1
"\/" v2)
"/\" ((v0
"/\" (v1
"\/" v2))
"\/" v1))) by
A448;
hence thesis by
A374;
end;
A458: for v0, v2, v1 holds (v0
"/\" ((v1
"\/" v2)
"/\" (v1
"\/" (v0
"/\" (v1
"\/" v2)))))
= (v0
"/\" (v1
"\/" v2))
proof
let v0, v2, v1;
((v0
"/\" (v1
"\/" v2))
"\/" v1)
= (v1
"\/" (v0
"/\" (v1
"\/" v2))) by
A8;
hence thesis by
A456;
end;
A461: for v0, v1, v102, v100 holds (v100
"/\" (v0
"/\" (v1
"/\" (v100
"/\" v102))))
= ((v0
"/\" v1)
"/\" (v100
"/\" v102))
proof
let v0, v1, v102, v100;
((v0
"/\" v1)
"/\" (v100
"/\" v102))
= (v0
"/\" (v1
"/\" (v100
"/\" v102))) by
A3;
hence thesis by
A162;
end;
A464: for v1, v2, v3, v0 holds (v0
"/\" (v1
"/\" (v2
"/\" (v0
"/\" v3))))
= (v1
"/\" (v2
"/\" (v0
"/\" v3)))
proof
let v1, v2, v3, v0;
((v1
"/\" v2)
"/\" (v0
"/\" v3))
= (v1
"/\" (v2
"/\" (v0
"/\" v3))) by
A3;
hence thesis by
A461;
end;
A467: for v101, v0, v100, v1 holds (v100
"/\" (v101
"/\" (v0
"/\" (v1
"/\" v100))))
= (v101
"/\" ((v0
"/\" v1)
"/\" v100))
proof
let v101, v0, v100, v1;
((v0
"/\" v1)
"/\" v100)
= (v0
"/\" (v1
"/\" v100)) by
A3;
hence thesis by
A180;
end;
A470: for v1, v2, v0, v3 holds (v0
"/\" (v1
"/\" (v2
"/\" (v3
"/\" v0))))
= (v1
"/\" (v2
"/\" (v3
"/\" v0)))
proof
let v1, v2, v0, v3;
((v2
"/\" v3)
"/\" v0)
= (v2
"/\" (v3
"/\" v0)) by
A3;
hence thesis by
A467;
end;
A473: for v101, v1, v102 holds ((v1
"/\" (v102
"\/" v1))
"/\" (v101
"/\" (v102
"/\" v1)))
= (v101
"/\" (v102
"/\" (v1
"/\" (v102
"\/" v1))))
proof
let v101, v1, v102;
(v102
"/\" (v1
"/\" (v102
"\/" v1)))
= (v102
"/\" v1) by
A298;
hence thesis by
A180;
end;
A476: for v2, v0, v1 holds (v0
"/\" ((v1
"\/" v0)
"/\" (v2
"/\" (v1
"/\" v0))))
= (v2
"/\" (v1
"/\" (v0
"/\" (v1
"\/" v0))))
proof
let v2, v0, v1;
((v0
"/\" (v1
"\/" v0))
"/\" (v2
"/\" (v1
"/\" v0)))
= (v0
"/\" ((v1
"\/" v0)
"/\" (v2
"/\" (v1
"/\" v0)))) by
A3;
hence thesis by
A473;
end;
A478: for v2, v0, v1 holds ((v1
"\/" v0)
"/\" (v2
"/\" (v1
"/\" v0)))
= (v2
"/\" (v1
"/\" (v0
"/\" (v1
"\/" v0))))
proof
let v2, v0, v1;
(v0
"/\" ((v1
"\/" v0)
"/\" (v2
"/\" (v1
"/\" v0))))
= ((v1
"\/" v0)
"/\" (v2
"/\" (v1
"/\" v0))) by
A470;
hence thesis by
A476;
end;
A481: for v2, v1, v0 holds ((v0
"\/" v1)
"/\" (v2
"/\" (v0
"/\" v1)))
= (v2
"/\" (v0
"/\" v1))
proof
let v2, v1, v0;
(v0
"/\" (v1
"/\" (v0
"\/" v1)))
= (v0
"/\" v1) by
A298;
hence thesis by
A478;
end;
A484: for v0, v1, v102, v100 holds (v100
"\/" (v0
"\/" (v1
"\/" (v100
"\/" v102))))
= ((v0
"\/" v1)
"\/" (v100
"\/" v102))
proof
let v0, v1, v102, v100;
((v0
"\/" v1)
"\/" (v100
"\/" v102))
= (v0
"\/" (v1
"\/" (v100
"\/" v102))) by
A7;
hence thesis by
A195;
end;
A487: for v1, v2, v3, v0 holds (v0
"\/" (v1
"\/" (v2
"\/" (v0
"\/" v3))))
= (v1
"\/" (v2
"\/" (v0
"\/" v3)))
proof
let v1, v2, v3, v0;
((v1
"\/" v2)
"\/" (v0
"\/" v3))
= (v1
"\/" (v2
"\/" (v0
"\/" v3))) by
A7;
hence thesis by
A484;
end;
A490: for v101, v102, v1 holds ((v1
"/\" (v1
"\/" v102))
"\/" (v101
"\/" (v1
"\/" v102)))
= (v101
"\/" (v102
"\/" (v1
"/\" (v1
"\/" v102))))
proof
let v101, v102, v1;
(v102
"\/" (v1
"/\" (v1
"\/" v102)))
= (v1
"\/" v102) by
A344;
hence thesis by
A205;
end;
A493: for v2, v1, v0 holds (v0
"\/" ((v0
"/\" v1)
"\/" (v2
"\/" (v0
"\/" v1))))
= (v2
"\/" (v1
"\/" (v0
"/\" (v0
"\/" v1))))
proof
let v2, v1, v0;
((v0
"/\" (v0
"\/" v1))
"\/" (v2
"\/" (v0
"\/" v1)))
= (v0
"\/" ((v0
"/\" v1)
"\/" (v2
"\/" (v0
"\/" v1)))) by
A403;
hence thesis by
A490;
end;
A495: for v2, v1, v0 holds ((v0
"/\" v1)
"\/" (v2
"\/" (v0
"\/" v1)))
= (v2
"\/" (v1
"\/" (v0
"/\" (v0
"\/" v1))))
proof
let v2, v1, v0;
(v0
"\/" ((v0
"/\" v1)
"\/" (v2
"\/" (v0
"\/" v1))))
= ((v0
"/\" v1)
"\/" (v2
"\/" (v0
"\/" v1))) by
A487;
hence thesis by
A493;
end;
A497: for v2, v1, v0 holds ((v0
"/\" v1)
"\/" (v2
"\/" (v0
"\/" v1)))
= (v2
"\/" (v0
"\/" v1))
proof
let v2, v1, v0;
(v1
"\/" (v0
"/\" (v0
"\/" v1)))
= (v0
"\/" v1) by
A344;
hence thesis by
A495;
end;
A500: for v1, v2, v0 holds (v1
"\/" (v0
"/\" v2))
= ((v1
"\/" (v0
"/\" v2))
"/\" (v0
"\/" v1))
proof
let v1, v2, v0;
((v0
"\/" v1)
"/\" (v1
"\/" (v0
"/\" v2)))
= (v1
"\/" (v0
"/\" v2)) by
A86;
hence thesis by
A4;
end;
A505: for v102, v1, v100 holds ((v100
"\/" v1)
"/\" ((v100
"\/" v1)
"\/" (v100
"/\" v102)))
= ((v100
"\/" v1)
"\/" (v100
"/\" v102))
proof
let v102, v1, v100;
(v100
"\/" (v100
"\/" v1))
= (v100
"\/" v1) by
A43;
hence thesis by
A86;
end;
A508: for v1, v2, v0 holds ((v0
"\/" v1)
"/\" (v0
"\/" (v1
"\/" (v0
"/\" v2))))
= ((v0
"\/" v1)
"\/" (v0
"/\" v2))
proof
let v1, v2, v0;
((v0
"\/" v1)
"\/" (v0
"/\" v2))
= (v0
"\/" (v1
"\/" (v0
"/\" v2))) by
A7;
hence thesis by
A505;
end;
A510: for v1, v2, v0 holds ((v0
"\/" v1)
"/\" (v0
"\/" (v1
"\/" (v0
"/\" v2))))
= (v0
"\/" (v1
"\/" (v0
"/\" v2)))
proof
let v1, v2, v0;
((v0
"\/" v1)
"\/" (v0
"/\" v2))
= (v0
"\/" (v1
"\/" (v0
"/\" v2))) by
A7;
hence thesis by
A508;
end;
A513: for v0, v2, v1 holds (v0
"/\" (v1
"\/" v2))
= (((v1
"\/" v2)
"/\" v0)
"\/" (v0
"/\" v1))
proof
let v0, v2, v1;
((v0
"/\" v1)
"\/" ((v1
"\/" v2)
"/\" v0))
= (v0
"/\" (v1
"\/" v2)) by
A131;
hence thesis by
A8;
end;
A517: for v0, v1, v2 holds ((v0
"/\" v1)
"\/" ((v2
"\/" v1)
"/\" v0))
= (v0
"/\" (v1
"\/" v2))
proof
let v0, v1, v2;
(v1
"\/" v2)
= (v2
"\/" v1) by
A8;
hence thesis by
A131;
end;
A520: for v101, v2, v1 holds (((v101
"\/" v1)
"/\" v101)
"\/" ((v101
"\/" v1)
"/\" ((v1
"/\" v2)
"\/" v101)))
= (v101
"\/" (v1
"/\" v2))
proof
let v101, v2, v1;
((v101
"\/" v1)
"/\" (v101
"\/" (v1
"/\" v2)))
= (v101
"\/" (v1
"/\" v2)) by
A11;
hence thesis by
A133;
end;
A523: for v0, v2, v1 holds ((v0
"/\" (v0
"\/" v1))
"\/" ((v0
"\/" v1)
"/\" ((v1
"/\" v2)
"\/" v0)))
= (v0
"\/" (v1
"/\" v2))
proof
let v0, v2, v1;
((v0
"\/" v1)
"/\" v0)
= (v0
"/\" (v0
"\/" v1)) by
A4;
hence thesis by
A520;
end;
A525: for v0, v2, v1 holds ((v0
"/\" (v0
"\/" v1))
"\/" ((v1
"/\" v2)
"\/" v0))
= (v0
"\/" (v1
"/\" v2))
proof
let v0, v2, v1;
((v0
"\/" v1)
"/\" ((v1
"/\" v2)
"\/" v0))
= ((v1
"/\" v2)
"\/" v0) by
A432;
hence thesis by
A523;
end;
A527: for v0, v2, v1 holds (v0
"\/" ((v0
"/\" v1)
"\/" ((v1
"/\" v2)
"\/" v0)))
= (v0
"\/" (v1
"/\" v2))
proof
let v0, v2, v1;
((v0
"/\" (v0
"\/" v1))
"\/" ((v1
"/\" v2)
"\/" v0))
= (v0
"\/" ((v0
"/\" v1)
"\/" ((v1
"/\" v2)
"\/" v0))) by
A403;
hence thesis by
A525;
end;
A529: for v0, v2, v1 holds ((v0
"/\" v1)
"\/" ((v1
"/\" v2)
"\/" v0))
= (v0
"\/" (v1
"/\" v2))
proof
let v0, v2, v1;
(v0
"\/" ((v0
"/\" v1)
"\/" ((v1
"/\" v2)
"\/" v0)))
= ((v0
"/\" v1)
"\/" ((v1
"/\" v2)
"\/" v0)) by
A205;
hence thesis by
A527;
end;
A532: for v100, v1, v102 holds ((v100
"/\" (v1
"/\" (v102
"\/" v1)))
"\/" (v100
"/\" (v102
"\/" v1)))
= (v100
"/\" ((v1
"/\" (v102
"\/" v1))
"\/" v102))
proof
let v100, v1, v102;
(v102
"\/" (v1
"/\" (v102
"\/" v1)))
= (v102
"\/" v1) by
A386;
hence thesis by
A133;
end;
A535: for v0, v1, v2 holds ((v0
"/\" (v2
"\/" v1))
"\/" (v0
"/\" (v1
"/\" (v2
"\/" v1))))
= (v0
"/\" ((v1
"/\" (v2
"\/" v1))
"\/" v2))
proof
let v0, v1, v2;
((v0
"/\" (v1
"/\" (v2
"\/" v1)))
"\/" (v0
"/\" (v2
"\/" v1)))
= ((v0
"/\" (v2
"\/" v1))
"\/" (v0
"/\" (v1
"/\" (v2
"\/" v1)))) by
A8;
hence thesis by
A532;
end;
A538: for v0, v2, v1 holds (v0
"/\" ((v1
"\/" v2)
"/\" ((v0
"/\" (v1
"\/" v2))
"\/" v2)))
= (v0
"/\" ((v2
"/\" (v1
"\/" v2))
"\/" v1))
proof
let v0, v2, v1;
((v0
"/\" (v1
"\/" v2))
"\/" (v0
"/\" (v2
"/\" (v1
"\/" v2))))
= (v0
"/\" ((v1
"\/" v2)
"/\" ((v0
"/\" (v1
"\/" v2))
"\/" v2))) by
A448;
hence thesis by
A535;
end;
A540: for v0, v2, v1 holds (v0
"/\" ((v1
"\/" v2)
"/\" (v2
"\/" (v0
"/\" (v1
"\/" v2)))))
= (v0
"/\" ((v2
"/\" (v1
"\/" v2))
"\/" v1))
proof
let v0, v2, v1;
((v0
"/\" (v1
"\/" v2))
"\/" v2)
= (v2
"\/" (v0
"/\" (v1
"\/" v2))) by
A8;
hence thesis by
A538;
end;
A542: for v0, v2, v1 holds (v0
"/\" ((v1
"\/" v2)
"/\" (v2
"\/" (v0
"/\" (v1
"\/" v2)))))
= (v0
"/\" (v1
"\/" (v2
"/\" (v1
"\/" v2))))
proof
let v0, v2, v1;
((v2
"/\" (v1
"\/" v2))
"\/" v1)
= (v1
"\/" (v2
"/\" (v1
"\/" v2))) by
A8;
hence thesis by
A540;
end;
A544: for v0, v2, v1 holds (v0
"/\" ((v1
"\/" v2)
"/\" (v2
"\/" (v0
"/\" (v1
"\/" v2)))))
= (v0
"/\" (v1
"\/" v2))
proof
let v0, v2, v1;
(v1
"\/" (v2
"/\" (v1
"\/" v2)))
= (v1
"\/" v2) by
A386;
hence thesis by
A542;
end;
A547: for v101, v1, v2 holds (((v101
"\/" v1)
"/\" v101)
"\/" (v101
"\/" (v2
"/\" v1)))
= ((v101
"\/" v1)
"/\" (v101
"\/" (v2
"/\" v1)))
proof
let v101, v1, v2;
((v101
"\/" v1)
"/\" (v101
"\/" (v2
"/\" v1)))
= (v101
"\/" (v2
"/\" v1)) by
A186;
hence thesis by
A23;
end;
A550: for v0, v1, v2 holds ((v0
"/\" (v0
"\/" v1))
"\/" (v0
"\/" (v2
"/\" v1)))
= ((v0
"\/" v1)
"/\" (v0
"\/" (v2
"/\" v1)))
proof
let v0, v1, v2;
((v0
"\/" v1)
"/\" v0)
= (v0
"/\" (v0
"\/" v1)) by
A4;
hence thesis by
A547;
end;
A552: for v2, v1, v0 holds (v0
"\/" ((v0
"/\" (v0
"\/" v1))
"\/" (v2
"/\" v1)))
= ((v0
"\/" v1)
"/\" (v0
"\/" (v2
"/\" v1)))
proof
let v2, v1, v0;
((v0
"/\" (v0
"\/" v1))
"\/" (v0
"\/" (v2
"/\" v1)))
= (v0
"\/" ((v0
"/\" (v0
"\/" v1))
"\/" (v2
"/\" v1))) by
A57;
hence thesis by
A550;
end;
A554: for v2, v1, v0 holds (v0
"\/" (v0
"\/" ((v0
"/\" v1)
"\/" (v2
"/\" v1))))
= ((v0
"\/" v1)
"/\" (v0
"\/" (v2
"/\" v1)))
proof
let v2, v1, v0;
((v0
"/\" (v0
"\/" v1))
"\/" (v2
"/\" v1))
= (v0
"\/" ((v0
"/\" v1)
"\/" (v2
"/\" v1))) by
A403;
hence thesis by
A552;
end;
A556: for v2, v1, v0 holds (v0
"\/" ((v0
"/\" v1)
"\/" (v2
"/\" v1)))
= ((v0
"\/" v1)
"/\" (v0
"\/" (v2
"/\" v1)))
proof
let v2, v1, v0;
(v0
"\/" (v0
"\/" ((v0
"/\" v1)
"\/" (v2
"/\" v1))))
= (v0
"\/" ((v0
"/\" v1)
"\/" (v2
"/\" v1))) by
A43;
hence thesis by
A554;
end;
A558: for v2, v1, v0 holds (v0
"\/" ((v0
"/\" v1)
"\/" (v2
"/\" v1)))
= (v0
"\/" (v2
"/\" v1))
proof
let v2, v1, v0;
((v0
"\/" v1)
"/\" (v0
"\/" (v2
"/\" v1)))
= (v0
"\/" (v2
"/\" v1)) by
A186;
hence thesis by
A556;
end;
A561: for v102, v1, v100 holds ((v100
"\/" v1)
"/\" (v100
"\/" (v102
"/\" (v100
"\/" v1))))
= (v100
"\/" (v102
"/\" (v100
"\/" v1)))
proof
let v102, v1, v100;
(v100
"\/" (v100
"\/" v1))
= (v100
"\/" v1) by
A43;
hence thesis by
A186;
end;
A565: for v102, v100, v1 holds ((v1
"\/" v100)
"/\" (v100
"\/" (v102
"/\" (v1
"\/" v100))))
= (v100
"\/" (v102
"/\" (v1
"\/" v100)))
proof
let v102, v100, v1;
(v100
"\/" (v1
"\/" v100))
= (v1
"\/" v100) by
A51;
hence thesis by
A186;
end;
A568: for v0, v2, v1 holds (v0
"/\" (v1
"\/" (v0
"/\" (v1
"\/" v2))))
= (v0
"/\" (v1
"\/" v2))
proof
let v0, v2, v1;
((v1
"\/" v2)
"/\" (v1
"\/" (v0
"/\" (v1
"\/" v2))))
= (v1
"\/" (v0
"/\" (v1
"\/" v2))) by
A561;
hence thesis by
A458;
end;
A570: for v0, v2, v1 holds (v0
"/\" (v2
"\/" (v0
"/\" (v1
"\/" v2))))
= (v0
"/\" (v1
"\/" v2))
proof
let v0, v2, v1;
((v1
"\/" v2)
"/\" (v2
"\/" (v0
"/\" (v1
"\/" v2))))
= (v2
"\/" (v0
"/\" (v1
"\/" v2))) by
A565;
hence thesis by
A544;
end;
A574: for v101, v1, v2, v102 holds ((v1
"/\" ((v1
"\/" v102)
"/\" v2))
"/\" (v101
"/\" (v1
"/\" (v102
"/\" v2))))
= (v101
"/\" (v102
"/\" (v1
"/\" ((v1
"\/" v102)
"/\" v2))))
proof
let v101, v1, v2, v102;
(v102
"/\" (v1
"/\" ((v1
"\/" v102)
"/\" v2)))
= (v1
"/\" (v102
"/\" v2)) by
A323;
hence thesis by
A180;
end;
A577: for v3, v0, v2, v1 holds (v0
"/\" (((v0
"\/" v1)
"/\" v2)
"/\" (v3
"/\" (v0
"/\" (v1
"/\" v2)))))
= (v3
"/\" (v1
"/\" (v0
"/\" ((v0
"\/" v1)
"/\" v2))))
proof
let v3, v0, v2, v1;
((v0
"/\" ((v0
"\/" v1)
"/\" v2))
"/\" (v3
"/\" (v0
"/\" (v1
"/\" v2))))
= (v0
"/\" (((v0
"\/" v1)
"/\" v2)
"/\" (v3
"/\" (v0
"/\" (v1
"/\" v2))))) by
A3;
hence thesis by
A574;
end;
A579: for v3, v0, v2, v1 holds (v0
"/\" ((v0
"\/" v1)
"/\" (v2
"/\" (v3
"/\" (v0
"/\" (v1
"/\" v2))))))
= (v3
"/\" (v1
"/\" (v0
"/\" ((v0
"\/" v1)
"/\" v2))))
proof
let v3, v0, v2, v1;
(((v0
"\/" v1)
"/\" v2)
"/\" (v3
"/\" (v0
"/\" (v1
"/\" v2))))
= ((v0
"\/" v1)
"/\" (v2
"/\" (v3
"/\" (v0
"/\" (v1
"/\" v2))))) by
A3;
hence thesis by
A577;
end;
A581: for v3, v0, v2, v1 holds (v0
"/\" ((v0
"\/" v1)
"/\" (v3
"/\" (v0
"/\" (v1
"/\" v2)))))
= (v3
"/\" (v1
"/\" (v0
"/\" ((v0
"\/" v1)
"/\" v2))))
proof
let v3, v0, v2, v1;
(v2
"/\" (v3
"/\" (v0
"/\" (v1
"/\" v2))))
= (v3
"/\" (v0
"/\" (v1
"/\" v2))) by
A470;
hence thesis by
A579;
end;
A584: for v2, v0, v3, v1 holds ((v0
"\/" v1)
"/\" (v2
"/\" (v0
"/\" (v1
"/\" v3))))
= (v2
"/\" (v1
"/\" (v0
"/\" ((v0
"\/" v1)
"/\" v3))))
proof
let v2, v0, v3, v1;
(v0
"/\" ((v0
"\/" v1)
"/\" (v2
"/\" (v0
"/\" (v1
"/\" v3)))))
= ((v0
"\/" v1)
"/\" (v2
"/\" (v0
"/\" (v1
"/\" v3)))) by
A464;
hence thesis by
A581;
end;
A586: for v2, v0, v3, v1 holds ((v0
"\/" v1)
"/\" (v2
"/\" (v0
"/\" (v1
"/\" v3))))
= (v2
"/\" (v0
"/\" (v1
"/\" v3)))
proof
let v2, v0, v3, v1;
(v1
"/\" (v0
"/\" ((v0
"\/" v1)
"/\" v3)))
= (v0
"/\" (v1
"/\" v3)) by
A323;
hence thesis by
A584;
end;
A589: for v102, v1, v100 holds (v100
"/\" ((v100
"/\" (v100
"\/" v1))
"/\" ((v100
"/\" v1)
"\/" v102)))
= (v100
"/\" ((v100
"/\" v1)
"\/" v102))
proof
let v102, v1, v100;
(v100
"\/" (v100
"/\" v1))
= (v100
"/\" (v100
"\/" v1)) by
A347;
hence thesis by
A351;
end;
A592: for v2, v1, v0 holds (v0
"/\" (v0
"/\" ((v0
"\/" v1)
"/\" ((v0
"/\" v1)
"\/" v2))))
= (v0
"/\" ((v0
"/\" v1)
"\/" v2))
proof
let v2, v1, v0;
((v0
"/\" (v0
"\/" v1))
"/\" ((v0
"/\" v1)
"\/" v2))
= (v0
"/\" ((v0
"\/" v1)
"/\" ((v0
"/\" v1)
"\/" v2))) by
A3;
hence thesis by
A589;
end;
A594: for v2, v1, v0 holds (v0
"/\" ((v0
"\/" v1)
"/\" ((v0
"/\" v1)
"\/" v2)))
= (v0
"/\" ((v0
"/\" v1)
"\/" v2))
proof
let v2, v1, v0;
(v0
"/\" (v0
"/\" ((v0
"\/" v1)
"/\" ((v0
"/\" v1)
"\/" v2))))
= (v0
"/\" ((v0
"\/" v1)
"/\" ((v0
"/\" v1)
"\/" v2))) by
A26;
hence thesis by
A592;
end;
A597: for v100, v1, v2, v101 holds (v100
"/\" ((v100
"\/" v101)
"/\" (v1
"\/" (v101
"\/" v2))))
= (v100
"/\" (v101
"\/" (v1
"\/" (v101
"\/" v2))))
proof
let v100, v1, v2, v101;
(v101
"\/" (v1
"\/" (v101
"\/" v2)))
= (v1
"\/" (v101
"\/" v2)) by
A195;
hence thesis by
A351;
end;
A600: for v0, v2, v3, v1 holds (v0
"/\" ((v0
"\/" v1)
"/\" (v2
"\/" (v1
"\/" v3))))
= (v0
"/\" (v2
"\/" (v1
"\/" v3)))
proof
let v0, v2, v3, v1;
(v1
"\/" (v2
"\/" (v1
"\/" v3)))
= (v2
"\/" (v1
"\/" v3)) by
A195;
hence thesis by
A597;
end;
A603: for v101, v1, v0 holds (v0
"\/" ((v0
"/\" v1)
"\/" v101))
= (v101
"\/" (v0
"/\" (v0
"\/" v1)))
proof
let v101, v1, v0;
((v0
"/\" (v0
"\/" v1))
"\/" v101)
= (v0
"\/" ((v0
"/\" v1)
"\/" v101)) by
A403;
hence thesis by
A8;
end;
A607: for v101, v1, v0 holds (v0
"\/" ((v0
"/\" v1)
"\/" ((v0
"/\" (v0
"\/" v1))
"/\" v101)))
= ((v0
"/\" (v0
"\/" v1))
"/\" ((v0
"/\" (v0
"\/" v1))
"\/" v101))
proof
let v101, v1, v0;
((v0
"/\" (v0
"\/" v1))
"\/" ((v0
"/\" (v0
"\/" v1))
"/\" v101))
= (v0
"\/" ((v0
"/\" v1)
"\/" ((v0
"/\" (v0
"\/" v1))
"/\" v101))) by
A403;
hence thesis by
A347;
end;
A610: for v2, v1, v0 holds (v0
"\/" ((v0
"/\" v1)
"\/" (v0
"/\" ((v0
"\/" v1)
"/\" v2))))
= ((v0
"/\" (v0
"\/" v1))
"/\" ((v0
"/\" (v0
"\/" v1))
"\/" v2))
proof
let v2, v1, v0;
((v0
"/\" (v0
"\/" v1))
"/\" v2)
= (v0
"/\" ((v0
"\/" v1)
"/\" v2)) by
A3;
hence thesis by
A607;
end;
A612: for v2, v1, v0 holds (v0
"\/" ((v0
"/\" v1)
"\/" (v0
"/\" ((v0
"\/" v1)
"/\" v2))))
= ((v0
"/\" (v0
"\/" v1))
"/\" (v0
"\/" ((v0
"/\" v1)
"\/" v2)))
proof
let v2, v1, v0;
((v0
"/\" (v0
"\/" v1))
"\/" v2)
= (v0
"\/" ((v0
"/\" v1)
"\/" v2)) by
A403;
hence thesis by
A610;
end;
A614: for v2, v1, v0 holds (v0
"\/" ((v0
"/\" v1)
"\/" (v0
"/\" ((v0
"\/" v1)
"/\" v2))))
= (v0
"/\" ((v0
"\/" v1)
"/\" (v0
"\/" ((v0
"/\" v1)
"\/" v2))))
proof
let v2, v1, v0;
((v0
"/\" (v0
"\/" v1))
"/\" (v0
"\/" ((v0
"/\" v1)
"\/" v2)))
= (v0
"/\" ((v0
"\/" v1)
"/\" (v0
"\/" ((v0
"/\" v1)
"\/" v2)))) by
A3;
hence thesis by
A612;
end;
A617: for v101, v1, v0 holds (v0
"\/" ((v0
"/\" v1)
"\/" (v101
"/\" (v0
"/\" (v0
"\/" v1)))))
= ((v0
"/\" (v0
"\/" v1))
"/\" ((v0
"/\" (v0
"\/" v1))
"\/" v101))
proof
let v101, v1, v0;
((v0
"/\" (v0
"\/" v1))
"\/" (v101
"/\" (v0
"/\" (v0
"\/" v1))))
= (v0
"\/" ((v0
"/\" v1)
"\/" (v101
"/\" (v0
"/\" (v0
"\/" v1))))) by
A403;
hence thesis by
A400;
end;
A620: for v2, v1, v0 holds (v0
"\/" ((v0
"/\" v1)
"\/" (v2
"/\" (v0
"/\" (v0
"\/" v1)))))
= ((v0
"/\" (v0
"\/" v1))
"/\" (v0
"\/" ((v0
"/\" v1)
"\/" v2)))
proof
let v2, v1, v0;
((v0
"/\" (v0
"\/" v1))
"\/" v2)
= (v0
"\/" ((v0
"/\" v1)
"\/" v2)) by
A403;
hence thesis by
A617;
end;
A622: for v2, v1, v0 holds (v0
"\/" ((v0
"/\" v1)
"\/" (v2
"/\" (v0
"/\" (v0
"\/" v1)))))
= (v0
"/\" ((v0
"\/" v1)
"/\" (v0
"\/" ((v0
"/\" v1)
"\/" v2))))
proof
let v2, v1, v0;
((v0
"/\" (v0
"\/" v1))
"/\" (v0
"\/" ((v0
"/\" v1)
"\/" v2)))
= (v0
"/\" ((v0
"\/" v1)
"/\" (v0
"\/" ((v0
"/\" v1)
"\/" v2)))) by
A3;
hence thesis by
A620;
end;
A625: for v101, v0, v102, v1 holds ((v0
"/\" v1)
"\/" (v101
"/\" (v0
"/\" (v1
"/\" v102))))
= ((v0
"/\" v1)
"/\" ((v0
"/\" v1)
"\/" (v101
"/\" v102)))
proof
let v101, v0, v102, v1;
((v0
"/\" v1)
"/\" v102)
= (v0
"/\" (v1
"/\" v102)) by
A3;
hence thesis by
A417;
end;
A628: for v2, v0, v3, v1 holds ((v0
"/\" v1)
"\/" (v2
"/\" (v0
"/\" (v1
"/\" v3))))
= (v0
"/\" (v1
"/\" ((v0
"/\" v1)
"\/" (v2
"/\" v3))))
proof
let v2, v0, v3, v1;
((v0
"/\" v1)
"/\" ((v0
"/\" v1)
"\/" (v2
"/\" v3)))
= (v0
"/\" (v1
"/\" ((v0
"/\" v1)
"\/" (v2
"/\" v3)))) by
A3;
hence thesis by
A625;
end;
A631: for v0, v1, v102, v100 holds (v100
"\/" (v0
"/\" (v1
"/\" (v100
"/\" v102))))
= (v100
"/\" (v100
"\/" ((v0
"/\" v1)
"/\" v102)))
proof
let v0, v1, v102, v100;
((v0
"/\" v1)
"/\" (v100
"/\" v102))
= (v0
"/\" (v1
"/\" (v100
"/\" v102))) by
A3;
hence thesis by
A417;
end;
A634: for v1, v2, v3, v0 holds (v0
"\/" (v1
"/\" (v2
"/\" (v0
"/\" v3))))
= (v0
"/\" (v0
"\/" (v1
"/\" (v2
"/\" v3))))
proof
let v1, v2, v3, v0;
((v1
"/\" v2)
"/\" v3)
= (v1
"/\" (v2
"/\" v3)) by
A3;
hence thesis by
A631;
end;
A637: for v1, v2, v102, v101 holds (((v1
"/\" ((v101
"/\" v102)
"/\" v2))
"/\" v101)
"\/" (v102
"/\" ((v101
"/\" v102)
"/\" ((v101
"/\" v102)
"\/" (v1
"/\" v2)))))
= ((v102
"\/" ((v101
"\/" v102)
"/\" (v1
"/\" ((v101
"/\" v102)
"/\" v2))))
"/\" ((v1
"/\" ((v101
"/\" v102)
"/\" v2))
"\/" v101))
proof
let v1, v2, v102, v101;
((v101
"/\" v102)
"\/" (v1
"/\" ((v101
"/\" v102)
"/\" v2)))
= ((v101
"/\" v102)
"/\" ((v101
"/\" v102)
"\/" (v1
"/\" v2))) by
A417;
hence thesis by
A20;
end;
A640: for v0, v1, v3, v2 holds (((v0
"/\" (v1
"/\" (v2
"/\" v3)))
"/\" v1)
"\/" (v2
"/\" ((v1
"/\" v2)
"/\" ((v1
"/\" v2)
"\/" (v0
"/\" v3)))))
= ((v2
"\/" ((v1
"\/" v2)
"/\" (v0
"/\" ((v1
"/\" v2)
"/\" v3))))
"/\" ((v0
"/\" ((v1
"/\" v2)
"/\" v3))
"\/" v1))
proof
let v0, v1, v3, v2;
((v1
"/\" v2)
"/\" v3)
= (v1
"/\" (v2
"/\" v3)) by
A3;
hence thesis by
A637;
end;
A642: for v0, v1, v3, v2 holds ((v1
"/\" (v0
"/\" (v1
"/\" (v2
"/\" v3))))
"\/" (v2
"/\" ((v1
"/\" v2)
"/\" ((v1
"/\" v2)
"\/" (v0
"/\" v3)))))
= ((v2
"\/" ((v1
"\/" v2)
"/\" (v0
"/\" ((v1
"/\" v2)
"/\" v3))))
"/\" ((v0
"/\" ((v1
"/\" v2)
"/\" v3))
"\/" v1))
proof
let v0, v1, v3, v2;
((v0
"/\" (v1
"/\" (v2
"/\" v3)))
"/\" v1)
= (v1
"/\" (v0
"/\" (v1
"/\" (v2
"/\" v3)))) by
A4;
hence thesis by
A640;
end;
A645: for v1, v0, v3, v2 holds ((v1
"/\" (v0
"/\" (v2
"/\" v3)))
"\/" (v2
"/\" ((v0
"/\" v2)
"/\" ((v0
"/\" v2)
"\/" (v1
"/\" v3)))))
= ((v2
"\/" ((v0
"\/" v2)
"/\" (v1
"/\" ((v0
"/\" v2)
"/\" v3))))
"/\" ((v1
"/\" ((v0
"/\" v2)
"/\" v3))
"\/" v0))
proof
let v1, v0, v3, v2;
(v0
"/\" (v1
"/\" (v0
"/\" (v2
"/\" v3))))
= (v1
"/\" (v0
"/\" (v2
"/\" v3))) by
A162;
hence thesis by
A642;
end;
A648: for v0, v1, v3, v2 holds ((v0
"/\" (v1
"/\" (v2
"/\" v3)))
"\/" (v2
"/\" (v1
"/\" (v2
"/\" ((v1
"/\" v2)
"\/" (v0
"/\" v3))))))
= ((v2
"\/" ((v1
"\/" v2)
"/\" (v0
"/\" ((v1
"/\" v2)
"/\" v3))))
"/\" ((v0
"/\" ((v1
"/\" v2)
"/\" v3))
"\/" v1))
proof
let v0, v1, v3, v2;
((v1
"/\" v2)
"/\" ((v1
"/\" v2)
"\/" (v0
"/\" v3)))
= (v1
"/\" (v2
"/\" ((v1
"/\" v2)
"\/" (v0
"/\" v3)))) by
A3;
hence thesis by
A645;
end;
A650: for v0, v1, v3, v2 holds ((v0
"/\" (v1
"/\" (v2
"/\" v3)))
"\/" (v1
"/\" (v2
"/\" ((v1
"/\" v2)
"\/" (v0
"/\" v3)))))
= ((v2
"\/" ((v1
"\/" v2)
"/\" (v0
"/\" ((v1
"/\" v2)
"/\" v3))))
"/\" ((v0
"/\" ((v1
"/\" v2)
"/\" v3))
"\/" v1))
proof
let v0, v1, v3, v2;
(v2
"/\" (v1
"/\" (v2
"/\" ((v1
"/\" v2)
"\/" (v0
"/\" v3)))))
= (v1
"/\" (v2
"/\" ((v1
"/\" v2)
"\/" (v0
"/\" v3)))) by
A162;
hence thesis by
A648;
end;
A652: for v0, v1, v3, v2 holds ((v0
"/\" (v1
"/\" (v2
"/\" v3)))
"\/" (v1
"/\" (v2
"/\" ((v1
"/\" v2)
"\/" (v0
"/\" v3)))))
= ((v2
"\/" ((v1
"\/" v2)
"/\" (v0
"/\" (v1
"/\" (v2
"/\" v3)))))
"/\" ((v0
"/\" ((v1
"/\" v2)
"/\" v3))
"\/" v1))
proof
let v0, v1, v3, v2;
((v1
"/\" v2)
"/\" v3)
= (v1
"/\" (v2
"/\" v3)) by
A3;
hence thesis by
A650;
end;
A654: for v0, v1, v3, v2 holds ((v0
"/\" (v1
"/\" (v2
"/\" v3)))
"\/" (v1
"/\" (v2
"/\" ((v1
"/\" v2)
"\/" (v0
"/\" v3)))))
= ((v2
"\/" (v0
"/\" (v1
"/\" (v2
"/\" v3))))
"/\" ((v0
"/\" ((v1
"/\" v2)
"/\" v3))
"\/" v1))
proof
let v0, v1, v3, v2;
((v1
"\/" v2)
"/\" (v0
"/\" (v1
"/\" (v2
"/\" v3))))
= (v0
"/\" (v1
"/\" (v2
"/\" v3))) by
A586;
hence thesis by
A652;
end;
A656: for v0, v1, v3, v2 holds ((v0
"/\" (v1
"/\" (v2
"/\" v3)))
"\/" (v1
"/\" (v2
"/\" ((v1
"/\" v2)
"\/" (v0
"/\" v3)))))
= ((v2
"/\" (v2
"\/" (v0
"/\" (v1
"/\" v3))))
"/\" ((v0
"/\" ((v1
"/\" v2)
"/\" v3))
"\/" v1))
proof
let v0, v1, v3, v2;
(v2
"\/" (v0
"/\" (v1
"/\" (v2
"/\" v3))))
= (v2
"/\" (v2
"\/" (v0
"/\" (v1
"/\" v3)))) by
A634;
hence thesis by
A654;
end;
A658: for v0, v1, v3, v2 holds ((v0
"/\" (v1
"/\" (v2
"/\" v3)))
"\/" (v1
"/\" (v2
"/\" ((v1
"/\" v2)
"\/" (v0
"/\" v3)))))
= ((v2
"/\" (v2
"\/" (v0
"/\" (v1
"/\" v3))))
"/\" ((v0
"/\" (v1
"/\" (v2
"/\" v3)))
"\/" v1))
proof
let v0, v1, v3, v2;
((v1
"/\" v2)
"/\" v3)
= (v1
"/\" (v2
"/\" v3)) by
A3;
hence thesis by
A656;
end;
A660: for v0, v1, v3, v2 holds ((v0
"/\" (v1
"/\" (v2
"/\" v3)))
"\/" (v1
"/\" (v2
"/\" ((v1
"/\" v2)
"\/" (v0
"/\" v3)))))
= ((v2
"/\" (v2
"\/" (v0
"/\" (v1
"/\" v3))))
"/\" (v1
"\/" (v0
"/\" (v1
"/\" (v2
"/\" v3)))))
proof
let v0, v1, v3, v2;
((v0
"/\" (v1
"/\" (v2
"/\" v3)))
"\/" v1)
= (v1
"\/" (v0
"/\" (v1
"/\" (v2
"/\" v3)))) by
A8;
hence thesis by
A658;
end;
A662: for v0, v1, v3, v2 holds ((v0
"/\" (v1
"/\" (v2
"/\" v3)))
"\/" (v1
"/\" (v2
"/\" ((v1
"/\" v2)
"\/" (v0
"/\" v3)))))
= ((v2
"/\" (v2
"\/" (v0
"/\" (v1
"/\" v3))))
"/\" (v1
"/\" (v1
"\/" (v0
"/\" (v2
"/\" v3)))))
proof
let v0, v1, v3, v2;
(v1
"\/" (v0
"/\" (v1
"/\" (v2
"/\" v3))))
= (v1
"/\" (v1
"\/" (v0
"/\" (v2
"/\" v3)))) by
A417;
hence thesis by
A660;
end;
A664: for v0, v1, v3, v2 holds ((v0
"/\" (v1
"/\" (v2
"/\" v3)))
"\/" (v1
"/\" (v2
"/\" ((v1
"/\" v2)
"\/" (v0
"/\" v3)))))
= (v1
"/\" ((v2
"/\" (v2
"\/" (v0
"/\" (v1
"/\" v3))))
"/\" (v1
"\/" (v0
"/\" (v2
"/\" v3)))))
proof
let v0, v1, v3, v2;
((v2
"/\" (v2
"\/" (v0
"/\" (v1
"/\" v3))))
"/\" (v1
"/\" (v1
"\/" (v0
"/\" (v2
"/\" v3)))))
= (v1
"/\" ((v2
"/\" (v2
"\/" (v0
"/\" (v1
"/\" v3))))
"/\" (v1
"\/" (v0
"/\" (v2
"/\" v3))))) by
A40;
hence thesis by
A662;
end;
A666: for v0, v1, v3, v2 holds ((v0
"/\" (v1
"/\" (v2
"/\" v3)))
"\/" (v1
"/\" (v2
"/\" ((v1
"/\" v2)
"\/" (v0
"/\" v3)))))
= (v1
"/\" (v2
"/\" ((v2
"\/" (v0
"/\" (v1
"/\" v3)))
"/\" (v1
"\/" (v0
"/\" (v2
"/\" v3))))))
proof
let v0, v1, v3, v2;
((v2
"/\" (v2
"\/" (v0
"/\" (v1
"/\" v3))))
"/\" (v1
"\/" (v0
"/\" (v2
"/\" v3))))
= (v2
"/\" ((v2
"\/" (v0
"/\" (v1
"/\" v3)))
"/\" (v1
"\/" (v0
"/\" (v2
"/\" v3))))) by
A3;
hence thesis by
A664;
end;
A669: for v0, v100, v1 holds (v100
"\/" ((v100
"\/" ((v1
"\/" v100)
"/\" v0))
"/\" (v0
"\/" v1)))
= ((v0
"/\" v1)
"\/" (v100
"/\" (v100
"\/" ((v1
"/\" v100)
"\/" v0))))
proof
let v0, v100, v1;
((v0
"/\" v1)
"\/" (v100
"/\" ((v1
"/\" v100)
"\/" v0)))
= ((v100
"\/" ((v1
"\/" v100)
"/\" v0))
"/\" (v0
"\/" v1)) by
A20;
hence thesis by
A421;
end;
A675: for v101, v1, v2, v100 holds ((v100
"/\" v101)
"\/" (v101
"/\" (v1
"\/" (v100
"/\" (v100
"\/" v2)))))
= (v101
"/\" (v100
"\/" (v1
"\/" (v100
"/\" v2))))
proof
let v101, v1, v2, v100;
(v100
"\/" (v1
"\/" (v100
"/\" v2)))
= (v1
"\/" (v100
"/\" (v100
"\/" v2))) by
A421;
hence thesis by
A128;
end;
A679: for v101, v2, v100 holds ((v2
"\/" ((v100
"\/" v2)
"/\" (v100
"\/" v101)))
"/\" ((v100
"\/" v101)
"\/" v100))
= (v100
"\/" ((v100
"/\" v101)
"\/" (v2
"/\" ((v100
"/\" v2)
"\/" (v100
"\/" v101)))))
proof
let v101, v2, v100;
((v100
"/\" (v100
"\/" v101))
"\/" (v2
"/\" ((v100
"/\" v2)
"\/" (v100
"\/" v101))))
= ((v2
"\/" ((v100
"\/" v2)
"/\" (v100
"\/" v101)))
"/\" ((v100
"\/" v101)
"\/" v100)) by
A101;
hence thesis by
A403;
end;
A682: for v2, v0, v1 holds ((v0
"\/" ((v1
"\/" v0)
"/\" (v1
"\/" v2)))
"/\" (v1
"\/" (v1
"\/" v2)))
= (v1
"\/" ((v1
"/\" v2)
"\/" (v0
"/\" ((v1
"/\" v0)
"\/" (v1
"\/" v2)))))
proof
let v2, v0, v1;
((v1
"\/" v2)
"\/" v1)
= (v1
"\/" (v1
"\/" v2)) by
A8;
hence thesis by
A679;
end;
A684: for v2, v0, v1 holds ((v0
"\/" ((v1
"\/" v0)
"/\" (v1
"\/" v2)))
"/\" (v1
"\/" v2))
= (v1
"\/" ((v1
"/\" v2)
"\/" (v0
"/\" ((v1
"/\" v0)
"\/" (v1
"\/" v2)))))
proof
let v2, v0, v1;
(v1
"\/" (v1
"\/" v2))
= (v1
"\/" v2) by
A43;
hence thesis by
A682;
end;
A686: for v2, v0, v1 holds ((v1
"\/" v2)
"/\" (v0
"\/" ((v1
"\/" v0)
"/\" (v1
"\/" v2))))
= (v1
"\/" ((v1
"/\" v2)
"\/" (v0
"/\" ((v1
"/\" v0)
"\/" (v1
"\/" v2)))))
proof
let v2, v0, v1;
((v0
"\/" ((v1
"\/" v0)
"/\" (v1
"\/" v2)))
"/\" (v1
"\/" v2))
= ((v1
"\/" v2)
"/\" (v0
"\/" ((v1
"\/" v0)
"/\" (v1
"\/" v2)))) by
A4;
hence thesis by
A684;
end;
A689: for v1, v2, v0 holds ((v0
"\/" v1)
"/\" (v2
"\/" ((v0
"\/" v2)
"/\" (v0
"\/" v1))))
= (v0
"\/" ((v0
"/\" v1)
"\/" (v2
"/\" (v0
"\/" ((v0
"/\" v2)
"\/" v1)))))
proof
let v1, v2, v0;
((v0
"/\" v2)
"\/" (v0
"\/" v1))
= (v0
"\/" ((v0
"/\" v2)
"\/" v1)) by
A57;
hence thesis by
A686;
end;
A693: for v102, v1, v100 holds (v100
"\/" (v100
"\/" ((v100
"/\" v1)
"\/" ((v100
"\/" v1)
"/\" v102))))
= (v100
"\/" ((v100
"\/" v1)
"/\" v102))
proof
let v102, v1, v100;
((v100
"/\" (v100
"\/" v1))
"\/" ((v100
"\/" v1)
"/\" v102))
= (v100
"\/" ((v100
"/\" v1)
"\/" ((v100
"\/" v1)
"/\" v102))) by
A403;
hence thesis by
A426;
end;
A696: for v2, v1, v0 holds (v0
"\/" ((v0
"/\" v1)
"\/" ((v0
"\/" v1)
"/\" v2)))
= (v0
"\/" ((v0
"\/" v1)
"/\" v2))
proof
let v2, v1, v0;
(v0
"\/" (v0
"\/" ((v0
"/\" v1)
"\/" ((v0
"\/" v1)
"/\" v2))))
= (v0
"\/" ((v0
"/\" v1)
"\/" ((v0
"\/" v1)
"/\" v2))) by
A43;
hence thesis by
A693;
end;
A699: for v100, v1, v102 holds ((v100
"/\" (v102
"\/" v1))
"\/" (v102
"/\" (v102
"\/" ((v102
"/\" v1)
"\/" v100))))
= ((v102
"\/" (((v102
"\/" v1)
"\/" v102)
"/\" v100))
"/\" (v100
"\/" (v102
"\/" v1)))
proof
let v100, v1, v102;
((v102
"/\" (v102
"\/" v1))
"\/" v100)
= (v102
"\/" ((v102
"/\" v1)
"\/" v100)) by
A403;
hence thesis by
A104;
end;
A702: for v0, v2, v1 holds ((v0
"/\" (v1
"\/" v2))
"\/" (v1
"/\" (v1
"\/" ((v1
"/\" v2)
"\/" v0))))
= ((v1
"\/" ((v1
"\/" (v1
"\/" v2))
"/\" v0))
"/\" (v0
"\/" (v1
"\/" v2)))
proof
let v0, v2, v1;
((v1
"\/" v2)
"\/" v1)
= (v1
"\/" (v1
"\/" v2)) by
A8;
hence thesis by
A699;
end;
A704: for v0, v2, v1 holds ((v0
"/\" (v1
"\/" v2))
"\/" (v1
"/\" (v1
"\/" ((v1
"/\" v2)
"\/" v0))))
= ((v1
"\/" ((v1
"\/" v2)
"/\" v0))
"/\" (v0
"\/" (v1
"\/" v2)))
proof
let v0, v2, v1;
(v1
"\/" (v1
"\/" v2))
= (v1
"\/" v2) by
A43;
hence thesis by
A702;
end;
A707: for v102, v2, v101 holds ((v101
"/\" v2)
"\/" (v102
"/\" (v101
"/\" v2)))
= ((v101
"/\" v2)
"/\" ((v101
"/\" v2)
"\/" (v101
"/\" v102)))
proof
let v102, v2, v101;
(v101
"/\" (v102
"/\" (v101
"/\" v2)))
= (v102
"/\" (v101
"/\" v2)) by
A162;
hence thesis by
A441;
end;
A710: for v2, v1, v0 holds ((v0
"/\" v1)
"/\" ((v0
"/\" v1)
"\/" v2))
= ((v0
"/\" v1)
"/\" ((v0
"/\" v1)
"\/" (v0
"/\" v2)))
proof
let v2, v1, v0;
((v0
"/\" v1)
"\/" (v2
"/\" (v0
"/\" v1)))
= ((v0
"/\" v1)
"/\" ((v0
"/\" v1)
"\/" v2)) by
A400;
hence thesis by
A707;
end;
A712: for v2, v1, v0 holds (v0
"/\" (v1
"/\" ((v0
"/\" v1)
"\/" v2)))
= ((v0
"/\" v1)
"/\" ((v0
"/\" v1)
"\/" (v0
"/\" v2)))
proof
let v2, v1, v0;
((v0
"/\" v1)
"/\" ((v0
"/\" v1)
"\/" v2))
= (v0
"/\" (v1
"/\" ((v0
"/\" v1)
"\/" v2))) by
A3;
hence thesis by
A710;
end;
A714: for v2, v1, v0 holds (v0
"/\" (v1
"/\" ((v0
"/\" v1)
"\/" v2)))
= (v0
"/\" (v1
"/\" ((v0
"/\" v1)
"\/" (v0
"/\" v2))))
proof
let v2, v1, v0;
((v0
"/\" v1)
"/\" ((v0
"/\" v1)
"\/" (v0
"/\" v2)))
= (v0
"/\" (v1
"/\" ((v0
"/\" v1)
"\/" (v0
"/\" v2)))) by
A3;
hence thesis by
A712;
end;
A717: for v2, v1, v0 holds ((v1
"\/" v0)
"/\" (v2
"/\" (v0
"/\" v1)))
= (v2
"/\" (v0
"/\" v1))
proof
let v2, v1, v0;
(v0
"\/" v1)
= (v1
"\/" v0) by
A8;
hence thesis by
A481;
end;
A721: for v102, v1, v0 holds ((v0
"/\" v1)
"\/" (v102
"/\" (v0
"/\" v1)))
= ((v0
"/\" v1)
"/\" ((v0
"/\" v1)
"\/" ((v0
"\/" v1)
"/\" v102)))
proof
let v102, v1, v0;
((v0
"\/" v1)
"/\" (v102
"/\" (v0
"/\" v1)))
= (v102
"/\" (v0
"/\" v1)) by
A481;
hence thesis by
A441;
end;
A724: for v2, v1, v0 holds ((v0
"/\" v1)
"/\" ((v0
"/\" v1)
"\/" v2))
= ((v0
"/\" v1)
"/\" ((v0
"/\" v1)
"\/" ((v0
"\/" v1)
"/\" v2)))
proof
let v2, v1, v0;
((v0
"/\" v1)
"\/" (v2
"/\" (v0
"/\" v1)))
= ((v0
"/\" v1)
"/\" ((v0
"/\" v1)
"\/" v2)) by
A400;
hence thesis by
A721;
end;
A726: for v2, v1, v0 holds (v0
"/\" (v1
"/\" ((v0
"/\" v1)
"\/" v2)))
= ((v0
"/\" v1)
"/\" ((v0
"/\" v1)
"\/" ((v0
"\/" v1)
"/\" v2)))
proof
let v2, v1, v0;
((v0
"/\" v1)
"/\" ((v0
"/\" v1)
"\/" v2))
= (v0
"/\" (v1
"/\" ((v0
"/\" v1)
"\/" v2))) by
A3;
hence thesis by
A724;
end;
A728: for v2, v1, v0 holds (v0
"/\" (v1
"/\" ((v0
"/\" v1)
"\/" v2)))
= (v0
"/\" (v1
"/\" ((v0
"/\" v1)
"\/" ((v0
"\/" v1)
"/\" v2))))
proof
let v2, v1, v0;
((v0
"/\" v1)
"/\" ((v0
"/\" v1)
"\/" ((v0
"\/" v1)
"/\" v2)))
= (v0
"/\" (v1
"/\" ((v0
"/\" v1)
"\/" ((v0
"\/" v1)
"/\" v2)))) by
A3;
hence thesis by
A726;
end;
A732: for v2, v100, v1, v0 holds (v100
"/\" ((v100
"\/" (v0
"/\" v1))
"/\" (v2
"\/" (v0
"\/" v1))))
= (v100
"/\" ((v0
"/\" v1)
"\/" (v2
"\/" (v0
"\/" v1))))
proof
let v2, v100, v1, v0;
((v0
"/\" v1)
"\/" (v2
"\/" (v0
"\/" v1)))
= (v2
"\/" (v0
"\/" v1)) by
A497;
hence thesis by
A351;
end;
A735: for v3, v0, v2, v1 holds (v0
"/\" ((v0
"\/" (v1
"/\" v2))
"/\" (v3
"\/" (v1
"\/" v2))))
= (v0
"/\" (v3
"\/" (v1
"\/" v2)))
proof
let v3, v0, v2, v1;
((v1
"/\" v2)
"\/" (v3
"\/" (v1
"\/" v2)))
= (v3
"\/" (v1
"\/" v2)) by
A497;
hence thesis by
A732;
end;
A738: for v102, v0, v2, v1 holds ((v0
"\/" (v1
"/\" v2))
"/\" v102)
= ((v0
"\/" (v1
"/\" v2))
"/\" ((v1
"\/" v0)
"/\" v102))
proof
let v102, v0, v2, v1;
((v0
"\/" (v1
"/\" v2))
"/\" (v1
"\/" v0))
= (v0
"\/" (v1
"/\" v2)) by
A500;
hence thesis by
A3;
end;
A743: for v102, v1, v2, v100 holds (((v1
"\/" (v100
"/\" (v100
"\/" v2)))
"/\" v102)
"\/" (v102
"/\" v100))
= (v102
"/\" (v100
"\/" (v1
"\/" (v100
"/\" v2))))
proof
let v102, v1, v2, v100;
(v100
"\/" (v1
"\/" (v100
"/\" v2)))
= (v1
"\/" (v100
"/\" (v100
"\/" v2))) by
A421;
hence thesis by
A513;
end;
A747: for v101, v1, v102 holds ((v102
"\/" v1)
"\/" (v101
"/\" (v102
"\/" v1)))
= ((v102
"\/" v1)
"\/" (v101
"/\" v102))
proof
let v101, v1, v102;
(((v102
"\/" v1)
"/\" v101)
"\/" (v101
"/\" v102))
= (v101
"/\" (v102
"\/" v1)) by
A513;
hence thesis by
A426;
end;
A750: for v2, v1, v0 holds ((v0
"\/" v1)
"/\" ((v0
"\/" v1)
"\/" v2))
= ((v0
"\/" v1)
"\/" (v2
"/\" v0))
proof
let v2, v1, v0;
((v0
"\/" v1)
"\/" (v2
"/\" (v0
"\/" v1)))
= ((v0
"\/" v1)
"/\" ((v0
"\/" v1)
"\/" v2)) by
A400;
hence thesis by
A747;
end;
A752: for v0, v2, v1 holds ((v0
"\/" v1)
"/\" (v0
"\/" (v1
"\/" v2)))
= ((v0
"\/" v1)
"\/" (v2
"/\" v0))
proof
let v0, v2, v1;
((v0
"\/" v1)
"\/" v2)
= (v0
"\/" (v1
"\/" v2)) by
A7;
hence thesis by
A750;
end;
A754: for v0, v2, v1 holds ((v0
"\/" v1)
"/\" (v0
"\/" (v1
"\/" v2)))
= (v0
"\/" (v1
"\/" (v2
"/\" v0)))
proof
let v0, v2, v1;
((v0
"\/" v1)
"\/" (v2
"/\" v0))
= (v0
"\/" (v1
"\/" (v2
"/\" v0))) by
A7;
hence thesis by
A752;
end;
A758: for v102, v1, v100 holds (v100
"\/" (v102
"/\" (v100
"\/" v1)))
= (((v100
"\/" v1)
"/\" v102)
"\/" (v100
"/\" (v100
"\/" v102)))
proof
let v102, v1, v100;
(((v100
"\/" v1)
"/\" v102)
"\/" (v102
"/\" v100))
= (v102
"/\" (v100
"\/" v1)) by
A513;
hence thesis by
A451;
end;
A764: for v102, v1, v2, v100 holds (((v1
"\/" (v100
"/\" (v100
"\/" v2)))
"/\" v102)
"\/" (v102
"/\" v100))
= (v102
"/\" (v100
"\/" (v1
"\/" (v2
"/\" v100))))
proof
let v102, v1, v2, v100;
(v100
"\/" (v1
"\/" (v2
"/\" v100)))
= (v1
"\/" (v100
"/\" (v100
"\/" v2))) by
A451;
hence thesis by
A513;
end;
A767: for v3, v0, v2, v1 holds (v3
"/\" (v1
"\/" (v0
"\/" (v1
"/\" v2))))
= (v3
"/\" (v1
"\/" (v0
"\/" (v2
"/\" v1))))
proof
let v3, v0, v2, v1;
(((v0
"\/" (v1
"/\" (v1
"\/" v2)))
"/\" v3)
"\/" (v3
"/\" v1))
= (v3
"/\" (v1
"\/" (v0
"\/" (v1
"/\" v2)))) by
A743;
hence thesis by
A764;
end;
A770: for v0, v2, v3, v1 holds (v0
"/\" (v1
"\/" (v2
"\/" (v1
"/\" v3))))
= (v0
"/\" ((v1
"\/" v2)
"/\" (v1
"\/" (v2
"\/" v3))))
proof
let v0, v2, v3, v1;
(v1
"\/" (v2
"\/" (v3
"/\" v1)))
= ((v1
"\/" v2)
"/\" (v1
"\/" (v2
"\/" v3))) by
A754;
hence thesis by
A767;
end;
A772: for v1, v2, v3, v0 holds ((v0
"/\" v1)
"\/" (v1
"/\" (v2
"\/" (v0
"/\" (v0
"\/" v3)))))
= (v1
"/\" ((v0
"\/" v2)
"/\" (v0
"\/" (v2
"\/" v3))))
proof
let v1, v2, v3, v0;
(v1
"/\" (v0
"\/" (v2
"\/" (v0
"/\" v3))))
= (v1
"/\" ((v0
"\/" v2)
"/\" (v0
"\/" (v2
"\/" v3)))) by
A770;
hence thesis by
A675;
end;
A774: for v0, v2, v1 holds ((v0
"\/" v1)
"/\" ((v0
"\/" v1)
"/\" (v0
"\/" (v1
"\/" v2))))
= (v0
"\/" (v1
"\/" (v0
"/\" v2)))
proof
let v0, v2, v1;
((v0
"\/" v1)
"/\" (v0
"\/" (v1
"\/" (v0
"/\" v2))))
= ((v0
"\/" v1)
"/\" ((v0
"\/" v1)
"/\" (v0
"\/" (v1
"\/" v2)))) by
A770;
hence thesis by
A510;
end;
A776: for v0, v2, v1 holds ((v0
"\/" v1)
"/\" (v0
"\/" (v1
"\/" v2)))
= (v0
"\/" (v1
"\/" (v0
"/\" v2)))
proof
let v0, v2, v1;
((v0
"\/" v1)
"/\" ((v0
"\/" v1)
"/\" (v0
"\/" (v1
"\/" v2))))
= ((v0
"\/" v1)
"/\" (v0
"\/" (v1
"\/" v2))) by
A26;
hence thesis by
A774;
end;
A779: for v2, v1, v0 holds ((v0
"\/" (v0
"/\" v1))
"/\" (v0
"\/" ((v0
"/\" v1)
"\/" ((v0
"\/" v1)
"/\" v2))))
= (v0
"/\" ((v0
"\/" v1)
"/\" (v0
"\/" ((v0
"/\" v1)
"\/" v2))))
proof
let v2, v1, v0;
(v0
"\/" ((v0
"/\" v1)
"\/" (v0
"/\" ((v0
"\/" v1)
"/\" v2))))
= ((v0
"\/" (v0
"/\" v1))
"/\" (v0
"\/" ((v0
"/\" v1)
"\/" ((v0
"\/" v1)
"/\" v2)))) by
A776;
hence thesis by
A614;
end;
A781: for v2, v1, v0 holds ((v0
"/\" (v0
"\/" v1))
"/\" (v0
"\/" ((v0
"/\" v1)
"\/" ((v0
"\/" v1)
"/\" v2))))
= (v0
"/\" ((v0
"\/" v1)
"/\" (v0
"\/" ((v0
"/\" v1)
"\/" v2))))
proof
let v2, v1, v0;
(v0
"\/" (v0
"/\" v1))
= (v0
"/\" (v0
"\/" v1)) by
A347;
hence thesis by
A779;
end;
A783: for v2, v1, v0 holds ((v0
"/\" (v0
"\/" v1))
"/\" (v0
"\/" ((v0
"\/" v1)
"/\" v2)))
= (v0
"/\" ((v0
"\/" v1)
"/\" (v0
"\/" ((v0
"/\" v1)
"\/" v2))))
proof
let v2, v1, v0;
(v0
"\/" ((v0
"/\" v1)
"\/" ((v0
"\/" v1)
"/\" v2)))
= (v0
"\/" ((v0
"\/" v1)
"/\" v2)) by
A696;
hence thesis by
A781;
end;
A785: for v2, v1, v0 holds (v0
"/\" ((v0
"\/" v1)
"/\" (v0
"\/" ((v0
"\/" v1)
"/\" v2))))
= (v0
"/\" ((v0
"\/" v1)
"/\" (v0
"\/" ((v0
"/\" v1)
"\/" v2))))
proof
let v2, v1, v0;
((v0
"/\" (v0
"\/" v1))
"/\" (v0
"\/" ((v0
"\/" v1)
"/\" v2)))
= (v0
"/\" ((v0
"\/" v1)
"/\" (v0
"\/" ((v0
"\/" v1)
"/\" v2)))) by
A3;
hence thesis by
A783;
end;
A787: for v2, v1, v0 holds (v0
"/\" (v0
"\/" ((v0
"\/" v1)
"/\" v2)))
= (v0
"/\" ((v0
"\/" v1)
"/\" (v0
"\/" ((v0
"/\" v1)
"\/" v2))))
proof
let v2, v1, v0;
((v0
"\/" v1)
"/\" (v0
"\/" ((v0
"\/" v1)
"/\" v2)))
= (v0
"\/" ((v0
"\/" v1)
"/\" v2)) by
A198;
hence thesis by
A785;
end;
A790: for v2, v1, v0 holds (v0
"\/" ((v0
"/\" v1)
"\/" (v2
"/\" (v0
"/\" (v0
"\/" v1)))))
= (v0
"/\" (v0
"\/" ((v0
"\/" v1)
"/\" v2)))
proof
let v2, v1, v0;
(v0
"/\" ((v0
"\/" v1)
"/\" (v0
"\/" ((v0
"/\" v1)
"\/" v2))))
= (v0
"/\" (v0
"\/" ((v0
"\/" v1)
"/\" v2))) by
A787;
hence thesis by
A622;
end;
A793: for v102, v100, v1 holds ((v1
"/\" v100)
"\/" ((v102
"\/" (v1
"/\" (v1
"\/" v100)))
"/\" v100))
= (v100
"/\" ((v1
"/\" (v1
"\/" v100))
"\/" v102))
proof
let v102, v100, v1;
(v100
"/\" (v1
"/\" (v1
"\/" v100)))
= (v1
"/\" v100) by
A308;
hence thesis by
A517;
end;
A796: for v2, v1, v0 holds ((v0
"/\" v1)
"\/" (v1
"/\" (v2
"\/" (v0
"/\" (v0
"\/" v1)))))
= (v1
"/\" ((v0
"/\" (v0
"\/" v1))
"\/" v2))
proof
let v2, v1, v0;
((v2
"\/" (v0
"/\" (v0
"\/" v1)))
"/\" v1)
= (v1
"/\" (v2
"\/" (v0
"/\" (v0
"\/" v1)))) by
A4;
hence thesis by
A793;
end;
A798: for v0, v1, v2 holds (v1
"/\" ((v0
"\/" v2)
"/\" (v0
"\/" (v2
"\/" v1))))
= (v1
"/\" ((v0
"/\" (v0
"\/" v1))
"\/" v2))
proof
let v0, v1, v2;
((v0
"/\" v1)
"\/" (v1
"/\" (v2
"\/" (v0
"/\" (v0
"\/" v1)))))
= (v1
"/\" ((v0
"\/" v2)
"/\" (v0
"\/" (v2
"\/" v1)))) by
A772;
hence thesis by
A796;
end;
A801: for v0, v2, v1 holds ((v1
"\/" v2)
"/\" v0)
= (v0
"/\" ((v1
"/\" (v1
"\/" v0))
"\/" v2))
proof
let v0, v2, v1;
(v0
"/\" ((v1
"\/" v2)
"/\" (v1
"\/" (v2
"\/" v0))))
= ((v1
"\/" v2)
"/\" v0) by
A328;
hence thesis by
A798;
end;
A804: for v2, v1, v0 holds ((v0
"\/" v1)
"/\" v2)
= (v2
"/\" (v0
"\/" ((v0
"/\" v2)
"\/" v1)))
proof
let v2, v1, v0;
((v0
"/\" (v0
"\/" v2))
"\/" v1)
= (v0
"\/" ((v0
"/\" v2)
"\/" v1)) by
A403;
hence thesis by
A801;
end;
A808: for v2, v1, v0 holds (v0
"\/" ((v0
"/\" v1)
"\/" ((v0
"\/" v1)
"/\" v2)))
= ((v0
"\/" v1)
"/\" (v2
"\/" ((v0
"\/" v2)
"/\" (v0
"\/" v1))))
proof
let v2, v1, v0;
(v2
"/\" (v0
"\/" ((v0
"/\" v2)
"\/" v1)))
= ((v0
"\/" v1)
"/\" v2) by
A804;
hence thesis by
A689;
end;
A810: for v2, v1, v0 holds (v0
"\/" ((v0
"\/" v1)
"/\" v2))
= ((v0
"\/" v1)
"/\" (v2
"\/" ((v0
"\/" v2)
"/\" (v0
"\/" v1))))
proof
let v2, v1, v0;
(v0
"\/" ((v0
"/\" v1)
"\/" ((v0
"\/" v1)
"/\" v2)))
= (v0
"\/" ((v0
"\/" v1)
"/\" v2)) by
A696;
hence thesis by
A808;
end;
A813: for v1, v2, v102, v101 holds (((v1
"/\" ((v101
"/\" v102)
"/\" v2))
"/\" v101)
"\/" ((v101
"/\" v102)
"/\" ((v101
"/\" v102)
"\/" (v1
"/\" v2))))
= ((v1
"/\" ((v101
"/\" v102)
"/\" v2))
"\/" (v101
"/\" v102))
proof
let v1, v2, v102, v101;
((v101
"/\" v102)
"\/" (v1
"/\" ((v101
"/\" v102)
"/\" v2)))
= ((v101
"/\" v102)
"/\" ((v101
"/\" v102)
"\/" (v1
"/\" v2))) by
A417;
hence thesis by
A529;
end;
A816: for v0, v1, v3, v2 holds (((v0
"/\" (v1
"/\" (v2
"/\" v3)))
"/\" v1)
"\/" ((v1
"/\" v2)
"/\" ((v1
"/\" v2)
"\/" (v0
"/\" v3))))
= ((v0
"/\" ((v1
"/\" v2)
"/\" v3))
"\/" (v1
"/\" v2))
proof
let v0, v1, v3, v2;
((v1
"/\" v2)
"/\" v3)
= (v1
"/\" (v2
"/\" v3)) by
A3;
hence thesis by
A813;
end;
A818: for v0, v1, v3, v2 holds ((v1
"/\" (v0
"/\" (v1
"/\" (v2
"/\" v3))))
"\/" ((v1
"/\" v2)
"/\" ((v1
"/\" v2)
"\/" (v0
"/\" v3))))
= ((v0
"/\" ((v1
"/\" v2)
"/\" v3))
"\/" (v1
"/\" v2))
proof
let v0, v1, v3, v2;
((v0
"/\" (v1
"/\" (v2
"/\" v3)))
"/\" v1)
= (v1
"/\" (v0
"/\" (v1
"/\" (v2
"/\" v3)))) by
A4;
hence thesis by
A816;
end;
A821: for v1, v0, v3, v2 holds ((v1
"/\" (v0
"/\" (v2
"/\" v3)))
"\/" ((v0
"/\" v2)
"/\" ((v0
"/\" v2)
"\/" (v1
"/\" v3))))
= ((v1
"/\" ((v0
"/\" v2)
"/\" v3))
"\/" (v0
"/\" v2))
proof
let v1, v0, v3, v2;
(v0
"/\" (v1
"/\" (v0
"/\" (v2
"/\" v3))))
= (v1
"/\" (v0
"/\" (v2
"/\" v3))) by
A162;
hence thesis by
A818;
end;
A824: for v0, v1, v3, v2 holds ((v0
"/\" (v1
"/\" (v2
"/\" v3)))
"\/" (v1
"/\" (v2
"/\" ((v1
"/\" v2)
"\/" (v0
"/\" v3)))))
= ((v0
"/\" ((v1
"/\" v2)
"/\" v3))
"\/" (v1
"/\" v2))
proof
let v0, v1, v3, v2;
((v1
"/\" v2)
"/\" ((v1
"/\" v2)
"\/" (v0
"/\" v3)))
= (v1
"/\" (v2
"/\" ((v1
"/\" v2)
"\/" (v0
"/\" v3)))) by
A3;
hence thesis by
A821;
end;
A826: for v2, v0, v3, v1 holds (v1
"/\" (v2
"/\" ((v2
"\/" (v0
"/\" (v1
"/\" v3)))
"/\" (v1
"\/" (v0
"/\" (v2
"/\" v3))))))
= ((v0
"/\" ((v1
"/\" v2)
"/\" v3))
"\/" (v1
"/\" v2))
proof
let v2, v0, v3, v1;
((v0
"/\" (v1
"/\" (v2
"/\" v3)))
"\/" (v1
"/\" (v2
"/\" ((v1
"/\" v2)
"\/" (v0
"/\" v3)))))
= (v1
"/\" (v2
"/\" ((v2
"\/" (v0
"/\" (v1
"/\" v3)))
"/\" (v1
"\/" (v0
"/\" (v2
"/\" v3)))))) by
A666;
hence thesis by
A824;
end;
A829: for v1, v2, v3, v0 holds (v0
"/\" (v1
"/\" ((v1
"\/" (v2
"/\" (v0
"/\" v3)))
"/\" (v0
"\/" (v2
"/\" (v1
"/\" v3))))))
= ((v2
"/\" (v0
"/\" (v1
"/\" v3)))
"\/" (v0
"/\" v1))
proof
let v1, v2, v3, v0;
((v0
"/\" v1)
"/\" v3)
= (v0
"/\" (v1
"/\" v3)) by
A3;
hence thesis by
A826;
end;
A831: for v1, v2, v3, v0 holds (v0
"/\" (v1
"/\" ((v1
"\/" (v2
"/\" (v0
"/\" v3)))
"/\" (v0
"\/" (v2
"/\" (v1
"/\" v3))))))
= ((v0
"/\" v1)
"\/" (v2
"/\" (v0
"/\" (v1
"/\" v3))))
proof
let v1, v2, v3, v0;
((v2
"/\" (v0
"/\" (v1
"/\" v3)))
"\/" (v0
"/\" v1))
= ((v0
"/\" v1)
"\/" (v2
"/\" (v0
"/\" (v1
"/\" v3)))) by
A8;
hence thesis by
A829;
end;
A833: for v1, v2, v3, v0 holds (v0
"/\" (v1
"/\" ((v1
"\/" (v2
"/\" (v0
"/\" v3)))
"/\" (v0
"\/" (v2
"/\" (v1
"/\" v3))))))
= (v0
"/\" (v1
"/\" ((v0
"/\" v1)
"\/" (v2
"/\" v3))))
proof
let v1, v2, v3, v0;
((v0
"/\" v1)
"\/" (v2
"/\" (v0
"/\" (v1
"/\" v3))))
= (v0
"/\" (v1
"/\" ((v0
"/\" v1)
"\/" (v2
"/\" v3)))) by
A628;
hence thesis by
A831;
end;
A836: for v2, v1, v0 holds (v0
"/\" (v1
"/\" ((v1
"\/" ((v0
"\/" v1)
"/\" (v0
"/\" v2)))
"/\" (v0
"\/" ((v0
"\/" v1)
"/\" (v1
"/\" v2))))))
= (v0
"/\" (v1
"/\" ((v0
"/\" v1)
"\/" v2)))
proof
let v2, v1, v0;
(v0
"/\" (v1
"/\" ((v0
"/\" v1)
"\/" ((v0
"\/" v1)
"/\" v2))))
= (v0
"/\" (v1
"/\" ((v1
"\/" ((v0
"\/" v1)
"/\" (v0
"/\" v2)))
"/\" (v0
"\/" ((v0
"\/" v1)
"/\" (v1
"/\" v2)))))) by
A833;
hence thesis by
A728;
end;
A838: for v2, v1, v0 holds (v0
"/\" (v1
"/\" ((v1
"\/" (v0
"/\" ((v0
"\/" v1)
"/\" v2)))
"/\" (v0
"\/" ((v0
"\/" v1)
"/\" (v1
"/\" v2))))))
= (v0
"/\" (v1
"/\" ((v0
"/\" v1)
"\/" v2)))
proof
let v2, v1, v0;
((v0
"\/" v1)
"/\" (v0
"/\" v2))
= (v0
"/\" ((v0
"\/" v1)
"/\" v2)) by
A40;
hence thesis by
A836;
end;
A840: for v2, v1, v0 holds (v0
"/\" (v1
"/\" ((v1
"\/" (v0
"/\" ((v0
"\/" v1)
"/\" v2)))
"/\" (v0
"\/" (v1
"/\" ((v0
"\/" v1)
"/\" v2))))))
= (v0
"/\" (v1
"/\" ((v0
"/\" v1)
"\/" v2)))
proof
let v2, v1, v0;
((v0
"\/" v1)
"/\" (v1
"/\" v2))
= (v1
"/\" ((v0
"\/" v1)
"/\" v2)) by
A40;
hence thesis by
A838;
end;
A843: for v2, v1, v0 holds (v0
"/\" (v1
"/\" ((v1
"\/" (v0
"/\" ((v0
"\/" v1)
"/\" (v0
"/\" v2))))
"/\" (v0
"\/" (v1
"/\" ((v0
"\/" v1)
"/\" (v0
"/\" v2)))))))
= (v0
"/\" (v1
"/\" ((v0
"/\" v1)
"\/" v2)))
proof
let v2, v1, v0;
(v0
"/\" (v1
"/\" ((v0
"/\" v1)
"\/" (v0
"/\" v2))))
= (v0
"/\" (v1
"/\" ((v1
"\/" (v0
"/\" ((v0
"\/" v1)
"/\" (v0
"/\" v2))))
"/\" (v0
"\/" (v1
"/\" ((v0
"\/" v1)
"/\" (v0
"/\" v2))))))) by
A840;
hence thesis by
A714;
end;
A845: for v2, v1, v0 holds (v0
"/\" (v1
"/\" ((v1
"\/" (v0
"/\" (v0
"/\" ((v0
"\/" v1)
"/\" v2))))
"/\" (v0
"\/" (v1
"/\" ((v0
"\/" v1)
"/\" (v0
"/\" v2)))))))
= (v0
"/\" (v1
"/\" ((v0
"/\" v1)
"\/" v2)))
proof
let v2, v1, v0;
((v0
"\/" v1)
"/\" (v0
"/\" v2))
= (v0
"/\" ((v0
"\/" v1)
"/\" v2)) by
A40;
hence thesis by
A843;
end;
A847: for v2, v1, v0 holds (v0
"/\" (v1
"/\" ((v1
"\/" (v0
"/\" ((v0
"\/" v1)
"/\" v2)))
"/\" (v0
"\/" (v1
"/\" ((v0
"\/" v1)
"/\" (v0
"/\" v2)))))))
= (v0
"/\" (v1
"/\" ((v0
"/\" v1)
"\/" v2)))
proof
let v2, v1, v0;
(v0
"/\" (v0
"/\" ((v0
"\/" v1)
"/\" v2)))
= (v0
"/\" ((v0
"\/" v1)
"/\" v2)) by
A26;
hence thesis by
A845;
end;
A849: for v2, v1, v0 holds (v0
"/\" (v1
"/\" ((v1
"\/" (v0
"/\" ((v0
"\/" v1)
"/\" v2)))
"/\" (v0
"\/" (v1
"/\" (v0
"/\" ((v0
"\/" v1)
"/\" v2)))))))
= (v0
"/\" (v1
"/\" ((v0
"/\" v1)
"\/" v2)))
proof
let v2, v1, v0;
((v0
"\/" v1)
"/\" (v0
"/\" v2))
= (v0
"/\" ((v0
"\/" v1)
"/\" v2)) by
A40;
hence thesis by
A847;
end;
A851: for v2, v1, v0 holds (v0
"/\" (v1
"/\" ((v1
"\/" (v0
"/\" ((v0
"\/" v1)
"/\" v2)))
"/\" (v0
"\/" (v0
"/\" (v1
"/\" v2))))))
= (v0
"/\" (v1
"/\" ((v0
"/\" v1)
"\/" v2)))
proof
let v2, v1, v0;
(v1
"/\" (v0
"/\" ((v0
"\/" v1)
"/\" v2)))
= (v0
"/\" (v1
"/\" v2)) by
A323;
hence thesis by
A849;
end;
A853: for v2, v1, v0 holds (v0
"/\" (v1
"/\" ((v1
"\/" (v0
"/\" ((v0
"\/" v1)
"/\" v2)))
"/\" (v0
"/\" (v0
"\/" (v1
"/\" v2))))))
= (v0
"/\" (v1
"/\" ((v0
"/\" v1)
"\/" v2)))
proof
let v2, v1, v0;
(v0
"\/" (v0
"/\" (v1
"/\" v2)))
= (v0
"/\" (v0
"\/" (v1
"/\" v2))) by
A347;
hence thesis by
A851;
end;
A855: for v2, v1, v0 holds (v0
"/\" (v1
"/\" (v0
"/\" ((v1
"\/" (v0
"/\" ((v0
"\/" v1)
"/\" v2)))
"/\" (v0
"\/" (v1
"/\" v2))))))
= (v0
"/\" (v1
"/\" ((v0
"/\" v1)
"\/" v2)))
proof
let v2, v1, v0;
((v1
"\/" (v0
"/\" ((v0
"\/" v1)
"/\" v2)))
"/\" (v0
"/\" (v0
"\/" (v1
"/\" v2))))
= (v0
"/\" ((v1
"\/" (v0
"/\" ((v0
"\/" v1)
"/\" v2)))
"/\" (v0
"\/" (v1
"/\" v2)))) by
A40;
hence thesis by
A853;
end;
A857: for v2, v1, v0 holds (v1
"/\" (v0
"/\" ((v1
"\/" (v0
"/\" ((v0
"\/" v1)
"/\" v2)))
"/\" (v0
"\/" (v1
"/\" v2)))))
= (v0
"/\" (v1
"/\" ((v0
"/\" v1)
"\/" v2)))
proof
let v2, v1, v0;
(v0
"/\" (v1
"/\" (v0
"/\" ((v1
"\/" (v0
"/\" ((v0
"\/" v1)
"/\" v2)))
"/\" (v0
"\/" (v1
"/\" v2))))))
= (v1
"/\" (v0
"/\" ((v1
"\/" (v0
"/\" ((v0
"\/" v1)
"/\" v2)))
"/\" (v0
"\/" (v1
"/\" v2))))) by
A162;
hence thesis by
A855;
end;
A860: for v2, v0, v1 holds (v0
"/\" (v1
"/\" ((v0
"\/" (v1
"/\" ((v1
"\/" v0)
"/\" v2)))
"/\" (v1
"\/" (v0
"/\" v2)))))
= (v1
"/\" (v0
"/\" ((v0
"\/" (v1
"/\" ((v1
"\/" v0)
"/\" v2)))
"/\" (v1
"\/" (v0
"/\" ((v1
"\/" v0)
"/\" v2))))))
proof
let v2, v0, v1;
(v1
"/\" (v0
"/\" ((v1
"/\" v0)
"\/" v2)))
= (v1
"/\" (v0
"/\" ((v0
"\/" (v1
"/\" ((v1
"\/" v0)
"/\" v2)))
"/\" (v1
"\/" (v0
"/\" ((v1
"\/" v0)
"/\" v2)))))) by
A840;
hence thesis by
A857;
end;
A864: for v2, v1, v0 holds (v0
"/\" (v1
"/\" ((v0
"/\" v1)
"\/" v2)))
= (v1
"/\" (v0
"/\" ((v1
"\/" (v0
"/\" ((v0
"\/" v1)
"/\" v2)))
"/\" (v0
"\/" (v1
"/\" v2)))))
proof
let v2, v1, v0;
(v0
"/\" (v1
"/\" ((v1
"\/" (v0
"/\" ((v0
"\/" v1)
"/\" v2)))
"/\" (v0
"\/" (v1
"/\" ((v0
"\/" v1)
"/\" v2))))))
= (v1
"/\" (v0
"/\" ((v1
"\/" (v0
"/\" ((v0
"\/" v1)
"/\" v2)))
"/\" (v0
"\/" (v1
"/\" v2))))) by
A860;
hence thesis by
A840;
end;
A867: for v102, v1, v100 holds (v100
"\/" ((v100
"/\" v1)
"\/" (v102
"/\" (v100
"/\" v1))))
= (v100
"\/" (v102
"/\" (v100
"/\" v1)))
proof
let v102, v1, v100;
(v100
"/\" (v100
"/\" v1))
= (v100
"/\" v1) by
A26;
hence thesis by
A558;
end;
A870: for v2, v1, v0 holds (v0
"\/" ((v0
"/\" v1)
"/\" ((v0
"/\" v1)
"\/" v2)))
= (v0
"\/" (v2
"/\" (v0
"/\" v1)))
proof
let v2, v1, v0;
((v0
"/\" v1)
"\/" (v2
"/\" (v0
"/\" v1)))
= ((v0
"/\" v1)
"/\" ((v0
"/\" v1)
"\/" v2)) by
A400;
hence thesis by
A867;
end;
A872: for v2, v1, v0 holds (v0
"\/" (v0
"/\" (v1
"/\" ((v0
"/\" v1)
"\/" v2))))
= (v0
"\/" (v2
"/\" (v0
"/\" v1)))
proof
let v2, v1, v0;
((v0
"/\" v1)
"/\" ((v0
"/\" v1)
"\/" v2))
= (v0
"/\" (v1
"/\" ((v0
"/\" v1)
"\/" v2))) by
A3;
hence thesis by
A870;
end;
A874: for v2, v1, v0 holds (v0
"\/" (v1
"/\" (v0
"/\" ((v1
"\/" (v0
"/\" ((v0
"\/" v1)
"/\" v2)))
"/\" (v0
"\/" (v1
"/\" v2))))))
= (v0
"\/" (v2
"/\" (v0
"/\" v1)))
proof
let v2, v1, v0;
(v0
"/\" (v1
"/\" ((v0
"/\" v1)
"\/" v2)))
= (v1
"/\" (v0
"/\" ((v1
"\/" (v0
"/\" ((v0
"\/" v1)
"/\" v2)))
"/\" (v0
"\/" (v1
"/\" v2))))) by
A864;
hence thesis by
A872;
end;
A876: for v2, v1, v0 holds (v0
"/\" (v0
"\/" (v1
"/\" ((v1
"\/" (v0
"/\" ((v0
"\/" v1)
"/\" v2)))
"/\" (v0
"\/" (v1
"/\" v2))))))
= (v0
"\/" (v2
"/\" (v0
"/\" v1)))
proof
let v2, v1, v0;
(v0
"\/" (v1
"/\" (v0
"/\" ((v1
"\/" (v0
"/\" ((v0
"\/" v1)
"/\" v2)))
"/\" (v0
"\/" (v1
"/\" v2))))))
= (v0
"/\" (v0
"\/" (v1
"/\" ((v1
"\/" (v0
"/\" ((v0
"\/" v1)
"/\" v2)))
"/\" (v0
"\/" (v1
"/\" v2)))))) by
A417;
hence thesis by
A874;
end;
A878: for v2, v1, v0 holds (v0
"/\" (v0
"\/" (v1
"/\" ((v1
"\/" (v0
"/\" ((v0
"\/" v1)
"/\" v2)))
"/\" (v0
"\/" (v1
"/\" v2))))))
= (v0
"/\" (v0
"\/" (v2
"/\" v1)))
proof
let v2, v1, v0;
(v0
"\/" (v2
"/\" (v0
"/\" v1)))
= (v0
"/\" (v0
"\/" (v2
"/\" v1))) by
A417;
hence thesis by
A876;
end;
A881: for v102, v1, v100 holds (v100
"\/" (v100
"\/" ((v100
"/\" v1)
"\/" (v102
"/\" (v100
"\/" v1)))))
= (v100
"\/" (v102
"/\" (v100
"\/" v1)))
proof
let v102, v1, v100;
((v100
"/\" (v100
"\/" v1))
"\/" (v102
"/\" (v100
"\/" v1)))
= (v100
"\/" ((v100
"/\" v1)
"\/" (v102
"/\" (v100
"\/" v1)))) by
A403;
hence thesis by
A558;
end;
A884: for v2, v1, v0 holds (v0
"\/" ((v0
"/\" v1)
"\/" (v2
"/\" (v0
"\/" v1))))
= (v0
"\/" (v2
"/\" (v0
"\/" v1)))
proof
let v2, v1, v0;
(v0
"\/" (v0
"\/" ((v0
"/\" v1)
"\/" (v2
"/\" (v0
"\/" v1)))))
= (v0
"\/" ((v0
"/\" v1)
"\/" (v2
"/\" (v0
"\/" v1)))) by
A43;
hence thesis by
A881;
end;
A887: for v101, v2, v1 holds ((v1
"\/" (v101
"/\" (v1
"\/" v2)))
"\/" (v101
"/\" (v1
"\/" v2)))
= ((v1
"\/" (v101
"/\" (v1
"\/" v2)))
"/\" ((v1
"\/" (v101
"/\" (v1
"\/" v2)))
"\/" v101))
proof
let v101, v2, v1;
(v101
"/\" (v1
"\/" (v101
"/\" (v1
"\/" v2))))
= (v101
"/\" (v1
"\/" v2)) by
A568;
hence thesis by
A400;
end;
A890: for v1, v2, v0 holds ((v1
"/\" (v0
"\/" v2))
"\/" (v0
"\/" (v1
"/\" (v0
"\/" v2))))
= ((v0
"\/" (v1
"/\" (v0
"\/" v2)))
"/\" ((v0
"\/" (v1
"/\" (v0
"\/" v2)))
"\/" v1))
proof
let v1, v2, v0;
((v0
"\/" (v1
"/\" (v0
"\/" v2)))
"\/" (v1
"/\" (v0
"\/" v2)))
= ((v1
"/\" (v0
"\/" v2))
"\/" (v0
"\/" (v1
"/\" (v0
"\/" v2)))) by
A8;
hence thesis by
A887;
end;
A893: for v0, v2, v1 holds (v1
"\/" (v0
"/\" (v1
"\/" v2)))
= ((v1
"\/" (v0
"/\" (v1
"\/" v2)))
"/\" ((v1
"\/" (v0
"/\" (v1
"\/" v2)))
"\/" v0))
proof
let v0, v2, v1;
((v0
"/\" (v1
"\/" v2))
"\/" (v1
"\/" (v0
"/\" (v1
"\/" v2))))
= (v1
"\/" (v0
"/\" (v1
"\/" v2))) by
A51;
hence thesis by
A890;
end;
A896: for v1, v2, v0 holds (v0
"\/" (v1
"/\" (v0
"\/" v2)))
= ((v0
"\/" (v1
"/\" (v0
"\/" v2)))
"/\" (v1
"\/" (v0
"\/" (v1
"/\" (v0
"\/" v2)))))
proof
let v1, v2, v0;
((v0
"\/" (v1
"/\" (v0
"\/" v2)))
"\/" v1)
= (v1
"\/" (v0
"\/" (v1
"/\" (v0
"\/" v2)))) by
A8;
hence thesis by
A893;
end;
A898: for v1, v2, v0 holds (v0
"\/" (v1
"/\" (v0
"\/" v2)))
= ((v0
"\/" (v1
"/\" (v0
"\/" v2)))
"/\" ((v1
"\/" v0)
"/\" (v1
"\/" (v0
"\/" (v0
"\/" v2)))))
proof
let v1, v2, v0;
(v1
"\/" (v0
"\/" (v1
"/\" (v0
"\/" v2))))
= ((v1
"\/" v0)
"/\" (v1
"\/" (v0
"\/" (v0
"\/" v2)))) by
A776;
hence thesis by
A896;
end;
A900: for v1, v2, v0 holds (v0
"\/" (v1
"/\" (v0
"\/" v2)))
= ((v0
"\/" (v1
"/\" (v0
"\/" v2)))
"/\" ((v1
"\/" v0)
"/\" (v1
"\/" (v0
"\/" v2))))
proof
let v1, v2, v0;
(v0
"\/" (v0
"\/" v2))
= (v0
"\/" v2) by
A43;
hence thesis by
A898;
end;
A902: for v1, v2, v0 holds (v0
"\/" (v1
"/\" (v0
"\/" v2)))
= ((v0
"\/" (v1
"/\" (v0
"\/" v2)))
"/\" (v1
"\/" (v0
"\/" v2)))
proof
let v1, v2, v0;
((v0
"\/" (v1
"/\" (v0
"\/" v2)))
"/\" ((v1
"\/" v0)
"/\" (v1
"\/" (v0
"\/" v2))))
= ((v0
"\/" (v1
"/\" (v0
"\/" v2)))
"/\" (v1
"\/" (v0
"\/" v2))) by
A738;
hence thesis by
A900;
end;
A905: for v0, v1, v2 holds (v0
"/\" (v1
"\/" ((v2
"\/" v1)
"/\" v0)))
= (v0
"/\" (v2
"\/" v1))
proof
let v0, v1, v2;
(v0
"/\" (v2
"\/" v1))
= ((v2
"\/" v1)
"/\" v0) by
A4;
hence thesis by
A570;
end;
A908: for v100, v101, v1, v2 holds (v100
"/\" (v101
"/\" (v2
"\/" v1)))
= (v101
"/\" (v100
"/\" (v1
"\/" (v101
"/\" (v2
"\/" v1)))))
proof
let v100, v101, v1, v2;
(v101
"/\" (v1
"\/" (v101
"/\" (v2
"\/" v1))))
= (v101
"/\" (v2
"\/" v1)) by
A570;
hence thesis by
A40;
end;
A913: for v2, v1, v0 holds (v0
"\/" ((v0
"\/" v1)
"/\" v2))
= ((v0
"\/" v1)
"/\" (v0
"\/" v2))
proof
let v2, v1, v0;
((v0
"\/" v1)
"/\" (v2
"\/" ((v0
"\/" v2)
"/\" (v0
"\/" v1))))
= ((v0
"\/" v1)
"/\" (v0
"\/" v2)) by
A905;
hence thesis by
A810;
end;
A915: for v0, v2, v1 holds ((v0
"/\" v1)
"\/" (v2
"/\" (v0
"/\" (v0
"\/" (v1
"/\" v2)))))
= ((v0
"\/" v1)
"/\" (v0
"/\" ((v0
"/\" v1)
"\/" v2)))
proof
let v0, v2, v1;
(v0
"/\" ((v0
"\/" v1)
"/\" (v2
"\/" (v0
"/\" ((v0
"/\" v1)
"\/" v2)))))
= ((v0
"\/" v1)
"/\" (v0
"/\" ((v0
"/\" v1)
"\/" v2))) by
A908;
hence thesis by
A357;
end;
A917: for v0, v2, v1 holds ((v0
"/\" v1)
"\/" (v2
"/\" (v0
"/\" (v0
"\/" (v1
"/\" v2)))))
= (v0
"/\" ((v0
"\/" v1)
"/\" ((v0
"/\" v1)
"\/" v2)))
proof
let v0, v2, v1;
((v0
"\/" v1)
"/\" (v0
"/\" ((v0
"/\" v1)
"\/" v2)))
= (v0
"/\" ((v0
"\/" v1)
"/\" ((v0
"/\" v1)
"\/" v2))) by
A40;
hence thesis by
A915;
end;
A919: for v0, v2, v1 holds ((v0
"/\" v1)
"\/" (v2
"/\" (v0
"/\" (v0
"\/" (v1
"/\" v2)))))
= (v0
"/\" ((v0
"/\" v1)
"\/" v2))
proof
let v0, v2, v1;
(v0
"/\" ((v0
"\/" v1)
"/\" ((v0
"/\" v1)
"\/" v2)))
= (v0
"/\" ((v0
"/\" v1)
"\/" v2)) by
A594;
hence thesis by
A917;
end;
A921: for v2, v1, v0 holds (v0
"\/" ((v0
"/\" v1)
"\/" (v2
"/\" (v0
"/\" (v0
"\/" v1)))))
= (v0
"/\" ((v0
"\/" v1)
"/\" (v0
"\/" v2)))
proof
let v2, v1, v0;
(v0
"\/" ((v0
"\/" v1)
"/\" v2))
= ((v0
"\/" v1)
"/\" (v0
"\/" v2)) by
A913;
hence thesis by
A790;
end;
A923: for v0, v2, v1 holds ((v0
"/\" (v1
"\/" v2))
"\/" (v1
"/\" (v1
"\/" ((v1
"/\" v2)
"\/" v0))))
= (((v1
"\/" v2)
"/\" (v1
"\/" v0))
"/\" (v0
"\/" (v1
"\/" v2)))
proof
let v0, v2, v1;
(v1
"\/" ((v1
"\/" v2)
"/\" v0))
= ((v1
"\/" v2)
"/\" (v1
"\/" v0)) by
A913;
hence thesis by
A704;
end;
A925: for v0, v2, v1 holds ((v0
"/\" (v1
"\/" v2))
"\/" (v1
"/\" (v1
"\/" ((v1
"/\" v2)
"\/" v0))))
= ((v1
"\/" v2)
"/\" ((v1
"\/" v0)
"/\" (v0
"\/" (v1
"\/" v2))))
proof
let v0, v2, v1;
(((v1
"\/" v2)
"/\" (v1
"\/" v0))
"/\" (v0
"\/" (v1
"\/" v2)))
= ((v1
"\/" v2)
"/\" ((v1
"\/" v0)
"/\" (v0
"\/" (v1
"\/" v2)))) by
A3;
hence thesis by
A923;
end;
A927: for v0, v2, v1 holds ((v0
"/\" v1)
"\/" (v2
"/\" (v2
"\/" ((v1
"/\" v2)
"\/" v0))))
= ((v2
"\/" ((v1
"\/" v2)
"/\" v0))
"/\" (v2
"\/" (v0
"\/" v1)))
proof
let v0, v2, v1;
(v2
"\/" ((v2
"\/" ((v1
"\/" v2)
"/\" v0))
"/\" (v0
"\/" v1)))
= ((v2
"\/" ((v1
"\/" v2)
"/\" v0))
"/\" (v2
"\/" (v0
"\/" v1))) by
A913;
hence thesis by
A669;
end;
A929: for v2, v1, v0 holds (((v0
"/\" v1)
"\/" v2)
"/\" ((v2
"\/" ((v1
"\/" v2)
"/\" v0))
"/\" (v0
"\/" v1)))
= ((v2
"\/" ((v2
"\/" v1)
"/\" v0))
"/\" (v0
"\/" v1))
proof
let v2, v1, v0;
(v1
"\/" v2)
= (v2
"\/" v1) by
A8;
hence thesis by
A110;
end;
A931: for v0, v2, v1 holds ((v2
"\/" ((v1
"\/" v2)
"/\" v0))
"/\" (v0
"\/" v1))
= ((v2
"\/" ((v2
"\/" v1)
"/\" v0))
"/\" (v0
"\/" v1))
proof
let v0, v2, v1;
(((v0
"/\" v1)
"\/" v2)
"/\" ((v2
"\/" ((v1
"\/" v2)
"/\" v0))
"/\" (v0
"\/" v1)))
= ((v2
"\/" ((v1
"\/" v2)
"/\" v0))
"/\" (v0
"\/" v1)) by
A110;
hence thesis by
A929;
end;
A934: for v2, v0, v1 holds ((v0
"\/" ((v1
"\/" v0)
"/\" v2))
"/\" (v2
"\/" v1))
= (((v0
"\/" v1)
"/\" (v0
"\/" v2))
"/\" (v2
"\/" v1))
proof
let v2, v0, v1;
(v0
"\/" ((v0
"\/" v1)
"/\" v2))
= ((v0
"\/" v1)
"/\" (v0
"\/" v2)) by
A913;
hence thesis by
A931;
end;
A936: for v2, v0, v1 holds ((v0
"\/" ((v1
"\/" v0)
"/\" v2))
"/\" (v2
"\/" v1))
= ((v0
"\/" v1)
"/\" ((v0
"\/" v2)
"/\" (v2
"\/" v1)))
proof
let v2, v0, v1;
(((v0
"\/" v1)
"/\" (v0
"\/" v2))
"/\" (v2
"\/" v1))
= ((v0
"\/" v1)
"/\" ((v0
"\/" v2)
"/\" (v2
"\/" v1))) by
A3;
hence thesis by
A934;
end;
A938: for v2, v1, v0 holds (((v0
"/\" v1)
"\/" v2)
"/\" ((v2
"\/" ((v1
"\/" v2)
"/\" v0))
"/\" (v0
"\/" v1)))
= ((((v1
"\/" v2)
"/\" v0)
"\/" v2)
"/\" (v0
"\/" v1))
proof
let v2, v1, v0;
(v2
"\/" ((v1
"\/" v2)
"/\" v0))
= (((v1
"\/" v2)
"/\" v0)
"\/" v2) by
A8;
hence thesis by
A110;
end;
A940: for v2, v1, v0 holds (((v0
"/\" v1)
"\/" v2)
"/\" ((v2
"\/" v1)
"/\" ((v2
"\/" v0)
"/\" (v0
"\/" v1))))
= ((((v1
"\/" v2)
"/\" v0)
"\/" v2)
"/\" (v0
"\/" v1))
proof
let v2, v1, v0;
((v2
"\/" ((v1
"\/" v2)
"/\" v0))
"/\" (v0
"\/" v1))
= ((v2
"\/" v1)
"/\" ((v2
"\/" v0)
"/\" (v0
"\/" v1))) by
A936;
hence thesis by
A938;
end;
A942: for v2, v1, v0 holds (((v0
"/\" v1)
"\/" v2)
"/\" ((v2
"\/" v1)
"/\" ((v2
"\/" v0)
"/\" (v0
"\/" v1))))
= ((v2
"\/" ((v1
"\/" v2)
"/\" v0))
"/\" (v0
"\/" v1))
proof
let v2, v1, v0;
(((v1
"\/" v2)
"/\" v0)
"\/" v2)
= (v2
"\/" ((v1
"\/" v2)
"/\" v0)) by
A8;
hence thesis by
A940;
end;
A944: for v2, v1, v0 holds (((v0
"/\" v1)
"\/" v2)
"/\" ((v2
"\/" v1)
"/\" ((v2
"\/" v0)
"/\" (v0
"\/" v1))))
= ((v2
"\/" v1)
"/\" ((v2
"\/" v0)
"/\" (v0
"\/" v1)))
proof
let v2, v1, v0;
((v2
"\/" ((v1
"\/" v2)
"/\" v0))
"/\" (v0
"\/" v1))
= ((v2
"\/" v1)
"/\" ((v2
"\/" v0)
"/\" (v0
"\/" v1))) by
A936;
hence thesis by
A942;
end;
A947: for v100, v1, v0 holds ((v100
"\/" (v0
"/\" v1))
"/\" ((v100
"\/" ((v1
"\/" v100)
"/\" v0))
"/\" (v0
"\/" v1)))
= (((v0
"/\" v1)
"\/" v100)
"/\" ((v100
"\/" ((v1
"\/" v100)
"/\" v0))
"/\" (v0
"\/" v1)))
proof
let v100, v1, v0;
(((v0
"/\" v1)
"\/" v100)
"/\" ((v100
"\/" ((v1
"\/" v100)
"/\" v0))
"/\" (v0
"\/" v1)))
= ((v100
"\/" ((v1
"\/" v100)
"/\" v0))
"/\" (v0
"\/" v1)) by
A110;
hence thesis by
A392;
end;
A950: for v0, v2, v1 holds ((v0
"\/" (v1
"/\" v2))
"/\" ((v0
"\/" v2)
"/\" ((v0
"\/" v1)
"/\" (v1
"\/" v2))))
= (((v1
"/\" v2)
"\/" v0)
"/\" ((v0
"\/" ((v2
"\/" v0)
"/\" v1))
"/\" (v1
"\/" v2)))
proof
let v0, v2, v1;
((v0
"\/" ((v2
"\/" v0)
"/\" v1))
"/\" (v1
"\/" v2))
= ((v0
"\/" v2)
"/\" ((v0
"\/" v1)
"/\" (v1
"\/" v2))) by
A936;
hence thesis by
A947;
end;
A952: for v0, v2, v1 holds ((v0
"\/" v2)
"/\" ((v0
"\/" (v1
"/\" v2))
"/\" ((v0
"\/" v1)
"/\" (v1
"\/" v2))))
= (((v1
"/\" v2)
"\/" v0)
"/\" ((v0
"\/" ((v2
"\/" v0)
"/\" v1))
"/\" (v1
"\/" v2)))
proof
let v0, v2, v1;
((v0
"\/" (v1
"/\" v2))
"/\" ((v0
"\/" v2)
"/\" ((v0
"\/" v1)
"/\" (v1
"\/" v2))))
= ((v0
"\/" v2)
"/\" ((v0
"\/" (v1
"/\" v2))
"/\" ((v0
"\/" v1)
"/\" (v1
"\/" v2)))) by
A40;
hence thesis by
A950;
end;
A955: for v0, v1, v2 holds ((v0
"\/" v1)
"/\" ((v0
"\/" v2)
"/\" ((v0
"\/" (v2
"/\" v1))
"/\" (v2
"\/" v1))))
= (((v2
"/\" v1)
"\/" v0)
"/\" ((v0
"\/" ((v1
"\/" v0)
"/\" v2))
"/\" (v2
"\/" v1)))
proof
let v0, v1, v2;
((v0
"\/" (v2
"/\" v1))
"/\" ((v0
"\/" v2)
"/\" (v2
"\/" v1)))
= ((v0
"\/" v2)
"/\" ((v0
"\/" (v2
"/\" v1))
"/\" (v2
"\/" v1))) by
A40;
hence thesis by
A952;
end;
A957: for v0, v1, v2 holds ((v0
"\/" v1)
"/\" ((v0
"\/" (v2
"/\" v1))
"/\" (v2
"\/" v1)))
= (((v2
"/\" v1)
"\/" v0)
"/\" ((v0
"\/" ((v1
"\/" v0)
"/\" v2))
"/\" (v2
"\/" v1)))
proof
let v0, v1, v2;
((v0
"\/" v2)
"/\" ((v0
"\/" (v2
"/\" v1))
"/\" (v2
"\/" v1)))
= ((v0
"\/" (v2
"/\" v1))
"/\" (v2
"\/" v1)) by
A60;
hence thesis by
A955;
end;
A959: for v0, v1, v2 holds ((v0
"\/" (v2
"/\" v1))
"/\" (v2
"\/" v1))
= (((v2
"/\" v1)
"\/" v0)
"/\" ((v0
"\/" ((v1
"\/" v0)
"/\" v2))
"/\" (v2
"\/" v1)))
proof
let v0, v1, v2;
((v0
"\/" v1)
"/\" ((v0
"\/" (v2
"/\" v1))
"/\" (v2
"\/" v1)))
= ((v0
"\/" (v2
"/\" v1))
"/\" (v2
"\/" v1)) by
A315;
hence thesis by
A957;
end;
A962: for v0, v2, v1 holds ((v0
"\/" (v1
"/\" v2))
"/\" (v1
"\/" v2))
= (((v1
"/\" v2)
"\/" v0)
"/\" ((v0
"\/" v2)
"/\" ((v0
"\/" v1)
"/\" (v1
"\/" v2))))
proof
let v0, v2, v1;
((v0
"\/" ((v2
"\/" v0)
"/\" v1))
"/\" (v1
"\/" v2))
= ((v0
"\/" v2)
"/\" ((v0
"\/" v1)
"/\" (v1
"\/" v2))) by
A936;
hence thesis by
A959;
end;
A964: for v0, v2, v1 holds ((v0
"\/" (v1
"/\" v2))
"/\" (v1
"\/" v2))
= ((v0
"\/" v2)
"/\" ((v0
"\/" v1)
"/\" (v1
"\/" v2)))
proof
let v0, v2, v1;
(((v1
"/\" v2)
"\/" v0)
"/\" ((v0
"\/" v2)
"/\" ((v0
"\/" v1)
"/\" (v1
"\/" v2))))
= ((v0
"\/" v2)
"/\" ((v0
"\/" v1)
"/\" (v1
"\/" v2))) by
A944;
hence thesis by
A962;
end;
A966: for v1, v2, v0 holds ((v0
"\/" (v0
"\/" v2))
"/\" ((v0
"\/" v1)
"/\" (v1
"\/" (v0
"\/" v2))))
= (v0
"\/" (v1
"/\" (v0
"\/" v2)))
proof
let v1, v2, v0;
((v0
"\/" (v1
"/\" (v0
"\/" v2)))
"/\" (v1
"\/" (v0
"\/" v2)))
= ((v0
"\/" (v0
"\/" v2))
"/\" ((v0
"\/" v1)
"/\" (v1
"\/" (v0
"\/" v2)))) by
A964;
hence thesis by
A902;
end;
A969: for v2, v1, v0 holds ((v0
"\/" v1)
"/\" ((v0
"\/" v2)
"/\" (v2
"\/" (v0
"\/" v1))))
= (v0
"\/" (v2
"/\" (v0
"\/" v1)))
proof
let v2, v1, v0;
(v0
"\/" (v0
"\/" v1))
= (v0
"\/" v1) by
A43;
hence thesis by
A966;
end;
A974: for v1, v101, v100 holds (v100
"\/" ((v100
"/\" v101)
"/\" ((v100
"/\" v101)
"\/" v1)))
= (((v100
"/\" v101)
"/\" v1)
"\/" (v100
"/\" (v100
"\/" v101)))
proof
let v1, v101, v100;
((v100
"/\" v101)
"\/" ((v100
"/\" v101)
"/\" v1))
= ((v100
"/\" v101)
"/\" ((v100
"/\" v101)
"\/" v1)) by
A347;
hence thesis by
A603;
end;
A977: for v2, v1, v0 holds (v0
"\/" (v0
"/\" (v1
"/\" ((v0
"/\" v1)
"\/" v2))))
= (((v0
"/\" v1)
"/\" v2)
"\/" (v0
"/\" (v0
"\/" v1)))
proof
let v2, v1, v0;
((v0
"/\" v1)
"/\" ((v0
"/\" v1)
"\/" v2))
= (v0
"/\" (v1
"/\" ((v0
"/\" v1)
"\/" v2))) by
A3;
hence thesis by
A974;
end;
A979: for v2, v1, v0 holds (v0
"\/" (v1
"/\" (v0
"/\" ((v1
"\/" (v0
"/\" ((v0
"\/" v1)
"/\" v2)))
"/\" (v0
"\/" (v1
"/\" v2))))))
= (((v0
"/\" v1)
"/\" v2)
"\/" (v0
"/\" (v0
"\/" v1)))
proof
let v2, v1, v0;
(v0
"/\" (v1
"/\" ((v0
"/\" v1)
"\/" v2)))
= (v1
"/\" (v0
"/\" ((v1
"\/" (v0
"/\" ((v0
"\/" v1)
"/\" v2)))
"/\" (v0
"\/" (v1
"/\" v2))))) by
A864;
hence thesis by
A977;
end;
A981: for v2, v1, v0 holds (v0
"/\" (v0
"\/" (v1
"/\" ((v1
"\/" (v0
"/\" ((v0
"\/" v1)
"/\" v2)))
"/\" (v0
"\/" (v1
"/\" v2))))))
= (((v0
"/\" v1)
"/\" v2)
"\/" (v0
"/\" (v0
"\/" v1)))
proof
let v2, v1, v0;
(v0
"\/" (v1
"/\" (v0
"/\" ((v1
"\/" (v0
"/\" ((v0
"\/" v1)
"/\" v2)))
"/\" (v0
"\/" (v1
"/\" v2))))))
= (v0
"/\" (v0
"\/" (v1
"/\" ((v1
"\/" (v0
"/\" ((v0
"\/" v1)
"/\" v2)))
"/\" (v0
"\/" (v1
"/\" v2)))))) by
A417;
hence thesis by
A979;
end;
A983: for v0, v1, v2 holds (v0
"/\" (v0
"\/" (v2
"/\" v1)))
= (((v0
"/\" v1)
"/\" v2)
"\/" (v0
"/\" (v0
"\/" v1)))
proof
let v0, v1, v2;
(v0
"/\" (v0
"\/" (v1
"/\" ((v1
"\/" (v0
"/\" ((v0
"\/" v1)
"/\" v2)))
"/\" (v0
"\/" (v1
"/\" v2))))))
= (v0
"/\" (v0
"\/" (v2
"/\" v1))) by
A878;
hence thesis by
A981;
end;
A986: for v0, v2, v1 holds (v0
"/\" (v0
"\/" (v1
"/\" v2)))
= ((v0
"/\" (v2
"/\" v1))
"\/" (v0
"/\" (v0
"\/" v2)))
proof
let v0, v2, v1;
((v0
"/\" v2)
"/\" v1)
= (v0
"/\" (v2
"/\" v1)) by
A3;
hence thesis by
A983;
end;
A991: for v100, v1, v102 holds (v100
"\/" (v102
"\/" ((v102
"/\" v1)
"\/" (v100
"/\" (v102
"\/" v1)))))
= (v100
"\/" (v102
"/\" (v102
"\/" v1)))
proof
let v100, v1, v102;
(v102
"\/" ((v102
"/\" v1)
"\/" (v100
"/\" (v102
"\/" v1))))
= ((v100
"/\" (v102
"\/" v1))
"\/" (v102
"/\" (v102
"\/" v1))) by
A603;
hence thesis by
A558;
end;
A994: for v0, v2, v1 holds (v0
"\/" (v1
"\/" (v0
"/\" (v1
"\/" v2))))
= (v0
"\/" (v1
"/\" (v1
"\/" v2)))
proof
let v0, v2, v1;
(v1
"\/" ((v1
"/\" v2)
"\/" (v0
"/\" (v1
"\/" v2))))
= (v1
"\/" (v0
"/\" (v1
"\/" v2))) by
A884;
hence thesis by
A991;
end;
A996: for v0, v2, v1 holds ((v0
"\/" v1)
"/\" (v0
"\/" (v1
"\/" (v1
"\/" v2))))
= (v0
"\/" (v1
"/\" (v1
"\/" v2)))
proof
let v0, v2, v1;
(v0
"\/" (v1
"\/" (v0
"/\" (v1
"\/" v2))))
= ((v0
"\/" v1)
"/\" (v0
"\/" (v1
"\/" (v1
"\/" v2)))) by
A776;
hence thesis by
A994;
end;
A998: for v0, v2, v1 holds ((v0
"\/" v1)
"/\" (v0
"\/" (v1
"\/" v2)))
= (v0
"\/" (v1
"/\" (v1
"\/" v2)))
proof
let v0, v2, v1;
(v1
"\/" (v1
"\/" v2))
= (v1
"\/" v2) by
A43;
hence thesis by
A996;
end;
A1001: for v0, v2, v1 holds (((v0
"/\" (v1
"/\" v2))
"\/" v0)
"/\" ((v0
"/\" (v1
"/\" v2))
"\/" (v0
"\/" v1)))
= (v0
"/\" (v0
"\/" (v2
"/\" v1)))
proof
let v0, v2, v1;
((v0
"/\" (v1
"/\" v2))
"\/" (v0
"/\" (v0
"\/" v1)))
= (((v0
"/\" (v1
"/\" v2))
"\/" v0)
"/\" ((v0
"/\" (v1
"/\" v2))
"\/" (v0
"\/" v1))) by
A998;
hence thesis by
A986;
end;
A1003: for v0, v2, v1 holds ((v0
"\/" (v0
"/\" (v1
"/\" v2)))
"/\" ((v0
"/\" (v1
"/\" v2))
"\/" (v0
"\/" v1)))
= (v0
"/\" (v0
"\/" (v2
"/\" v1)))
proof
let v0, v2, v1;
((v0
"/\" (v1
"/\" v2))
"\/" v0)
= (v0
"\/" (v0
"/\" (v1
"/\" v2))) by
A8;
hence thesis by
A1001;
end;
A1005: for v0, v2, v1 holds ((v0
"/\" (v0
"\/" (v1
"/\" v2)))
"/\" ((v0
"/\" (v1
"/\" v2))
"\/" (v0
"\/" v1)))
= (v0
"/\" (v0
"\/" (v2
"/\" v1)))
proof
let v0, v2, v1;
(v0
"\/" (v0
"/\" (v1
"/\" v2)))
= (v0
"/\" (v0
"\/" (v1
"/\" v2))) by
A347;
hence thesis by
A1003;
end;
A1007: for v0, v2, v1 holds ((v0
"/\" (v0
"\/" (v1
"/\" v2)))
"/\" (v0
"\/" ((v0
"/\" (v1
"/\" v2))
"\/" v1)))
= (v0
"/\" (v0
"\/" (v2
"/\" v1)))
proof
let v0, v2, v1;
((v0
"/\" (v1
"/\" v2))
"\/" (v0
"\/" v1))
= (v0
"\/" ((v0
"/\" (v1
"/\" v2))
"\/" v1)) by
A57;
hence thesis by
A1005;
end;
A1009: for v0, v2, v1 holds ((v0
"/\" (v0
"\/" (v1
"/\" v2)))
"/\" (v0
"\/" (v1
"\/" (v0
"/\" (v1
"/\" v2)))))
= (v0
"/\" (v0
"\/" (v2
"/\" v1)))
proof
let v0, v2, v1;
((v0
"/\" (v1
"/\" v2))
"\/" v1)
= (v1
"\/" (v0
"/\" (v1
"/\" v2))) by
A8;
hence thesis by
A1007;
end;
A1011: for v0, v2, v1 holds ((v0
"/\" (v0
"\/" (v1
"/\" v2)))
"/\" (v0
"\/" (v1
"/\" (v1
"\/" (v0
"/\" v2)))))
= (v0
"/\" (v0
"\/" (v2
"/\" v1)))
proof
let v0, v2, v1;
(v1
"\/" (v0
"/\" (v1
"/\" v2)))
= (v1
"/\" (v1
"\/" (v0
"/\" v2))) by
A417;
hence thesis by
A1009;
end;
A1013: for v0, v2, v1 holds ((v0
"/\" (v0
"\/" (v1
"/\" v2)))
"/\" ((v0
"\/" v1)
"/\" (v0
"\/" (v1
"\/" (v0
"/\" v2)))))
= (v0
"/\" (v0
"\/" (v2
"/\" v1)))
proof
let v0, v2, v1;
(v0
"\/" (v1
"/\" (v1
"\/" (v0
"/\" v2))))
= ((v0
"\/" v1)
"/\" (v0
"\/" (v1
"\/" (v0
"/\" v2)))) by
A998;
hence thesis by
A1011;
end;
A1015: for v0, v2, v1 holds ((v0
"/\" (v0
"\/" (v1
"/\" v2)))
"/\" ((v0
"\/" v1)
"/\" ((v0
"\/" v1)
"/\" (v0
"\/" (v1
"\/" v2)))))
= (v0
"/\" (v0
"\/" (v2
"/\" v1)))
proof
let v0, v2, v1;
(v0
"\/" (v1
"\/" (v0
"/\" v2)))
= ((v0
"\/" v1)
"/\" (v0
"\/" (v1
"\/" v2))) by
A776;
hence thesis by
A1013;
end;
A1017: for v0, v2, v1 holds ((v0
"/\" (v0
"\/" (v1
"/\" v2)))
"/\" ((v0
"\/" v1)
"/\" (v0
"\/" (v1
"\/" v2))))
= (v0
"/\" (v0
"\/" (v2
"/\" v1)))
proof
let v0, v2, v1;
((v0
"\/" v1)
"/\" ((v0
"\/" v1)
"/\" (v0
"\/" (v1
"\/" v2))))
= ((v0
"\/" v1)
"/\" (v0
"\/" (v1
"\/" v2))) by
A26;
hence thesis by
A1015;
end;
A1019: for v0, v2, v1 holds ((v0
"\/" v1)
"/\" ((v0
"/\" (v0
"\/" (v1
"/\" v2)))
"/\" (v0
"\/" (v1
"\/" v2))))
= (v0
"/\" (v0
"\/" (v2
"/\" v1)))
proof
let v0, v2, v1;
((v0
"/\" (v0
"\/" (v1
"/\" v2)))
"/\" ((v0
"\/" v1)
"/\" (v0
"\/" (v1
"\/" v2))))
= ((v0
"\/" v1)
"/\" ((v0
"/\" (v0
"\/" (v1
"/\" v2)))
"/\" (v0
"\/" (v1
"\/" v2)))) by
A40;
hence thesis by
A1017;
end;
A1021: for v0, v2, v1 holds ((v0
"\/" v1)
"/\" (v0
"/\" ((v0
"\/" (v1
"/\" v2))
"/\" (v0
"\/" (v1
"\/" v2)))))
= (v0
"/\" (v0
"\/" (v2
"/\" v1)))
proof
let v0, v2, v1;
((v0
"/\" (v0
"\/" (v1
"/\" v2)))
"/\" (v0
"\/" (v1
"\/" v2)))
= (v0
"/\" ((v0
"\/" (v1
"/\" v2))
"/\" (v0
"\/" (v1
"\/" v2)))) by
A3;
hence thesis by
A1019;
end;
A1023: for v0, v2, v1 holds ((v0
"\/" v1)
"/\" (v0
"/\" (v0
"\/" (v1
"\/" v2))))
= (v0
"/\" (v0
"\/" (v2
"/\" v1)))
proof
let v0, v2, v1;
(v0
"/\" ((v0
"\/" (v1
"/\" v2))
"/\" (v0
"\/" (v1
"\/" v2))))
= (v0
"/\" (v0
"\/" (v1
"\/" v2))) by
A735;
hence thesis by
A1021;
end;
A1025: for v0, v2, v1 holds (v0
"/\" ((v0
"\/" v1)
"/\" (v0
"\/" (v1
"\/" v2))))
= (v0
"/\" (v0
"\/" (v2
"/\" v1)))
proof
let v0, v2, v1;
((v0
"\/" v1)
"/\" (v0
"/\" (v0
"\/" (v1
"\/" v2))))
= (v0
"/\" ((v0
"\/" v1)
"/\" (v0
"\/" (v1
"\/" v2)))) by
A40;
hence thesis by
A1023;
end;
A1027: for v0, v2, v1 holds (v0
"/\" (v0
"\/" (v1
"\/" v2)))
= (v0
"/\" (v0
"\/" (v2
"/\" v1)))
proof
let v0, v2, v1;
(v0
"/\" ((v0
"\/" v1)
"/\" (v0
"\/" (v1
"\/" v2))))
= (v0
"/\" (v0
"\/" (v1
"\/" v2))) by
A600;
hence thesis by
A1025;
end;
A1029: for v0, v2, v1 holds ((v0
"/\" v1)
"\/" (v2
"/\" (v2
"\/" (v0
"/\" (v1
"/\" v2)))))
= ((v2
"\/" ((v1
"\/" v2)
"/\" v0))
"/\" (v2
"\/" (v0
"\/" v1)))
proof
let v0, v2, v1;
(v2
"/\" (v2
"\/" ((v1
"/\" v2)
"\/" v0)))
= (v2
"/\" (v2
"\/" (v0
"/\" (v1
"/\" v2)))) by
A1027;
hence thesis by
A927;
end;
A1031: for v2, v1, v0 holds ((v0
"/\" v1)
"\/" (v2
"/\" (v2
"/\" (v2
"\/" (v0
"/\" v1)))))
= ((v2
"\/" ((v1
"\/" v2)
"/\" v0))
"/\" (v2
"\/" (v0
"\/" v1)))
proof
let v2, v1, v0;
(v2
"\/" (v0
"/\" (v1
"/\" v2)))
= (v2
"/\" (v2
"\/" (v0
"/\" v1))) by
A441;
hence thesis by
A1029;
end;
A1033: for v2, v1, v0 holds (((v0
"/\" v1)
"\/" (v2
"/\" (v2
"\/" (v0
"/\" v1))))
= ((v2
"\/" ((v1
"\/" v2)
"/\" v0))
"/\" (v2
"\/" (v0
"\/" v1))))
proof
let v2, v1, v0;
(v2
"/\" (v2
"/\" (v2
"\/" (v0
"/\" v1))))
= (v2
"/\" (v2
"\/" (v0
"/\" v1))) by
A26;
hence thesis by
A1031;
end;
A1035: for v2, v1, v0 holds (v2
"\/" (v0
"/\" v1))
= ((v2
"\/" ((v1
"\/" v2)
"/\" v0))
"/\" (v2
"\/" (v0
"\/" v1)))
proof
let v2, v1, v0;
((v0
"/\" v1)
"\/" (v2
"/\" (v2
"\/" (v0
"/\" v1))))
= (v2
"\/" (v0
"/\" v1)) by
A344;
hence thesis by
A1033;
end;
A1040: for v0, v2, v1 holds ((v0
"/\" (v1
"\/" v2))
"\/" (v1
"/\" (v1
"\/" (v0
"/\" (v1
"/\" v2)))))
= ((v1
"\/" v2)
"/\" ((v1
"\/" v0)
"/\" (v0
"\/" (v1
"\/" v2))))
proof
let v0, v2, v1;
(v1
"/\" (v1
"\/" ((v1
"/\" v2)
"\/" v0)))
= (v1
"/\" (v1
"\/" (v0
"/\" (v1
"/\" v2)))) by
A1027;
hence thesis by
A925;
end;
A1042: for v0, v2, v1 holds ((v0
"/\" (v1
"\/" v2))
"\/" (v1
"/\" (v1
"/\" (v1
"\/" (v0
"/\" v2)))))
= ((v1
"\/" v2)
"/\" ((v1
"\/" v0)
"/\" (v0
"\/" (v1
"\/" v2))))
proof
let v0, v2, v1;
(v1
"\/" (v0
"/\" (v1
"/\" v2)))
= (v1
"/\" (v1
"\/" (v0
"/\" v2))) by
A417;
hence thesis by
A1040;
end;
A1044: for v0, v2, v1 holds ((v0
"/\" (v1
"\/" v2))
"\/" (v1
"/\" (v1
"\/" (v0
"/\" v2))))
= ((v1
"\/" v2)
"/\" ((v1
"\/" v0)
"/\" (v0
"\/" (v1
"\/" v2))))
proof
let v0, v2, v1;
(v1
"/\" (v1
"/\" (v1
"\/" (v0
"/\" v2))))
= (v1
"/\" (v1
"\/" (v0
"/\" v2))) by
A26;
hence thesis by
A1042;
end;
A1046: for v0, v2, v1 holds (((v0
"/\" (v1
"\/" v2))
"\/" v1)
"/\" ((v0
"/\" (v1
"\/" v2))
"\/" (v1
"\/" (v0
"/\" v2))))
= ((v1
"\/" v2)
"/\" ((v1
"\/" v0)
"/\" (v0
"\/" (v1
"\/" v2))))
proof
let v0, v2, v1;
((v0
"/\" (v1
"\/" v2))
"\/" (v1
"/\" (v1
"\/" (v0
"/\" v2))))
= (((v0
"/\" (v1
"\/" v2))
"\/" v1)
"/\" ((v0
"/\" (v1
"\/" v2))
"\/" (v1
"\/" (v0
"/\" v2)))) by
A998;
hence thesis by
A1044;
end;
A1048: for v0, v2, v1 holds ((v1
"\/" (v0
"/\" (v1
"\/" v2)))
"/\" ((v0
"/\" (v1
"\/" v2))
"\/" (v1
"\/" (v0
"/\" v2))))
= ((v1
"\/" v2)
"/\" ((v1
"\/" v0)
"/\" (v0
"\/" (v1
"\/" v2))))
proof
let v0, v2, v1;
((v0
"/\" (v1
"\/" v2))
"\/" v1)
= (v1
"\/" (v0
"/\" (v1
"\/" v2))) by
A8;
hence thesis by
A1046;
end;
A1051: for v1, v2, v0 holds ((v0
"\/" (v1
"/\" (v0
"\/" v2)))
"/\" (v0
"\/" ((v1
"/\" (v0
"\/" v2))
"\/" (v1
"/\" v2))))
= ((v0
"\/" v2)
"/\" ((v0
"\/" v1)
"/\" (v1
"\/" (v0
"\/" v2))))
proof
let v1, v2, v0;
((v1
"/\" (v0
"\/" v2))
"\/" (v0
"\/" (v1
"/\" v2)))
= (v0
"\/" ((v1
"/\" (v0
"\/" v2))
"\/" (v1
"/\" v2))) by
A57;
hence thesis by
A1048;
end;
A1053: for v1, v2, v0 holds ((v0
"\/" (v1
"/\" (v0
"\/" v2)))
"/\" (v0
"\/" ((v1
"/\" v2)
"\/" (v1
"/\" (v0
"\/" v2)))))
= ((v0
"\/" v2)
"/\" ((v0
"\/" v1)
"/\" (v1
"\/" (v0
"\/" v2))))
proof
let v1, v2, v0;
((v1
"/\" (v0
"\/" v2))
"\/" (v1
"/\" v2))
= ((v1
"/\" v2)
"\/" (v1
"/\" (v0
"\/" v2))) by
A8;
hence thesis by
A1051;
end;
A1055: for v1, v2, v0 holds ((v0
"\/" (v1
"/\" (v0
"\/" v2)))
"/\" (v0
"\/" (v1
"/\" (v2
"\/" v0))))
= ((v0
"\/" v2)
"/\" ((v0
"\/" v1)
"/\" (v1
"\/" (v0
"\/" v2))))
proof
let v1, v2, v0;
((v1
"/\" v2)
"\/" (v1
"/\" (v0
"\/" v2)))
= (v1
"/\" (v2
"\/" v0)) by
A133;
hence thesis by
A1053;
end;
A1057: for v1, v0, v2 holds (v0
"\/" (v1
"/\" (v2
"\/" v0)))
= ((v0
"\/" v2)
"/\" ((v0
"\/" v1)
"/\" (v1
"\/" (v0
"\/" v2))))
proof
let v1, v0, v2;
((v0
"\/" (v1
"/\" (v0
"\/" v2)))
"/\" (v0
"\/" (v1
"/\" (v2
"\/" v0))))
= (v0
"\/" (v1
"/\" (v2
"\/" v0))) by
A438;
hence thesis by
A1055;
end;
A1059: for v2, v1, v0 holds ((((v0
"\/" v1)
"/\" v2)
"\/" v0)
"/\" (((v0
"\/" v1)
"/\" v2)
"\/" (v0
"\/" v2)))
= (v0
"\/" (v2
"/\" (v0
"\/" v1)))
proof
let v2, v1, v0;
(((v0
"\/" v1)
"/\" v2)
"\/" (v0
"/\" (v0
"\/" v2)))
= ((((v0
"\/" v1)
"/\" v2)
"\/" v0)
"/\" (((v0
"\/" v1)
"/\" v2)
"\/" (v0
"\/" v2))) by
A998;
hence thesis by
A758;
end;
A1061: for v2, v1, v0 holds ((v0
"\/" ((v0
"\/" v1)
"/\" v2))
"/\" (((v0
"\/" v1)
"/\" v2)
"\/" (v0
"\/" v2)))
= (v0
"\/" (v2
"/\" (v0
"\/" v1)))
proof
let v2, v1, v0;
(((v0
"\/" v1)
"/\" v2)
"\/" v0)
= (v0
"\/" ((v0
"\/" v1)
"/\" v2)) by
A8;
hence thesis by
A1059;
end;
A1063: for v2, v1, v0 holds (((v0
"\/" v1)
"/\" (v0
"\/" v2))
"/\" (((v0
"\/" v1)
"/\" v2)
"\/" (v0
"\/" v2)))
= (v0
"\/" (v2
"/\" (v0
"\/" v1)))
proof
let v2, v1, v0;
(v0
"\/" ((v0
"\/" v1)
"/\" v2))
= ((v0
"\/" v1)
"/\" (v0
"\/" v2)) by
A913;
hence thesis by
A1061;
end;
A1065: for v2, v1, v0 holds (((v0
"\/" v1)
"/\" (v0
"\/" v2))
"/\" (v0
"\/" (((v0
"\/" v1)
"/\" v2)
"\/" v2)))
= (v0
"\/" (v2
"/\" (v0
"\/" v1)))
proof
let v2, v1, v0;
(((v0
"\/" v1)
"/\" v2)
"\/" (v0
"\/" v2))
= (v0
"\/" (((v0
"\/" v1)
"/\" v2)
"\/" v2)) by
A57;
hence thesis by
A1063;
end;
A1067: for v2, v1, v0 holds (((v0
"\/" v1)
"/\" (v0
"\/" v2))
"/\" (v0
"\/" (v2
"\/" ((v0
"\/" v1)
"/\" v2))))
= (v0
"\/" (v2
"/\" (v0
"\/" v1)))
proof
let v2, v1, v0;
(((v0
"\/" v1)
"/\" v2)
"\/" v2)
= (v2
"\/" ((v0
"\/" v1)
"/\" v2)) by
A8;
hence thesis by
A1065;
end;
A1069: for v2, v1, v0 holds (((v0
"\/" v1)
"/\" (v0
"\/" v2))
"/\" (v0
"\/" (v2
"/\" (v2
"\/" (v0
"\/" v1)))))
= (v0
"\/" (v2
"/\" (v0
"\/" v1)))
proof
let v2, v1, v0;
(v2
"\/" ((v0
"\/" v1)
"/\" v2))
= (v2
"/\" (v2
"\/" (v0
"\/" v1))) by
A400;
hence thesis by
A1067;
end;
A1071: for v2, v1, v0 holds (((v0
"\/" v1)
"/\" (v0
"\/" v2))
"/\" (v0
"\/" (v2
"/\" (v2
"\/" (v1
"/\" v0)))))
= (v0
"\/" (v2
"/\" (v0
"\/" v1)))
proof
let v2, v1, v0;
(v2
"/\" (v2
"\/" (v0
"\/" v1)))
= (v2
"/\" (v2
"\/" (v1
"/\" v0))) by
A1027;
hence thesis by
A1069;
end;
A1073: for v2, v1, v0 holds (((v0
"\/" v1)
"/\" (v0
"\/" v2))
"/\" ((v0
"\/" v2)
"/\" (v0
"\/" (v2
"\/" (v1
"/\" v0)))))
= (v0
"\/" (v2
"/\" (v0
"\/" v1)))
proof
let v2, v1, v0;
(v0
"\/" (v2
"/\" (v2
"\/" (v1
"/\" v0))))
= ((v0
"\/" v2)
"/\" (v0
"\/" (v2
"\/" (v1
"/\" v0)))) by
A998;
hence thesis by
A1071;
end;
A1075: for v2, v1, v0 holds (((v0
"\/" v1)
"/\" (v0
"\/" v2))
"/\" ((v0
"\/" v2)
"/\" ((v0
"\/" v2)
"/\" (v0
"\/" (v2
"\/" v1)))))
= (v0
"\/" (v2
"/\" (v0
"\/" v1)))
proof
let v2, v1, v0;
(v0
"\/" (v2
"\/" (v1
"/\" v0)))
= ((v0
"\/" v2)
"/\" (v0
"\/" (v2
"\/" v1))) by
A754;
hence thesis by
A1073;
end;
A1077: for v2, v1, v0 holds (((v0
"\/" v1)
"/\" (v0
"\/" v2))
"/\" ((v0
"\/" v2)
"/\" (v0
"\/" (v2
"\/" v1))))
= (v0
"\/" (v2
"/\" (v0
"\/" v1)))
proof
let v2, v1, v0;
((v0
"\/" v2)
"/\" ((v0
"\/" v2)
"/\" (v0
"\/" (v2
"\/" v1))))
= ((v0
"\/" v2)
"/\" (v0
"\/" (v2
"\/" v1))) by
A26;
hence thesis by
A1075;
end;
A1079: for v2, v1, v0 holds ((v0
"\/" v2)
"/\" (((v0
"\/" v1)
"/\" (v0
"\/" v2))
"/\" (v0
"\/" (v2
"\/" v1))))
= (v0
"\/" (v2
"/\" (v0
"\/" v1)))
proof
let v2, v1, v0;
(((v0
"\/" v1)
"/\" (v0
"\/" v2))
"/\" ((v0
"\/" v2)
"/\" (v0
"\/" (v2
"\/" v1))))
= ((v0
"\/" v2)
"/\" (((v0
"\/" v1)
"/\" (v0
"\/" v2))
"/\" (v0
"\/" (v2
"\/" v1)))) by
A40;
hence thesis by
A1077;
end;
A1082: for v0, v2, v1 holds ((v0
"\/" v1)
"/\" ((v0
"\/" v2)
"/\" ((v0
"\/" v1)
"/\" (v0
"\/" (v1
"\/" v2)))))
= (v0
"\/" (v1
"/\" (v0
"\/" v2)))
proof
let v0, v2, v1;
(((v0
"\/" v2)
"/\" (v0
"\/" v1))
"/\" (v0
"\/" (v1
"\/" v2)))
= ((v0
"\/" v2)
"/\" ((v0
"\/" v1)
"/\" (v0
"\/" (v1
"\/" v2)))) by
A3;
hence thesis by
A1079;
end;
A1084: for v0, v2, v1 holds ((v0
"\/" v2)
"/\" ((v0
"\/" v1)
"/\" (v0
"\/" (v1
"\/" v2))))
= (v0
"\/" (v1
"/\" (v0
"\/" v2)))
proof
let v0, v2, v1;
((v0
"\/" v1)
"/\" ((v0
"\/" v2)
"/\" ((v0
"\/" v1)
"/\" (v0
"\/" (v1
"\/" v2)))))
= ((v0
"\/" v2)
"/\" ((v0
"\/" v1)
"/\" (v0
"\/" (v1
"\/" v2)))) by
A162;
hence thesis by
A1082;
end;
A1089: for v2, v1, v0 holds (v0
"/\" (v1
"/\" (v1
"\/" (v2
"/\" (v0
"/\" v1)))))
= (v0
"/\" (v1
"/\" (v1
"\/" v2)))
proof
let v2, v1, v0;
(v1
"/\" (v1
"\/" ((v0
"/\" v1)
"\/" v2)))
= (v1
"/\" (v1
"\/" (v2
"/\" (v0
"/\" v1)))) by
A1027;
hence thesis by
A414;
end;
A1091: for v1, v0, v2 holds (v0
"/\" (v1
"/\" (v1
"/\" (v1
"\/" (v2
"/\" v0)))))
= (v0
"/\" (v1
"/\" (v1
"\/" v2)))
proof
let v1, v0, v2;
(v1
"\/" (v2
"/\" (v0
"/\" v1)))
= (v1
"/\" (v1
"\/" (v2
"/\" v0))) by
A441;
hence thesis by
A1089;
end;
A1093: for v1, v0, v2 holds (v0
"/\" (v1
"/\" (v1
"\/" (v2
"/\" v0))))
= (v0
"/\" (v1
"/\" (v1
"\/" v2)))
proof
let v1, v0, v2;
(v1
"/\" (v1
"/\" (v1
"\/" (v2
"/\" v0))))
= (v1
"/\" (v1
"\/" (v2
"/\" v0))) by
A26;
hence thesis by
A1091;
end;
A1095: for v2, v1, v0 holds ((v0
"/\" v1)
"\/" (v2
"/\" (v0
"/\" (v0
"\/" v1))))
= (v0
"/\" ((v0
"/\" v1)
"\/" v2))
proof
let v2, v1, v0;
(v2
"/\" (v0
"/\" (v0
"\/" (v1
"/\" v2))))
= (v2
"/\" (v0
"/\" (v0
"\/" v1))) by
A1093;
hence thesis by
A919;
end;
A1097: for v2, v1, v0 holds (v0
"\/" (v0
"/\" ((v0
"/\" v1)
"\/" v2)))
= (v0
"/\" ((v0
"\/" v1)
"/\" (v0
"\/" v2)))
proof
let v2, v1, v0;
((v0
"/\" v1)
"\/" (v2
"/\" (v0
"/\" (v0
"\/" v1))))
= (v0
"/\" ((v0
"/\" v1)
"\/" v2)) by
A1095;
hence thesis by
A921;
end;
A1099: for v2, v1, v0 holds (v0
"/\" (v0
"\/" ((v0
"/\" v1)
"\/" v2)))
= (v0
"/\" ((v0
"\/" v1)
"/\" (v0
"\/" v2)))
proof
let v2, v1, v0;
(v0
"\/" (v0
"/\" ((v0
"/\" v1)
"\/" v2)))
= (v0
"/\" (v0
"\/" ((v0
"/\" v1)
"\/" v2))) by
A347;
hence thesis by
A1097;
end;
A1101: for v2, v1, v0 holds (v0
"/\" (v0
"\/" (v2
"/\" (v0
"/\" v1))))
= (v0
"/\" ((v0
"\/" v1)
"/\" (v0
"\/" v2)))
proof
let v2, v1, v0;
(v0
"/\" (v0
"\/" ((v0
"/\" v1)
"\/" v2)))
= (v0
"/\" (v0
"\/" (v2
"/\" (v0
"/\" v1)))) by
A1027;
hence thesis by
A1099;
end;
A1104: for v0, v2, v1 holds (v0
"/\" (v0
"/\" (v0
"\/" (v1
"/\" v2))))
= (v0
"/\" ((v0
"\/" v2)
"/\" (v0
"\/" v1)))
proof
let v0, v2, v1;
(v0
"\/" (v1
"/\" (v0
"/\" v2)))
= (v0
"/\" (v0
"\/" (v1
"/\" v2))) by
A417;
hence thesis by
A1101;
end;
A1106: for v0, v2, v1 holds (v0
"/\" (v0
"\/" (v1
"/\" v2)))
= (v0
"/\" ((v0
"\/" v2)
"/\" (v0
"\/" v1)))
proof
let v0, v2, v1;
(v0
"/\" (v0
"/\" (v0
"\/" (v1
"/\" v2))))
= (v0
"/\" (v0
"\/" (v1
"/\" v2))) by
A26;
hence thesis by
A1104;
end;
A1108: for v0, v2, v1 holds (v0
"/\" (v0
"\/" (v1
"\/" v2)))
= (v0
"/\" ((v0
"\/" v1)
"/\" (v0
"\/" v2)))
proof
let v0, v2, v1;
(v0
"/\" (v0
"\/" (v2
"/\" v1)))
= (v0
"/\" ((v0
"\/" v1)
"/\" (v0
"\/" v2))) by
A1106;
hence thesis by
A1027;
end;
A1111: for v100, v0, v1 holds ((v100
"\/" (v1
"\/" v0))
"/\" ((v1
"\/" v0)
"/\" v100))
= ((v0
"\/" v1)
"/\" ((v1
"\/" v0)
"/\" v100))
proof
let v100, v0, v1;
((v0
"\/" v1)
"/\" ((v1
"\/" v0)
"/\" v100))
= ((v1
"\/" v0)
"/\" v100) by
A392;
hence thesis by
A717;
end;
A1114: for v0, v2, v1 holds (((v1
"\/" v2)
"/\" v0)
"/\" (v0
"\/" (v1
"\/" v2)))
= ((v2
"\/" v1)
"/\" ((v1
"\/" v2)
"/\" v0))
proof
let v0, v2, v1;
((v0
"\/" (v1
"\/" v2))
"/\" ((v1
"\/" v2)
"/\" v0))
= (((v1
"\/" v2)
"/\" v0)
"/\" (v0
"\/" (v1
"\/" v2))) by
A4;
hence thesis by
A1111;
end;
A1117: for v2, v1, v0 holds ((v0
"\/" v1)
"/\" (v2
"/\" (v2
"\/" (v0
"\/" v1))))
= ((v1
"\/" v0)
"/\" ((v0
"\/" v1)
"/\" v2))
proof
let v2, v1, v0;
(((v0
"\/" v1)
"/\" v2)
"/\" (v2
"\/" (v0
"\/" v1)))
= ((v0
"\/" v1)
"/\" (v2
"/\" (v2
"\/" (v0
"\/" v1)))) by
A3;
hence thesis by
A1114;
end;
A1119: for v1, v0, v2 holds ((v0
"\/" v1)
"/\" (v2
"/\" ((v2
"\/" v0)
"/\" (v2
"\/" v1))))
= ((v1
"\/" v0)
"/\" ((v0
"\/" v1)
"/\" v2))
proof
let v1, v0, v2;
(v2
"/\" (v2
"\/" (v0
"\/" v1)))
= (v2
"/\" ((v2
"\/" v0)
"/\" (v2
"\/" v1))) by
A1108;
hence thesis by
A1117;
end;
A1121: for v1, v0, v2 holds ((v0
"\/" v1)
"/\" (v2
"/\" ((v2
"\/" v0)
"/\" (v2
"\/" v1))))
= ((v0
"\/" v1)
"/\" v2)
proof
let v1, v0, v2;
((v1
"\/" v0)
"/\" ((v0
"\/" v1)
"/\" v2))
= ((v0
"\/" v1)
"/\" v2) by
A392;
hence thesis by
A1119;
end;
A1123: for v2, v0, v1 holds ((v0
"\/" v1)
"/\" (v1
"\/" ((v1
"\/" v0)
"/\" v2)))
= (v1
"\/" ((v0
"\/" v1)
"/\" v2))
proof
let v2, v0, v1;
(v0
"\/" v1)
= (v1
"\/" v0) by
A8;
hence thesis by
A208;
end;
A1125: for v2, v0, v1 holds ((v0
"\/" v1)
"/\" ((v1
"\/" v0)
"/\" (v1
"\/" v2)))
= (v1
"\/" ((v0
"\/" v1)
"/\" v2))
proof
let v2, v0, v1;
(v1
"\/" ((v1
"\/" v0)
"/\" v2))
= ((v1
"\/" v0)
"/\" (v1
"\/" v2)) by
A913;
hence thesis by
A1123;
end;
A1127: for v2, v0, v1 holds ((v1
"\/" v0)
"/\" (v1
"\/" v2))
= (v1
"\/" ((v0
"\/" v1)
"/\" v2))
proof
let v2, v0, v1;
((v0
"\/" v1)
"/\" ((v1
"\/" v0)
"/\" (v1
"\/" v2)))
= ((v1
"\/" v0)
"/\" (v1
"\/" v2)) by
A392;
hence thesis by
A1125;
end;
A1132: for v1, v101, v100 holds ((v100
"\/" v101)
"/\" (v101
"\/" (v1
"/\" (v100
"\/" v101))))
= (v101
"\/" ((v100
"\/" v101)
"/\" (v1
"/\" (v1
"\/" (v100
"\/" v101)))))
proof
let v1, v101, v100;
((v100
"\/" v101)
"/\" (v1
"/\" (v1
"\/" (v100
"\/" v101))))
= (v1
"/\" (v100
"\/" v101)) by
A308;
hence thesis by
A208;
end;
A1135: for v2, v1, v0 holds (v1
"\/" (v2
"/\" (v0
"\/" v1)))
= (v1
"\/" ((v0
"\/" v1)
"/\" (v2
"/\" (v2
"\/" (v0
"\/" v1)))))
proof
let v2, v1, v0;
((v0
"\/" v1)
"/\" (v1
"\/" (v2
"/\" (v0
"\/" v1))))
= (v1
"\/" (v2
"/\" (v0
"\/" v1))) by
A565;
hence thesis by
A1132;
end;
A1138: for v1, v0, v2 holds (v0
"\/" (v1
"/\" (v2
"\/" v0)))
= (v0
"\/" ((v2
"\/" v0)
"/\" (v1
"/\" ((v1
"\/" v2)
"/\" (v1
"\/" v0)))))
proof
let v1, v0, v2;
(v1
"/\" (v1
"\/" (v2
"\/" v0)))
= (v1
"/\" ((v1
"\/" v2)
"/\" (v1
"\/" v0))) by
A1108;
hence thesis by
A1135;
end;
A1140: for v1, v0, v2 holds (v0
"\/" (v1
"/\" (v2
"\/" v0)))
= (v0
"\/" ((v2
"\/" v0)
"/\" v1))
proof
let v1, v0, v2;
((v2
"\/" v0)
"/\" (v1
"/\" ((v1
"\/" v2)
"/\" (v1
"\/" v0))))
= ((v2
"\/" v0)
"/\" v1) by
A1121;
hence thesis by
A1138;
end;
A1142: for v1, v0, v2 holds (v0
"\/" (v1
"/\" (v2
"\/" v0)))
= ((v0
"\/" v2)
"/\" (v0
"\/" v1))
proof
let v1, v0, v2;
(v0
"\/" ((v2
"\/" v0)
"/\" v1))
= ((v0
"\/" v2)
"/\" (v0
"\/" v1)) by
A1127;
hence thesis by
A1140;
end;
A1144: for v2, v1, v0 holds (((v0
"\/" v1)
"/\" (v0
"\/" v2))
"/\" (v0
"\/" (v2
"\/" v1)))
= (v0
"\/" (v2
"/\" v1))
proof
let v2, v1, v0;
(v0
"\/" ((v1
"\/" v0)
"/\" v2))
= ((v0
"\/" v1)
"/\" (v0
"\/" v2)) by
A1127;
hence thesis by
A1035;
end;
A1146: for v0, v1, v2 holds ((v0
"\/" v1)
"/\" ((v0
"\/" v2)
"/\" (v0
"\/" (v2
"\/" v1))))
= (v0
"\/" (v2
"/\" v1))
proof
let v0, v1, v2;
(((v0
"\/" v1)
"/\" (v0
"\/" v2))
"/\" (v0
"\/" (v2
"\/" v1)))
= ((v0
"\/" v1)
"/\" ((v0
"\/" v2)
"/\" (v0
"\/" (v2
"\/" v1)))) by
A3;
hence thesis by
A1144;
end;
A1148: for v1, v2, v0 holds ((v0
"\/" v2)
"/\" (v0
"\/" v1))
= ((v0
"\/" v2)
"/\" ((v0
"\/" v1)
"/\" (v1
"\/" (v0
"\/" v2))))
proof
let v1, v2, v0;
(v0
"\/" (v1
"/\" (v2
"\/" v0)))
= ((v0
"\/" v2)
"/\" (v0
"\/" v1)) by
A1142;
hence thesis by
A1057;
end;
A1152: for v1, v2, v0 holds (v0
"\/" (v1
"/\" (v0
"\/" v2)))
= (v0
"\/" (v1
"/\" v2))
proof
let v1, v2, v0;
((v0
"\/" v2)
"/\" ((v0
"\/" v1)
"/\" (v0
"\/" (v1
"\/" v2))))
= (v0
"\/" (v1
"/\" v2)) by
A1146;
hence thesis by
A1084;
end;
A1154: for v0, v2, v1 holds (v0
"\/" (v1
"/\" v2))
= ((v0
"\/" v2)
"/\" ((v0
"\/" v1)
"/\" (v1
"\/" (v0
"\/" v2))))
proof
let v0, v2, v1;
(v0
"\/" (v1
"/\" (v0
"\/" v2)))
= (v0
"\/" (v1
"/\" v2)) by
A1152;
hence thesis by
A969;
end;
Z: for v0, v2, v1 holds (v0
"\/" (v1
"/\" v2))
= ((v0
"\/" v2)
"/\" (v0
"\/" v1))
proof
let v0, v2, v1;
((v0
"\/" v2)
"/\" ((v0
"\/" v1)
"/\" (v1
"\/" (v0
"\/" v2))))
= ((v0
"\/" v2)
"/\" (v0
"\/" v1)) by
A1148;
hence thesis by
A1154;
end;
let v1, v2, v3;
(v1
"\/" (v2
"/\" v3))
= ((v1
"\/" v3)
"/\" (v1
"\/" v2)) by
Z;
hence thesis by
A4;
end;
Cluster3: L is
meet-idempotent
meet-associative
meet-commutative
satisfying_QLT1
join-idempotent
join-associative
join-commutative
satisfying_QLT2
QLT-selfdistributive implies L is
distributive'
proof
assume L is
meet-idempotent
meet-associative
meet-commutative
satisfying_QLT1
join-idempotent
join-associative
join-commutative
satisfying_QLT2
QLT-selfdistributive;
then (for v0 holds (v0
"/\" v0)
= v0) & (for v2, v1, v0 holds ((v0
"/\" v1)
"/\" v2)
= (v0
"/\" (v1
"/\" v2))) & (for v1, v0 holds (v0
"/\" v1)
= (v1
"/\" v0)) & (for v0, v2, v1 holds ((v0
"/\" (v1
"\/" v2))
"\/" (v0
"/\" v1))
= (v0
"/\" (v1
"\/" v2))) & (for v0 holds (v0
"\/" v0)
= v0) & (for v2, v1, v0 holds ((v0
"\/" v1)
"\/" v2)
= (v0
"\/" (v1
"\/" v2))) & (for v1, v0 holds (v0
"\/" v1)
= (v1
"\/" v0)) & (for v0, v2, v1 holds ((v0
"\/" (v1
"/\" v2))
"/\" (v0
"\/" v1))
= (v0
"\/" (v1
"/\" v2))) & (for v2, v1, v0 holds ((((v0
"/\" v1)
"\/" v2)
"/\" v1)
"\/" (v2
"/\" v0))
= ((((v0
"\/" v1)
"/\" v2)
"\/" v1)
"/\" (v2
"\/" v0))) by
LATTICES:def 4,
LATTICES:def 5,
LATTICES:def 6,
LATTICES:def 7,
SHEFFER1:def 9,
ROBBINS1:def 7;
then for v1, v2, v3 holds (v1
"\/" (v2
"/\" v3))
= ((v1
"\/" v2)
"/\" (v1
"\/" v3)) by
ThQLT3;
hence thesis by
SHEFFER1:def 5;
end;
registration
cluster
meet-idempotent
meet-associative
meet-commutative
satisfying_QLT1
join-idempotent
join-associative
join-commutative
satisfying_QLT2
QLT-selfdistributive ->
distributive' for non
empty
LattStr;
coherence by
Cluster3;
end
begin
definition
let L;
::
LATQUASI:def11
attr L is
satisfying_Bowden_inequality means for x,y,z be
Element of L holds ((x
"\/" y)
"/\" z)
[= (x
"\/" (y
"/\" z));
end
definition
let L be
join-commutative non
empty
LattStr;
:: original:
satisfying_Bowden_inequality
redefine
::
LATQUASI:def12
attr L is
satisfying_Bowden_inequality means
:
BowdenRedef: for v0,v2,v1 be
Element of L holds ((v0
"\/" (v1
"/\" v2))
"\/" ((v0
"\/" v1)
"/\" v2))
= (v0
"\/" (v1
"/\" v2));
compatibility
proof
thus L is
satisfying_Bowden_inequality implies for v0,v2,v1 be
Element of L holds ((v0
"\/" (v1
"/\" v2))
"\/" ((v0
"\/" v1)
"/\" v2))
= (v0
"\/" (v1
"/\" v2)) by
LATTICES:def 3;
assume
B1: for v0,v2,v1 be
Element of L holds ((v0
"\/" (v1
"/\" v2))
"\/" ((v0
"\/" v1)
"/\" v2))
= (v0
"\/" (v1
"/\" v2));
let x,y,z be
Element of L;
((x
"\/" (y
"/\" z))
"\/" ((x
"\/" y)
"/\" z))
= (x
"\/" (y
"/\" z)) by
B1;
hence thesis by
LATTICES:def 3;
end;
end
theorem ::
LATQUASI:9
ThQLT4: (for v0 holds (v0
"/\" v0)
= v0) & (for v1, v0 holds (v0
"/\" v1)
= (v1
"/\" v0)) & (for v0, v2, v1 holds ((v0
"/\" (v1
"\/" v2))
"\/" (v0
"/\" v1))
= (v0
"/\" (v1
"\/" v2))) & (for v0 holds (v0
"\/" v0)
= v0) & (for v2, v1, v0 holds ((v0
"\/" v1)
"\/" v2)
= (v0
"\/" (v1
"\/" v2))) & (for v1, v0 holds (v0
"\/" v1)
= (v1
"\/" v0)) & (for v0, v2, v1 holds ((v0
"\/" (v1
"/\" v2))
"/\" (v0
"\/" v1))
= (v0
"\/" (v1
"/\" v2))) & (for v0, v2, v1 holds ((v0
"\/" (v1
"/\" v2))
"\/" ((v0
"\/" v1)
"/\" v2))
= (v0
"\/" (v1
"/\" v2))) implies for v1, v2, v3 holds (v1
"\/" (v2
"/\" v3))
= ((v1
"\/" v2)
"/\" (v1
"\/" v3))
proof
assume
A1: for v0 holds (v0
"/\" v0)
= v0;
assume
A2: for v1, v0 holds (v0
"/\" v1)
= (v1
"/\" v0);
assume
A3: for v0, v2, v1 holds ((v0
"/\" (v1
"\/" v2))
"\/" (v0
"/\" v1))
= (v0
"/\" (v1
"\/" v2));
assume
A4: for v0 holds (v0
"\/" v0)
= v0;
assume
A5: for v2, v1, v0 holds ((v0
"\/" v1)
"\/" v2)
= (v0
"\/" (v1
"\/" v2));
assume
A6: for v1, v0 holds (v0
"\/" v1)
= (v1
"\/" v0);
assume
A7: for v0, v2, v1 holds ((v0
"\/" (v1
"/\" v2))
"/\" (v0
"\/" v1))
= (v0
"\/" (v1
"/\" v2));
A9: for v0, v2, v1 holds ((v0
"\/" v1)
"/\" (v0
"\/" (v1
"/\" v2)))
= (v0
"\/" (v1
"/\" v2))
proof
let v0, v2, v1;
((v0
"\/" (v1
"/\" v2))
"/\" (v0
"\/" v1))
= ((v0
"\/" v1)
"/\" (v0
"\/" (v1
"/\" v2))) by
A2;
hence thesis by
A7;
end;
assume
A10: for v0, v2, v1 holds ((v0
"\/" (v1
"/\" v2))
"\/" ((v0
"\/" v1)
"/\" v2))
= (v0
"\/" (v1
"/\" v2));
A12: for v2, v1, v0 holds (v0
"\/" ((v1
"/\" v2)
"\/" ((v0
"\/" v1)
"/\" v2)))
= (v0
"\/" (v1
"/\" v2))
proof
let v2, v1, v0;
((v0
"\/" (v1
"/\" v2))
"\/" ((v0
"\/" v1)
"/\" v2))
= (v0
"\/" ((v1
"/\" v2)
"\/" ((v0
"\/" v1)
"/\" v2))) by
A5;
hence thesis by
A10;
end;
A16: for v0, v2, v1 holds ((v0
"/\" v1)
"\/" (v0
"/\" (v1
"\/" v2)))
= (v0
"/\" (v1
"\/" v2))
proof
let v0, v2, v1;
((v0
"/\" (v1
"\/" v2))
"\/" (v0
"/\" v1))
= ((v0
"/\" v1)
"\/" (v0
"/\" (v1
"\/" v2))) by
A6;
hence thesis by
A3;
end;
A19: for v102, v100 holds ((v100
"\/" v102)
"\/" v102)
= (v100
"\/" v102)
proof
let v102, v100;
(v102
"\/" v102)
= v102 by
A4;
hence thesis by
A5;
end;
A22: for v1, v0 holds (v1
"\/" (v0
"\/" v1))
= (v0
"\/" v1)
proof
let v1, v0;
((v0
"\/" v1)
"\/" v1)
= (v1
"\/" (v0
"\/" v1)) by
A6;
hence thesis by
A19;
end;
A25: for v2, v0, v1 holds ((v1
"\/" v0)
"\/" v2)
= (v0
"\/" (v1
"\/" v2))
proof
let v2, v0, v1;
(v0
"\/" v1)
= (v1
"\/" v0) by
A6;
hence thesis by
A5;
end;
A28: for v0, v2, v1 holds (v0
"\/" (v1
"\/" v2))
= (v1
"\/" (v0
"\/" v2))
proof
let v0, v2, v1;
((v0
"\/" v1)
"\/" v2)
= (v0
"\/" (v1
"\/" v2)) by
A5;
hence thesis by
A25;
end;
A31: for v102, v101 holds (v101
"/\" (v101
"\/" (v101
"/\" v102)))
= (v101
"\/" (v101
"/\" v102))
proof
let v102, v101;
(v101
"\/" v101)
= v101 by
A4;
hence thesis by
A9;
end;
A34: for v2, v1, v0 holds (v0
"\/" ((v1
"/\" v2)
"\/" (v2
"/\" (v0
"\/" v1))))
= (v0
"\/" (v1
"/\" v2))
proof
let v2, v1, v0;
((v0
"\/" v1)
"/\" v2)
= (v2
"/\" (v0
"\/" v1)) by
A2;
hence thesis by
A12;
end;
A37: for v102, v101 holds (v101
"\/" (v101
"/\" (v101
"\/" v102)))
= (v101
"/\" (v101
"\/" v102))
proof
let v102, v101;
(v101
"/\" v101)
= v101 by
A1;
hence thesis by
A16;
end;
A40: for v0, v2, v1 holds ((v1
"/\" v0)
"\/" (v0
"/\" (v1
"\/" v2)))
= (v0
"/\" (v1
"\/" v2))
proof
let v0, v2, v1;
(v0
"/\" v1)
= (v1
"/\" v0) by
A2;
hence thesis by
A16;
end;
A44: for v101, v2, v1 holds (((v101
"\/" v1)
"/\" v101)
"\/" (v101
"\/" (v1
"/\" v2)))
= ((v101
"\/" v1)
"/\" (v101
"\/" (v1
"/\" v2)))
proof
let v101, v2, v1;
((v101
"\/" v1)
"/\" (v101
"\/" (v1
"/\" v2)))
= (v101
"\/" (v1
"/\" v2)) by
A9;
hence thesis by
A16;
end;
A47: for v0, v2, v1 holds ((v0
"/\" (v0
"\/" v1))
"\/" (v0
"\/" (v1
"/\" v2)))
= ((v0
"\/" v1)
"/\" (v0
"\/" (v1
"/\" v2)))
proof
let v0, v2, v1;
((v0
"\/" v1)
"/\" v0)
= (v0
"/\" (v0
"\/" v1)) by
A2;
hence thesis by
A44;
end;
A49: for v2, v1, v0 holds (v0
"\/" ((v0
"/\" (v0
"\/" v1))
"\/" (v1
"/\" v2)))
= ((v0
"\/" v1)
"/\" (v0
"\/" (v1
"/\" v2)))
proof
let v2, v1, v0;
((v0
"/\" (v0
"\/" v1))
"\/" (v0
"\/" (v1
"/\" v2)))
= (v0
"\/" ((v0
"/\" (v0
"\/" v1))
"\/" (v1
"/\" v2))) by
A28;
hence thesis by
A47;
end;
A51: for v2, v1, v0 holds (v0
"\/" ((v0
"/\" (v0
"\/" v1))
"\/" (v1
"/\" v2)))
= (v0
"\/" (v1
"/\" v2))
proof
let v2, v1, v0;
((v0
"\/" v1)
"/\" (v0
"\/" (v1
"/\" v2)))
= (v0
"\/" (v1
"/\" v2)) by
A9;
hence thesis by
A49;
end;
A53: for v0, v1 holds (v0
"/\" (v0
"\/" (v1
"/\" v0)))
= (v0
"\/" (v0
"/\" v1))
proof
let v0, v1;
(v0
"/\" v1)
= (v1
"/\" v0) by
A2;
hence thesis by
A31;
end;
A56: for v102, v1, v100 holds ((v100
"/\" (v100
"\/" v1))
"\/" v102)
= (v100
"\/" ((v100
"/\" (v100
"\/" v1))
"\/" v102))
proof
let v102, v1, v100;
(v100
"\/" (v100
"/\" (v100
"\/" v1)))
= (v100
"/\" (v100
"\/" v1)) by
A37;
hence thesis by
A5;
end;
A60: for v2, v1, v0 holds ((v0
"/\" (v0
"\/" v1))
"\/" (v1
"/\" v2))
= (v0
"\/" (v1
"/\" v2))
proof
let v2, v1, v0;
(v0
"\/" ((v0
"/\" (v0
"\/" v1))
"\/" (v1
"/\" v2)))
= ((v0
"/\" (v0
"\/" v1))
"\/" (v1
"/\" v2)) by
A56;
hence thesis by
A51;
end;
A63: for v101, v100, v1 holds ((v100
"/\" v101)
"\/" (v101
"/\" (v1
"\/" v100)))
= (v101
"/\" (v100
"\/" (v1
"\/" v100)))
proof
let v101, v100, v1;
(v100
"\/" (v1
"\/" v100))
= (v1
"\/" v100) by
A22;
hence thesis by
A40;
end;
A66: for v1, v0, v2 holds ((v0
"/\" v1)
"\/" (v1
"/\" (v2
"\/" v0)))
= (v1
"/\" (v2
"\/" v0))
proof
let v1, v0, v2;
(v0
"\/" (v2
"\/" v0))
= (v2
"\/" v0) by
A22;
hence thesis by
A63;
end;
A68: for v2, v1, v0 holds (v0
"\/" (v2
"/\" (v0
"\/" v1)))
= (v0
"\/" (v1
"/\" v2))
proof
let v2, v1, v0;
((v1
"/\" v2)
"\/" (v2
"/\" (v0
"\/" v1)))
= (v2
"/\" (v0
"\/" v1)) by
A66;
hence thesis by
A34;
end;
A71: for v1, v2, v0 holds (v0
"\/" ((v0
"\/" v2)
"/\" v1))
= (v0
"\/" (v2
"/\" v1))
proof
let v1, v2, v0;
(v1
"/\" (v0
"\/" v2))
= ((v0
"\/" v2)
"/\" v1) by
A2;
hence thesis by
A68;
end;
A74: for v1, v2, v0 holds (v0
"\/" (v1
"/\" (v0
"\/" v2)))
= (v0
"\/" (v1
"/\" v2))
proof
let v1, v2, v0;
(v2
"/\" v1)
= (v1
"/\" v2) by
A2;
hence thesis by
A68;
end;
A77: for v100, v2 holds (v100
"/\" (v100
"\/" (v2
"/\" v100)))
= (v100
"\/" (v100
"/\" (v100
"\/" v2)))
proof
let v100, v2;
(v100
"\/" (v100
"/\" (v100
"\/" v2)))
= (v100
"\/" (v2
"/\" v100)) by
A68;
hence thesis by
A31;
end;
A80: for v1, v0 holds (v0
"\/" (v0
"/\" v1))
= (v0
"\/" (v0
"/\" (v0
"\/" v1)))
proof
let v1, v0;
(v0
"/\" (v0
"\/" (v1
"/\" v0)))
= (v0
"\/" (v0
"/\" v1)) by
A53;
hence thesis by
A77;
end;
A82: for v1, v0 holds (v0
"\/" (v0
"/\" v1))
= (v0
"/\" (v0
"\/" v1))
proof
let v1, v0;
(v0
"\/" (v0
"/\" (v0
"\/" v1)))
= (v0
"/\" (v0
"\/" v1)) by
A37;
hence thesis by
A80;
end;
A85: for v2, v1, v0 holds ((v0
"\/" (v0
"/\" v1))
"\/" (v1
"/\" v2))
= (v0
"\/" (v1
"/\" v2))
proof
let v2, v1, v0;
(v0
"/\" (v0
"\/" v1))
= (v0
"\/" (v0
"/\" v1)) by
A82;
hence thesis by
A60;
end;
A87: for v2, v1, v0 holds (v0
"\/" ((v0
"/\" v1)
"\/" (v1
"/\" v2)))
= (v0
"\/" (v1
"/\" v2))
proof
let v2, v1, v0;
((v0
"\/" (v0
"/\" v1))
"\/" (v1
"/\" v2))
= (v0
"\/" ((v0
"/\" v1)
"\/" (v1
"/\" v2))) by
A5;
hence thesis by
A85;
end;
A90: for v102, v1, v100 holds ((v100
"\/" (v100
"/\" v1))
"\/" ((v100
"\/" v1)
"/\" (v100
"\/" v102)))
= ((v100
"\/" v1)
"/\" (v100
"\/" v102))
proof
let v102, v1, v100;
(v100
"/\" (v100
"\/" v1))
= (v100
"\/" (v100
"/\" v1)) by
A82;
hence thesis by
A40;
end;
A93: for v2, v1, v0 holds (v0
"\/" ((v0
"/\" v1)
"\/" ((v0
"\/" v1)
"/\" (v0
"\/" v2))))
= ((v0
"\/" v1)
"/\" (v0
"\/" v2))
proof
let v2, v1, v0;
((v0
"\/" (v0
"/\" v1))
"\/" ((v0
"\/" v1)
"/\" (v0
"\/" v2)))
= (v0
"\/" ((v0
"/\" v1)
"\/" ((v0
"\/" v1)
"/\" (v0
"\/" v2)))) by
A5;
hence thesis by
A90;
end;
A96: for v100, v101, v2, v1 holds (v100
"\/" (v101
"\/" (v1
"/\" v2)))
= (v101
"\/" (v100
"\/" ((v101
"\/" v1)
"/\" v2)))
proof
let v100, v101, v2, v1;
(v101
"\/" ((v101
"\/" v1)
"/\" v2))
= (v101
"\/" (v1
"/\" v2)) by
A71;
hence thesis by
A28;
end;
A101: for v1, v2, v0 holds ((v0
"/\" v1)
"\/" (v0
"\/" (v1
"/\" (v0
"\/" v2))))
= ((v0
"\/" v1)
"/\" (v0
"\/" v2))
proof
let v1, v2, v0;
(v0
"\/" ((v0
"/\" v1)
"\/" ((v0
"\/" v1)
"/\" (v0
"\/" v2))))
= ((v0
"/\" v1)
"\/" (v0
"\/" (v1
"/\" (v0
"\/" v2)))) by
A96;
hence thesis by
A93;
end;
A103: for v0, v2, v1 holds ((v0
"/\" v1)
"\/" (v0
"\/" (v1
"/\" v2)))
= ((v0
"\/" v1)
"/\" (v0
"\/" v2))
proof
let v0, v2, v1;
(v0
"\/" (v1
"/\" (v0
"\/" v2)))
= (v0
"\/" (v1
"/\" v2)) by
A74;
hence thesis by
A101;
end;
A105: for v2, v1, v0 holds (v0
"\/" ((v0
"/\" v1)
"\/" (v1
"/\" v2)))
= ((v0
"\/" v1)
"/\" (v0
"\/" v2))
proof
let v2, v1, v0;
((v0
"/\" v1)
"\/" (v0
"\/" (v1
"/\" v2)))
= (v0
"\/" ((v0
"/\" v1)
"\/" (v1
"/\" v2))) by
A28;
hence thesis by
A103;
end;
for v0, v2, v1 holds (v0
"\/" (v1
"/\" v2))
= ((v0
"\/" v1)
"/\" (v0
"\/" v2))
proof
let v0, v2, v1;
(v0
"\/" ((v0
"/\" v1)
"\/" (v1
"/\" v2)))
= (v0
"\/" (v1
"/\" v2)) by
A87;
hence thesis by
A105;
end;
hence thesis;
end;
Cluster4: L is
meet-idempotent
meet-associative
meet-commutative
satisfying_QLT1
join-idempotent
join-associative
join-commutative
satisfying_QLT2
satisfying_Bowden_inequality implies L is
distributive'
proof
assume
A0: L is
meet-idempotent
meet-associative
meet-commutative
satisfying_QLT1
join-idempotent
join-associative
join-commutative
satisfying_QLT2
satisfying_Bowden_inequality;
then
A2: (for v0 holds (v0
"/\" v0)
= v0) & (for v1, v0 holds (v0
"/\" v1)
= (v1
"/\" v0)) & (for v0, v2, v1 holds ((v0
"/\" (v1
"\/" v2))
"\/" (v0
"/\" v1))
= (v0
"/\" (v1
"\/" v2))) & (for v0 holds (v0
"\/" v0)
= v0) & (for v2, v1, v0 holds ((v0
"\/" v1)
"\/" v2)
= (v0
"\/" (v1
"\/" v2))) & (for v1, v0 holds (v0
"\/" v1)
= (v1
"\/" v0)) & (for v0, v2, v1 holds ((v0
"\/" (v1
"/\" v2))
"/\" (v0
"\/" v1))
= (v0
"\/" (v1
"/\" v2))) by
LATTICES:def 4,
LATTICES:def 5,
LATTICES:def 6,
SHEFFER1:def 9,
ROBBINS1:def 7;
(for v0, v2, v1 holds ((v0
"\/" (v1
"/\" v2))
"\/" ((v0
"\/" v1)
"/\" v2))
= (v0
"\/" (v1
"/\" v2))) by
BowdenRedef,
A0;
then for v1, v2, v3 holds (v1
"\/" (v2
"/\" v3))
= ((v1
"\/" v2)
"/\" (v1
"\/" v3)) by
ThQLT4,
A2;
hence thesis by
SHEFFER1:def 5;
end;
registration
cluster
meet-idempotent
meet-associative
meet-commutative
satisfying_QLT1
join-idempotent
join-associative
join-commutative
satisfying_QLT2
satisfying_Bowden_inequality ->
distributive' for non
empty
LattStr;
coherence by
Cluster4;
end
begin
definition
let L;
::
LATQUASI:def13
attr L is
QLT-selfmodular means for v2, v1, v0 holds ((v0
"/\" v1)
"\/" (v2
"/\" (v0
"\/" v1)))
= ((v0
"\/" v1)
"/\" (v2
"\/" (v0
"/\" v1)));
end
definition
let L be
join-idempotent non
empty
LattStr;
let a,b be
Element of L;
:: original:
[=
redefine
pred a
[= b;
reflexivity
proof
let a be
Element of L;
(a
"\/" a)
= a by
ROBBINS1:def 7;
hence thesis by
LATTICES:def 3;
end;
end
theorem ::
LATQUASI:10
QLTMod: (for v0 holds (v0
"/\" v0)
= v0) & (for v2, v1, v0 holds ((v0
"/\" v1)
"/\" v2)
= (v0
"/\" (v1
"/\" v2))) & (for v1, v0 holds (v0
"/\" v1)
= (v1
"/\" v0)) & (for v0, v2, v1 holds ((v0
"/\" (v1
"\/" v2))
"\/" (v0
"/\" v1))
= (v0
"/\" (v1
"\/" v2))) & (for v0 holds (v0
"\/" v0)
= v0) & (for v2, v1, v0 holds ((v0
"\/" v1)
"\/" v2)
= (v0
"\/" (v1
"\/" v2))) & (for v1, v0 holds (v0
"\/" v1)
= (v1
"\/" v0)) & (for v0, v2, v1 holds ((v0
"\/" (v1
"/\" v2))
"/\" (v0
"\/" v1))
= (v0
"\/" (v1
"/\" v2))) & (for v0, v1, v2 st (v0
"\/" v1)
= v1 holds (v0
"\/" (v2
"/\" v1))
= ((v0
"\/" v2)
"/\" v1)) implies for v1, v2, v3 holds ((v1
"/\" v2)
"\/" (v1
"/\" v3))
= (v1
"/\" (v2
"\/" (v1
"/\" v3)))
proof
assume
A2: for v0 holds (v0
"/\" v0)
= v0;
assume
A3: for v2, v1, v0 holds ((v0
"/\" v1)
"/\" v2)
= (v0
"/\" (v1
"/\" v2));
assume
A4: for v1, v0 holds (v0
"/\" v1)
= (v1
"/\" v0);
assume
A5: for v0, v2, v1 holds ((v0
"/\" (v1
"\/" v2))
"\/" (v0
"/\" v1))
= (v0
"/\" (v1
"\/" v2));
assume
A6: for v0 holds (v0
"\/" v0)
= v0;
assume
A7: for v2, v1, v0 holds ((v0
"\/" v1)
"\/" v2)
= (v0
"\/" (v1
"\/" v2));
assume
A8: for v1, v0 holds (v0
"\/" v1)
= (v1
"\/" v0);
assume
A9: for v0, v2, v1 holds ((v0
"\/" (v1
"/\" v2))
"/\" (v0
"\/" v1))
= (v0
"\/" (v1
"/\" v2));
A11: for v0, v2, v1 holds ((v0
"\/" v1)
"/\" (v0
"\/" (v1
"/\" v2)))
= (v0
"\/" (v1
"/\" v2))
proof
let v0, v2, v1;
((v0
"\/" (v1
"/\" v2))
"/\" (v0
"\/" v1))
= ((v0
"\/" v1)
"/\" (v0
"\/" (v1
"/\" v2))) by
A4;
hence thesis by
A9;
end;
assume
A12: for v0, v1, v2 st (v0
"\/" v1)
= v1 holds (v0
"\/" (v2
"/\" v1))
= ((v0
"\/" v2)
"/\" v1);
assume not thesis;
then
consider c1,c2,c3 be
Element of L such that
A14: ((c1
"/\" c2)
"\/" (c1
"/\" c3))
<> (c1
"/\" (c2
"\/" (c1
"/\" c3)));
A17: for v0, v2, v1 holds ((v0
"/\" v1)
"\/" (v0
"/\" (v1
"\/" v2)))
= (v0
"/\" (v1
"\/" v2))
proof
let v0, v2, v1;
((v0
"/\" (v1
"\/" v2))
"\/" (v0
"/\" v1))
= ((v0
"/\" v1)
"\/" (v0
"/\" (v1
"\/" v2))) by
A8;
hence thesis by
A5;
end;
A20: for v102, v101 holds (v101
"/\" v102)
= (v101
"/\" (v101
"/\" v102))
proof
let v102, v101;
(v101
"/\" v101)
= v101 by
A2;
hence thesis by
A3;
end;
A24: for v2, v0, v1 holds ((v1
"/\" v0)
"/\" v2)
= (v0
"/\" (v1
"/\" v2))
proof
let v2, v0, v1;
(v0
"/\" v1)
= (v1
"/\" v0) by
A4;
hence thesis by
A3;
end;
A27: for v0, v2, v1 holds (v0
"/\" (v1
"/\" v2))
= (v1
"/\" (v0
"/\" v2))
proof
let v0, v2, v1;
((v0
"/\" v1)
"/\" v2)
= (v0
"/\" (v1
"/\" v2)) by
A3;
hence thesis by
A24;
end;
A30: for v102, v101 holds (v101
"\/" v102)
= (v101
"\/" (v101
"\/" v102))
proof
let v102, v101;
(v101
"\/" v101)
= v101 by
A6;
hence thesis by
A7;
end;
A35: for v102, v100 holds ((v100
"\/" v102)
"\/" v102)
= (v100
"\/" v102)
proof
let v102, v100;
(v102
"\/" v102)
= v102 by
A6;
hence thesis by
A7;
end;
A38: for v1, v0 holds (v1
"\/" (v0
"\/" v1))
= (v0
"\/" v1)
proof
let v1, v0;
((v0
"\/" v1)
"\/" v1)
= (v1
"\/" (v0
"\/" v1)) by
A8;
hence thesis by
A35;
end;
A41: for v0, v2, v1 holds ((v1
"\/" v0)
"/\" (v0
"\/" (v1
"/\" v2)))
= (v0
"\/" (v1
"/\" v2))
proof
let v0, v2, v1;
(v0
"\/" v1)
= (v1
"\/" v0) by
A8;
hence thesis by
A11;
end;
A45: for v2, v100 holds ((v100
"\/" v2)
"/\" v100)
= (v100
"\/" (v2
"/\" v100))
proof
let v2, v100;
(v100
"\/" v100)
= v100 implies ((v100
"\/" v2)
"/\" v100)
= (v100
"\/" (v2
"/\" v100)) by
A12;
hence thesis by
A6;
end;
A48: for v1, v0 holds (v0
"/\" (v0
"\/" v1))
= (v0
"\/" (v1
"/\" v0))
proof
let v1, v0;
((v0
"\/" v1)
"/\" v0)
= (v0
"/\" (v0
"\/" v1)) by
A4;
hence thesis by
A45;
end;
A52: for v102, v1, v100 holds ((v100
"/\" v1)
"\/" (v100
"/\" ((v100
"/\" v1)
"\/" v102)))
= (v100
"/\" ((v100
"/\" v1)
"\/" v102))
proof
let v102, v1, v100;
(v100
"/\" (v100
"/\" v1))
= (v100
"/\" v1) by
A20;
hence thesis by
A17;
end;
A56: for v101, v2, v100 holds ((v100
"\/" v2)
"/\" (v100
"\/" v101))
= (v100
"\/" (v2
"/\" (v100
"\/" v101)))
proof
let v101, v2, v100;
(v100
"\/" (v100
"\/" v101))
= (v100
"\/" v101) implies ((v100
"\/" v2)
"/\" (v100
"\/" v101))
= (v100
"\/" (v2
"/\" (v100
"\/" v101))) by
A12;
hence thesis by
A30;
end;
A60: for v101, v2, v100 holds ((v100
"\/" v2)
"/\" (v101
"\/" v100))
= (v100
"\/" (v2
"/\" (v101
"\/" v100)))
proof
let v101, v2, v100;
(v100
"\/" (v101
"\/" v100))
= (v101
"\/" v100) implies ((v100
"\/" v2)
"/\" (v101
"\/" v100))
= (v100
"\/" (v2
"/\" (v101
"\/" v100))) by
A12;
hence thesis by
A38;
end;
A63: for v1, v0 holds (v0
"\/" (v0
"/\" v1))
= (v0
"/\" (v0
"\/" v1))
proof
let v1, v0;
(v1
"/\" v0)
= (v0
"/\" v1) by
A4;
hence thesis by
A48;
end;
A67: for v100, v1, v101 holds (v100
"/\" (v101
"\/" (v101
"/\" v1)))
= (v101
"/\" (v100
"/\" (v101
"\/" v1)))
proof
let v100, v1, v101;
(v101
"/\" (v101
"\/" v1))
= (v101
"\/" (v101
"/\" v1)) by
A63;
hence thesis by
A27;
end;
A71: for v1, v2, v0 holds (v1
"\/" (v0
"/\" v2))
= ((v1
"\/" (v0
"/\" v2))
"/\" (v0
"\/" v1))
proof
let v1, v2, v0;
((v0
"\/" v1)
"/\" (v1
"\/" (v0
"/\" v2)))
= (v1
"\/" (v0
"/\" v2)) by
A41;
hence thesis by
A4;
end;
A74: for v0, v2, v1 holds (v0
"\/" (v1
"/\" v2))
= (v0
"\/" ((v1
"/\" v2)
"/\" (v1
"\/" v0)))
proof
let v0, v2, v1;
((v0
"\/" (v1
"/\" v2))
"/\" (v1
"\/" v0))
= (v0
"\/" ((v1
"/\" v2)
"/\" (v1
"\/" v0))) by
A60;
hence thesis by
A71;
end;
A76: for v0, v2, v1 holds (v0
"\/" (v1
"/\" v2))
= (v0
"\/" (v1
"/\" (v2
"/\" (v1
"\/" v0))))
proof
let v0, v2, v1;
((v1
"/\" v2)
"/\" (v1
"\/" v0))
= (v1
"/\" (v2
"/\" (v1
"\/" v0))) by
A3;
hence thesis by
A74;
end;
A79: for v2, v0, v1 holds ((v1
"\/" v0)
"/\" (v0
"\/" v2))
= (v0
"\/" (v1
"/\" (v0
"\/" v2)))
proof
let v2, v0, v1;
(v0
"\/" v1)
= (v1
"\/" v0) by
A8;
hence thesis by
A56;
end;
A83: for v101, v2, v102 holds (((v102
"/\" v2)
"\/" v101)
"/\" (v102
"\/" (v102
"/\" v2)))
= ((v102
"/\" v2)
"\/" (v102
"/\" (v101
"/\" (v102
"\/" v2))))
proof
let v101, v2, v102;
(v101
"/\" (v102
"\/" (v102
"/\" v2)))
= (v102
"/\" (v101
"/\" (v102
"\/" v2))) by
A67;
hence thesis by
A60;
end;
A86: for v2, v1, v0 holds ((v0
"\/" (v0
"/\" v1))
"/\" ((v0
"/\" v1)
"\/" v2))
= ((v0
"/\" v1)
"\/" (v0
"/\" (v2
"/\" (v0
"\/" v1))))
proof
let v2, v1, v0;
(((v0
"/\" v1)
"\/" v2)
"/\" (v0
"\/" (v0
"/\" v1)))
= ((v0
"\/" (v0
"/\" v1))
"/\" ((v0
"/\" v1)
"\/" v2)) by
A4;
hence thesis by
A83;
end;
A88: for v2, v1, v0 holds ((v0
"/\" v1)
"\/" (v0
"/\" ((v0
"/\" v1)
"\/" v2)))
= ((v0
"/\" v1)
"\/" (v0
"/\" (v2
"/\" (v0
"\/" v1))))
proof
let v2, v1, v0;
((v0
"\/" (v0
"/\" v1))
"/\" ((v0
"/\" v1)
"\/" v2))
= ((v0
"/\" v1)
"\/" (v0
"/\" ((v0
"/\" v1)
"\/" v2))) by
A79;
hence thesis by
A86;
end;
A90: for v2, v1, v0 holds (v0
"/\" ((v0
"/\" v1)
"\/" v2))
= ((v0
"/\" v1)
"\/" (v0
"/\" (v2
"/\" (v0
"\/" v1))))
proof
let v2, v1, v0;
((v0
"/\" v1)
"\/" (v0
"/\" ((v0
"/\" v1)
"\/" v2)))
= (v0
"/\" ((v0
"/\" v1)
"\/" v2)) by
A52;
hence thesis by
A88;
end;
A94: for v102, v2, v101 holds ((v101
"/\" v2)
"\/" (v101
"/\" (v101
"/\" (v102
"/\" (v101
"\/" v2)))))
= ((v101
"/\" v2)
"\/" (v101
"/\" v102))
proof
let v102, v2, v101;
(v102
"/\" (v101
"\/" (v101
"/\" v2)))
= (v101
"/\" (v102
"/\" (v101
"\/" v2))) by
A67;
hence thesis by
A76;
end;
A97: for v2, v1, v0 holds ((v0
"/\" v1)
"\/" (v0
"/\" (v2
"/\" (v0
"\/" v1))))
= ((v0
"/\" v1)
"\/" (v0
"/\" v2))
proof
let v2, v1, v0;
(v0
"/\" (v0
"/\" (v2
"/\" (v0
"\/" v1))))
= (v0
"/\" (v2
"/\" (v0
"\/" v1))) by
A20;
hence thesis by
A94;
end;
A99: for v2, v1, v0 holds ((v0
"/\" v1)
"\/" (v0
"/\" v2))
= (v0
"/\" ((v0
"/\" v1)
"\/" v2))
proof
let v2, v1, v0;
((v0
"/\" v1)
"\/" (v0
"/\" (v2
"/\" (v0
"\/" v1))))
= ((v0
"/\" v1)
"\/" (v0
"/\" v2)) by
A97;
hence thesis by
A90;
end;
A102: for v2, v1, v0 holds (v0
"/\" (v2
"\/" (v0
"/\" v1)))
= ((v0
"/\" v1)
"\/" (v0
"/\" v2))
proof
let v2, v1, v0;
((v0
"/\" v1)
"\/" v2)
= (v2
"\/" (v0
"/\" v1)) by
A8;
hence thesis by
A99;
end;
((c1
"/\" c3)
"\/" (c1
"/\" c2))
<> ((c1
"/\" c2)
"\/" (c1
"/\" c3)) by
A102,
A14;
hence thesis by
A8;
end;
ClusterA: L is
meet-idempotent
join-idempotent
meet-commutative
join-commutative
meet-associative
join-associative
satisfying_QLT1
satisfying_QLT2
modular implies for v1, v2, v3 holds ((v1
"/\" v2)
"\/" (v1
"/\" v3))
= (v1
"/\" (v2
"\/" (v1
"/\" v3)))
proof
assume
A0: L is
meet-idempotent
join-idempotent
meet-commutative
join-commutative
meet-associative
join-associative
satisfying_QLT1
satisfying_QLT2
modular;
then
A1: (for v0 holds (v0
"/\" v0)
= v0) & (for v2, v1, v0 holds ((v0
"/\" v1)
"/\" v2)
= (v0
"/\" (v1
"/\" v2))) & (for v1, v0 holds (v0
"/\" v1)
= (v1
"/\" v0)) & (for v0, v2, v1 holds ((v0
"/\" (v1
"\/" v2))
"\/" (v0
"/\" v1))
= (v0
"/\" (v1
"\/" v2))) & (for v0 holds (v0
"\/" v0)
= v0) & (for v2, v1, v0 holds ((v0
"\/" v1)
"\/" v2)
= (v0
"\/" (v1
"\/" v2))) & (for v1, v0 holds (v0
"\/" v1)
= (v1
"\/" v0)) & (for v0, v2, v1 holds ((v0
"\/" (v1
"/\" v2))
"/\" (v0
"\/" v1))
= (v0
"\/" (v1
"/\" v2))) by
LATTICES:def 4,
LATTICES:def 5,
LATTICES:def 6,
LATTICES:def 7,
SHEFFER1:def 9,
ROBBINS1:def 7;
for v0, v1, v2 st (v0
"\/" v1)
= v1 holds (v0
"\/" (v2
"/\" v1))
= ((v0
"\/" v2)
"/\" v1) by
A0,
LATTICES:def 3,
LATTICES:def 12;
hence thesis by
QLTMod,
A1;
end;
theorem ::
LATQUASI:11
QLTMod2: (for v1, v0 holds (v0
"/\" v1)
= (v1
"/\" v0)) & (for v0, v2, v1 holds ((v0
"/\" (v1
"\/" v2))
"\/" (v0
"/\" v1))
= (v0
"/\" (v1
"\/" v2))) & (for v0 holds (v0
"\/" v0)
= v0) & (for v2, v1, v0 holds ((v0
"\/" v1)
"\/" v2)
= (v0
"\/" (v1
"\/" v2))) & (for v1, v0 holds (v0
"\/" v1)
= (v1
"\/" v0)) & (for v0, v2, v1 holds ((v0
"\/" (v1
"/\" v2))
"/\" (v0
"\/" v1))
= (v0
"\/" (v1
"/\" v2))) & (for v2, v1, v0 holds ((v0
"/\" v1)
"\/" (v0
"/\" v2))
= (v0
"/\" (v1
"\/" (v0
"/\" v2)))) implies for v1, v2, v3 st (v1
"\/" v2)
= v2 holds (v1
"\/" (v3
"/\" v2))
= ((v1
"\/" v3)
"/\" v2)
proof
assume
A1: for v1, v0 holds (v0
"/\" v1)
= (v1
"/\" v0);
assume
A2: for v0, v2, v1 holds ((v0
"/\" (v1
"\/" v2))
"\/" (v0
"/\" v1))
= (v0
"/\" (v1
"\/" v2));
assume
A3: for v0 holds (v0
"\/" v0)
= v0;
assume
A4: for v2, v1, v0 holds ((v0
"\/" v1)
"\/" v2)
= (v0
"\/" (v1
"\/" v2));
assume
A5: for v1, v0 holds (v0
"\/" v1)
= (v1
"\/" v0);
assume
A6: for v0, v2, v1 holds ((v0
"\/" (v1
"/\" v2))
"/\" (v0
"\/" v1))
= (v0
"\/" (v1
"/\" v2));
A8: for v0, v2, v1 holds ((v0
"\/" v1)
"/\" (v0
"\/" (v1
"/\" v2)))
= (v0
"\/" (v1
"/\" v2))
proof
let v0, v2, v1;
((v0
"\/" (v1
"/\" v2))
"/\" (v0
"\/" v1))
= ((v0
"\/" v1)
"/\" (v0
"\/" (v1
"/\" v2))) by
A1;
hence thesis by
A6;
end;
assume
A9: for v2, v1, v0 holds ((v0
"/\" v1)
"\/" (v0
"/\" v2))
= (v0
"/\" (v1
"\/" (v0
"/\" v2)));
assume not thesis;
then
consider c1,c2,c3 be
Element of L such that
A10: (c1
"\/" c2)
= c2 and
A11: (c1
"\/" (c3
"/\" c2))
<> ((c1
"\/" c3)
"/\" c2);
(c1
"\/" (c2
"/\" c3))
<> ((c1
"\/" c3)
"/\" c2) by
A1,
A11;
then
A15: (c1
"\/" (c2
"/\" c3))
<> (c2
"/\" (c1
"\/" c3)) by
A1;
A17: for v0, v2, v1 holds ((v0
"/\" v1)
"\/" (v0
"/\" (v1
"\/" v2)))
= (v0
"/\" (v1
"\/" v2))
proof
let v0, v2, v1;
((v0
"/\" (v1
"\/" v2))
"\/" (v0
"/\" v1))
= ((v0
"/\" v1)
"\/" (v0
"/\" (v1
"\/" v2))) by
A5;
hence thesis by
A2;
end;
A19: for v0, v2, v1 holds (v0
"/\" (v1
"\/" (v0
"/\" (v1
"\/" v2))))
= (v0
"/\" (v1
"\/" v2))
proof
let v0, v2, v1;
((v0
"/\" v1)
"\/" (v0
"/\" (v1
"\/" v2)))
= (v0
"/\" (v1
"\/" (v0
"/\" (v1
"\/" v2)))) by
A9;
hence thesis by
A17;
end;
A22: for v102, v101 holds (v101
"\/" v102)
= (v101
"\/" (v101
"\/" v102))
proof
let v102, v101;
(v101
"\/" v101)
= v101 by
A3;
hence thesis by
A4;
end;
A27: for v1, v2, v0 holds (v0
"/\" (v1
"\/" (v0
"/\" v2)))
= ((v0
"/\" v2)
"\/" (v0
"/\" v1))
proof
let v1, v2, v0;
((v0
"/\" v1)
"\/" (v0
"/\" v2))
= (v0
"/\" (v1
"\/" (v0
"/\" v2))) by
A9;
hence thesis by
A5;
end;
A29: for v1, v2, v0 holds (v0
"/\" (v1
"\/" (v0
"/\" v2)))
= (v0
"/\" (v2
"\/" (v0
"/\" v1)))
proof
let v1, v2, v0;
((v0
"/\" v2)
"\/" (v0
"/\" v1))
= (v0
"/\" (v2
"\/" (v0
"/\" v1))) by
A9;
hence thesis by
A27;
end;
A35: for v102 holds (c1
"\/" (c2
"/\" (c1
"\/" v102)))
= (c2
"/\" (c1
"\/" v102))
proof
let v102;
(c2
"/\" (c1
"\/" (c2
"/\" (c1
"\/" v102))))
= (c1
"\/" (c2
"/\" (c1
"\/" v102))) by
A10,
A8;
hence thesis by
A19;
end;
A38: for v0 holds (c1
"\/" (c2
"/\" (v0
"\/" c1)))
= (c2
"/\" (c1
"\/" v0))
proof
let v0;
(c1
"\/" v0)
= (v0
"\/" c1) by
A5;
hence thesis by
A35;
end;
A40: for v0, v2, v1 holds (v0
"/\" ((v1
"\/" v2)
"\/" (v0
"/\" v1)))
= (v0
"/\" (v1
"\/" v2))
proof
let v0, v2, v1;
(v0
"/\" (v1
"\/" (v0
"/\" (v1
"\/" v2))))
= (v0
"/\" ((v1
"\/" v2)
"\/" (v0
"/\" v1))) by
A29;
hence thesis by
A19;
end;
A42: for v2, v1, v0 holds (v0
"/\" (v1
"\/" (v2
"\/" (v0
"/\" v1))))
= (v0
"/\" (v1
"\/" v2))
proof
let v2, v1, v0;
((v1
"\/" v2)
"\/" (v0
"/\" v1))
= (v1
"\/" (v2
"\/" (v0
"/\" v1))) by
A4;
hence thesis by
A40;
end;
A44: for v0 holds (c2
"/\" (v0
"\/" (c2
"/\" c1)))
= (c1
"\/" (c2
"/\" v0))
proof
let v0;
(c2
"/\" (c1
"\/" (c2
"/\" v0)))
= (c2
"/\" (v0
"\/" (c2
"/\" c1))) by
A29;
hence thesis by
A10,
A8;
end;
A45: (c2
"/\" c1)
= (c1
"/\" c2) by
A1;
A50: for v101 holds (c2
"/\" (v101
"\/" (c1
"\/" (c2
"/\" v101))))
= (c2
"/\" (v101
"\/" (c1
"/\" c2)))
proof
let v101;
(c1
"\/" (c2
"/\" v101))
= (c2
"/\" (v101
"\/" (c1
"/\" c2))) by
A45,
A44;
hence thesis by
A19;
end;
A53: for v0 holds (c2
"/\" (v0
"\/" c1))
= (c2
"/\" (v0
"\/" (c1
"/\" c2)))
proof
let v0;
(c2
"/\" (v0
"\/" (c1
"\/" (c2
"/\" v0))))
= (c2
"/\" (v0
"\/" c1)) by
A42;
hence thesis by
A50;
end;
A57: for v0 holds (c1
"\/" (c2
"/\" (v0
"\/" (c1
"/\" c2))))
= (c1
"\/" (c2
"/\" v0))
proof
let v0;
(c1
"\/" (c2
"/\" v0))
= (c2
"/\" (v0
"\/" (c1
"/\" c2))) by
A45,
A44;
hence thesis by
A22;
end;
A59: for v0 holds (c1
"\/" (c2
"/\" (v0
"\/" c1)))
= (c1
"\/" (c2
"/\" v0))
proof
let v0;
(c2
"/\" (v0
"\/" (c1
"/\" c2)))
= (c2
"/\" (v0
"\/" c1)) by
A53;
hence thesis by
A57;
end;
for v0 holds (c2
"/\" (c1
"\/" v0))
= (c1
"\/" (c2
"/\" v0))
proof
let v0;
(c1
"\/" (c2
"/\" (v0
"\/" c1)))
= (c2
"/\" (c1
"\/" v0)) by
A38;
hence thesis by
A59;
end;
hence thesis by
A15;
end;
ClusterB: L is
meet-idempotent
join-idempotent
meet-commutative
join-commutative
meet-associative
join-associative
satisfying_QLT1
satisfying_QLT2 & (for v2,v1,v0 be
Element of L holds ((v0
"/\" v1)
"\/" (v0
"/\" v2))
= (v0
"/\" (v1
"\/" (v0
"/\" v2)))) implies L is
modular
proof
assume
A0: L is
meet-idempotent
join-idempotent
meet-commutative
join-commutative
meet-associative
join-associative
satisfying_QLT1
satisfying_QLT2;
A1: (for v1, v0 holds (v0
"/\" v1)
= (v1
"/\" v0)) & (for v0, v2, v1 holds ((v0
"/\" (v1
"\/" v2))
"\/" (v0
"/\" v1))
= (v0
"/\" (v1
"\/" v2))) & (for v0 holds (v0
"\/" v0)
= v0) & (for v2, v1, v0 holds ((v0
"\/" v1)
"\/" v2)
= (v0
"\/" (v1
"\/" v2))) & (for v1, v0 holds (v0
"\/" v1)
= (v1
"\/" v0)) & (for v0, v2, v1 holds ((v0
"\/" (v1
"/\" v2))
"/\" (v0
"\/" v1))
= (v0
"\/" (v1
"/\" v2))) & (for v2, v1, v0 holds ((v0
"/\" v1)
"\/" (v0
"/\" v2))
= (v0
"/\" (v1
"\/" (v0
"/\" v2)))) implies for v1, v2, v3 st (v1
"\/" v2)
= v2 holds (v1
"\/" (v3
"/\" v2))
= ((v1
"\/" v3)
"/\" v2) by
QLTMod2;
assume
AA: for v2,v1,v0 be
Element of L holds ((v0
"/\" v1)
"\/" (v0
"/\" v2))
= (v0
"/\" (v1
"\/" (v0
"/\" v2)));
for a,b,c be
Element of L st a
[= c holds (a
"\/" (b
"/\" c))
= ((a
"\/" b)
"/\" c)
proof
let a,b,c be
Element of L;
assume a
[= c;
then (a
"\/" c)
= c by
LATTICES:def 3;
hence thesis by
A1,
AA,
A0,
LATTICES:def 4,
LATTICES:def 5,
LATTICES:def 6,
ROBBINS1:def 7;
end;
hence thesis by
LATTICES:def 12;
end;
definition
let L be
meet-idempotent
join-idempotent
meet-commutative
join-commutative
meet-associative
join-associative
satisfying_QLT1
satisfying_QLT2 non
empty
LattStr;
:: original:
modular
redefine
::
LATQUASI:def14
attr L is
modular means
:
ModRedef: for v1,v2,v3 be
Element of L holds ((v1
"/\" v2)
"\/" (v1
"/\" v3))
= (v1
"/\" (v2
"\/" (v1
"/\" v3)));
compatibility
proof
thus L is
modular implies for v1,v2,v3 be
Element of L holds ((v1
"/\" v2)
"\/" (v1
"/\" v3))
= (v1
"/\" (v2
"\/" (v1
"/\" v3))) by
ClusterA;
SS: L is
meet-idempotent
join-idempotent
meet-commutative
join-commutative
meet-associative
join-associative
satisfying_QLT1
satisfying_QLT2 & (for v2,v1,v0 be
Element of L holds ((v0
"/\" v1)
"\/" (v0
"/\" v2))
= (v0
"/\" (v1
"\/" (v0
"/\" v2)))) implies L is
modular by
ClusterB;
assume for v1,v2,v3 be
Element of L holds ((v1
"/\" v2)
"\/" (v1
"/\" v3))
= (v1
"/\" (v2
"\/" (v1
"/\" v3)));
hence thesis by
SS;
end;
end
begin
theorem ::
LATQUASI:12
ThQLT5: (for v0 holds (v0
"/\" v0)
= v0) & (for v2, v1, v0 holds ((v0
"/\" v1)
"/\" v2)
= (v0
"/\" (v1
"/\" v2))) & (for v1, v0 holds (v0
"/\" v1)
= (v1
"/\" v0)) & (for v0, v2, v1 holds ((v0
"/\" (v1
"\/" v2))
"\/" (v0
"/\" v1))
= (v0
"/\" (v1
"\/" v2))) & (for v0 holds (v0
"\/" v0)
= v0) & (for v2, v1, v0 holds ((v0
"\/" v1)
"\/" v2)
= (v0
"\/" (v1
"\/" v2))) & (for v1, v0 holds (v0
"\/" v1)
= (v1
"\/" v0)) & (for v0, v2, v1 holds ((v0
"\/" (v1
"/\" v2))
"/\" (v0
"\/" v1))
= (v0
"\/" (v1
"/\" v2))) & (for v2, v1, v0 holds ((v0
"/\" v1)
"\/" (v2
"/\" (v0
"\/" v1)))
= ((v0
"\/" v1)
"/\" (v2
"\/" (v0
"/\" v1)))) implies for v1, v2, v3 holds ((v1
"/\" v2)
"\/" (v1
"/\" v3))
= (v1
"/\" (v2
"\/" (v1
"/\" v3)))
proof
assume
A1: for v0 holds (v0
"/\" v0)
= v0;
assume
A2: for v2, v1, v0 holds ((v0
"/\" v1)
"/\" v2)
= (v0
"/\" (v1
"/\" v2));
assume
A3: for v1, v0 holds (v0
"/\" v1)
= (v1
"/\" v0);
assume
A4: for v0, v2, v1 holds ((v0
"/\" (v1
"\/" v2))
"\/" (v0
"/\" v1))
= (v0
"/\" (v1
"\/" v2));
assume
A5: for v0 holds (v0
"\/" v0)
= v0;
assume
A6: for v2, v1, v0 holds ((v0
"\/" v1)
"\/" v2)
= (v0
"\/" (v1
"\/" v2));
assume
A7: for v1, v0 holds (v0
"\/" v1)
= (v1
"\/" v0);
assume
A8: for v0, v2, v1 holds ((v0
"\/" (v1
"/\" v2))
"/\" (v0
"\/" v1))
= (v0
"\/" (v1
"/\" v2));
A10: for v0, v2, v1 holds ((v0
"\/" v1)
"/\" (v0
"\/" (v1
"/\" v2)))
= (v0
"\/" (v1
"/\" v2))
proof
let v0, v2, v1;
((v0
"\/" (v1
"/\" v2))
"/\" (v0
"\/" v1))
= ((v0
"\/" v1)
"/\" (v0
"\/" (v1
"/\" v2))) by
A3;
hence thesis by
A8;
end;
assume
A11: for v2, v1, v0 holds ((v0
"/\" v1)
"\/" (v2
"/\" (v0
"\/" v1)))
= ((v0
"\/" v1)
"/\" (v2
"\/" (v0
"/\" v1)));
A14: for v0, v2, v1 holds ((v0
"/\" v1)
"\/" (v0
"/\" (v1
"\/" v2)))
= (v0
"/\" (v1
"\/" v2))
proof
let v0, v2, v1;
((v0
"/\" (v1
"\/" v2))
"\/" (v0
"/\" v1))
= ((v0
"/\" v1)
"\/" (v0
"/\" (v1
"\/" v2))) by
A7;
hence thesis by
A4;
end;
A17: for v102, v101 holds (v101
"/\" v102)
= (v101
"/\" (v101
"/\" v102))
proof
let v102, v101;
(v101
"/\" v101)
= v101 by
A1;
hence thesis by
A2;
end;
A22: for v102, v100 holds ((v100
"/\" v102)
"/\" v102)
= (v100
"/\" v102)
proof
let v102, v100;
(v102
"/\" v102)
= v102 by
A1;
hence thesis by
A2;
end;
A25: for v1, v0 holds (v1
"/\" (v0
"/\" v1))
= (v0
"/\" v1)
proof
let v1, v0;
((v0
"/\" v1)
"/\" v1)
= (v1
"/\" (v0
"/\" v1)) by
A3;
hence thesis by
A22;
end;
A28: for v2, v0, v1 holds ((v1
"/\" v0)
"/\" v2)
= (v0
"/\" (v1
"/\" v2))
proof
let v2, v0, v1;
(v0
"/\" v1)
= (v1
"/\" v0) by
A3;
hence thesis by
A2;
end;
A31: for v0, v2, v1 holds (v0
"/\" (v1
"/\" v2))
= (v1
"/\" (v0
"/\" v2))
proof
let v0, v2, v1;
((v0
"/\" v1)
"/\" v2)
= (v0
"/\" (v1
"/\" v2)) by
A2;
hence thesis by
A28;
end;
A34: for v102, v101 holds (v101
"\/" v102)
= (v101
"\/" (v101
"\/" v102))
proof
let v102, v101;
(v101
"\/" v101)
= v101 by
A5;
hence thesis by
A6;
end;
A39: for v102, v100 holds ((v100
"\/" v102)
"\/" v102)
= (v100
"\/" v102)
proof
let v102, v100;
(v102
"\/" v102)
= v102 by
A5;
hence thesis by
A6;
end;
A42: for v1, v0 holds (v1
"\/" (v0
"\/" v1))
= (v0
"\/" v1)
proof
let v1, v0;
((v0
"\/" v1)
"\/" v1)
= (v1
"\/" (v0
"\/" v1)) by
A7;
hence thesis by
A39;
end;
A45: for v2, v0, v1 holds ((v1
"\/" v0)
"\/" v2)
= (v0
"\/" (v1
"\/" v2))
proof
let v2, v0, v1;
(v0
"\/" v1)
= (v1
"\/" v0) by
A7;
hence thesis by
A6;
end;
A48: for v0, v2, v1 holds (v0
"\/" (v1
"\/" v2))
= (v1
"\/" (v0
"\/" v2))
proof
let v0, v2, v1;
((v0
"\/" v1)
"\/" v2)
= (v0
"\/" (v1
"\/" v2)) by
A6;
hence thesis by
A45;
end;
A51: for v102, v0, v2, v1 holds ((v0
"\/" (v1
"/\" v2))
"/\" v102)
= ((v0
"\/" v1)
"/\" ((v0
"\/" (v1
"/\" v2))
"/\" v102))
proof
let v102, v0, v2, v1;
((v0
"\/" v1)
"/\" (v0
"\/" (v1
"/\" v2)))
= (v0
"\/" (v1
"/\" v2)) by
A10;
hence thesis by
A2;
end;
A55: for v0, v2, v1 holds ((v1
"\/" v0)
"/\" (v0
"\/" (v1
"/\" v2)))
= (v0
"\/" (v1
"/\" v2))
proof
let v0, v2, v1;
(v0
"\/" v1)
= (v1
"\/" v0) by
A7;
hence thesis by
A10;
end;
A59: for v101, v102 holds (v101
"\/" (v102
"/\" (v101
"\/" v101)))
= ((v101
"\/" v101)
"/\" (v102
"\/" (v101
"/\" v101)))
proof
let v101, v102;
(v101
"/\" v101)
= v101 by
A1;
hence thesis by
A11;
end;
A62: for v0, v1 holds (v0
"\/" (v1
"/\" v0))
= ((v0
"\/" v0)
"/\" (v1
"\/" (v0
"/\" v0)))
proof
let v0, v1;
(v0
"\/" v0)
= v0 by
A5;
hence thesis by
A59;
end;
A64: for v0, v1 holds (v0
"\/" (v1
"/\" v0))
= (v0
"/\" (v1
"\/" (v0
"/\" v0)))
proof
let v0, v1;
(v0
"\/" v0)
= v0 by
A5;
hence thesis by
A62;
end;
A66: for v0, v1 holds (v0
"\/" (v1
"/\" v0))
= (v0
"/\" (v1
"\/" v0))
proof
let v0, v1;
(v0
"/\" v0)
= v0 by
A1;
hence thesis by
A64;
end;
A69: for v0, v1, v101, v100 holds ((v100
"/\" v101)
"\/" (v0
"/\" (v1
"/\" (v100
"\/" v101))))
= ((v100
"\/" v101)
"/\" ((v0
"/\" v1)
"\/" (v100
"/\" v101)))
proof
let v0, v1, v101, v100;
((v0
"/\" v1)
"/\" (v100
"\/" v101))
= (v0
"/\" (v1
"/\" (v100
"\/" v101))) by
A2;
hence thesis by
A11;
end;
A72: for v2, v1, v0 holds ((v0
"/\" v1)
"\/" (v2
"/\" (v0
"\/" v1)))
= ((v0
"\/" v1)
"/\" ((v0
"/\" v1)
"\/" v2))
proof
let v2, v1, v0;
(v2
"\/" (v0
"/\" v1))
= ((v0
"/\" v1)
"\/" v2) by
A7;
hence thesis by
A11;
end;
A74: for v0, v2, v1 holds ((v1
"/\" v0)
"\/" (v0
"/\" (v1
"\/" v2)))
= (v0
"/\" (v1
"\/" v2))
proof
let v0, v2, v1;
(v0
"/\" v1)
= (v1
"/\" v0) by
A3;
hence thesis by
A14;
end;
A77: for v0, v1, v2 holds ((v0
"/\" v1)
"\/" (v0
"/\" (v2
"\/" v1)))
= (v0
"/\" (v1
"\/" v2))
proof
let v0, v1, v2;
(v1
"\/" v2)
= (v2
"\/" v1) by
A7;
hence thesis by
A14;
end;
A80: for v101, v2, v1 holds (((v101
"/\" v1)
"\/" v101)
"/\" (v101
"/\" (v1
"\/" v2)))
= ((v101
"/\" v1)
"\/" (v101
"/\" (v1
"\/" v2)))
proof
let v101, v2, v1;
((v101
"/\" v1)
"\/" (v101
"/\" (v1
"\/" v2)))
= (v101
"/\" (v1
"\/" v2)) by
A14;
hence thesis by
A10;
end;
A83: for v0, v2, v1 holds ((v0
"\/" (v0
"/\" v1))
"/\" (v0
"/\" (v1
"\/" v2)))
= ((v0
"/\" v1)
"\/" (v0
"/\" (v1
"\/" v2)))
proof
let v0, v2, v1;
((v0
"/\" v1)
"\/" v0)
= (v0
"\/" (v0
"/\" v1)) by
A7;
hence thesis by
A80;
end;
A85: for v2, v1, v0 holds (v0
"/\" ((v0
"\/" (v0
"/\" v1))
"/\" (v1
"\/" v2)))
= ((v0
"/\" v1)
"\/" (v0
"/\" (v1
"\/" v2)))
proof
let v2, v1, v0;
((v0
"\/" (v0
"/\" v1))
"/\" (v0
"/\" (v1
"\/" v2)))
= (v0
"/\" ((v0
"\/" (v0
"/\" v1))
"/\" (v1
"\/" v2))) by
A31;
hence thesis by
A83;
end;
A87: for v2, v1, v0 holds (v0
"/\" ((v0
"\/" (v0
"/\" v1))
"/\" (v1
"\/" v2)))
= (v0
"/\" (v1
"\/" v2))
proof
let v2, v1, v0;
((v0
"/\" v1)
"\/" (v0
"/\" (v1
"\/" v2)))
= (v0
"/\" (v1
"\/" v2)) by
A14;
hence thesis by
A85;
end;
A90: for v102, v0, v2, v1 holds ((v0
"\/" (v1
"/\" v2))
"\/" ((v0
"\/" v1)
"/\" ((v0
"\/" (v1
"/\" v2))
"\/" v102)))
= ((v0
"\/" v1)
"/\" ((v0
"\/" (v1
"/\" v2))
"\/" v102))
proof
let v102, v0, v2, v1;
((v0
"\/" v1)
"/\" (v0
"\/" (v1
"/\" v2)))
= (v0
"\/" (v1
"/\" v2)) by
A10;
hence thesis by
A14;
end;
A93: for v0, v3, v2, v1 holds ((v0
"\/" (v1
"/\" v2))
"\/" ((v0
"\/" v1)
"/\" (v0
"\/" ((v1
"/\" v2)
"\/" v3))))
= ((v0
"\/" v1)
"/\" ((v0
"\/" (v1
"/\" v2))
"\/" v3))
proof
let v0, v3, v2, v1;
((v0
"\/" (v1
"/\" v2))
"\/" v3)
= (v0
"\/" ((v1
"/\" v2)
"\/" v3)) by
A6;
hence thesis by
A90;
end;
A95: for v0, v3, v2, v1 holds (v0
"\/" ((v1
"/\" v2)
"\/" ((v0
"\/" v1)
"/\" (v0
"\/" ((v1
"/\" v2)
"\/" v3)))))
= ((v0
"\/" v1)
"/\" ((v0
"\/" (v1
"/\" v2))
"\/" v3))
proof
let v0, v3, v2, v1;
((v0
"\/" (v1
"/\" v2))
"\/" ((v0
"\/" v1)
"/\" (v0
"\/" ((v1
"/\" v2)
"\/" v3))))
= (v0
"\/" ((v1
"/\" v2)
"\/" ((v0
"\/" v1)
"/\" (v0
"\/" ((v1
"/\" v2)
"\/" v3))))) by
A6;
hence thesis by
A93;
end;
A97: for v0, v3, v2, v1 holds (v0
"\/" ((v1
"/\" v2)
"\/" ((v0
"\/" v1)
"/\" (v0
"\/" ((v1
"/\" v2)
"\/" v3)))))
= ((v0
"\/" v1)
"/\" (v0
"\/" ((v1
"/\" v2)
"\/" v3)))
proof
let v0, v3, v2, v1;
((v0
"\/" (v1
"/\" v2))
"\/" v3)
= (v0
"\/" ((v1
"/\" v2)
"\/" v3)) by
A6;
hence thesis by
A95;
end;
A100: for v102, v1, v100 holds ((v100
"/\" v1)
"\/" (v100
"/\" ((v100
"/\" v1)
"\/" v102)))
= (v100
"/\" ((v100
"/\" v1)
"\/" v102))
proof
let v102, v1, v100;
(v100
"/\" (v100
"/\" v1))
= (v100
"/\" v1) by
A17;
hence thesis by
A14;
end;
A104: for v0, v100, v1 holds (v100
"\/" (v0
"\/" (v1
"\/" v100)))
= ((v0
"\/" v1)
"\/" v100)
proof
let v0, v100, v1;
((v0
"\/" v1)
"\/" v100)
= (v0
"\/" (v1
"\/" v100)) by
A6;
hence thesis by
A42;
end;
A107: for v1, v0, v2 holds (v0
"\/" (v1
"\/" (v2
"\/" v0)))
= (v1
"\/" (v2
"\/" v0))
proof
let v1, v0, v2;
((v1
"\/" v2)
"\/" v0)
= (v1
"\/" (v2
"\/" v0)) by
A6;
hence thesis by
A104;
end;
A109: for v1, v0 holds (v0
"\/" (v0
"/\" v1))
= (v0
"/\" (v1
"\/" v0))
proof
let v1, v0;
(v1
"/\" v0)
= (v0
"/\" v1) by
A3;
hence thesis by
A66;
end;
A112: for v102, v100, v1 holds ((v100
"/\" (v1
"\/" v100))
"\/" v102)
= (v100
"\/" ((v1
"/\" v100)
"\/" v102))
proof
let v102, v100, v1;
(v100
"\/" (v1
"/\" v100))
= (v100
"/\" (v1
"\/" v100)) by
A66;
hence thesis by
A6;
end;
A115: for v0, v1 holds (v0
"\/" (v1
"/\" v0))
= (v0
"/\" (v0
"\/" v1))
proof
let v0, v1;
(v1
"\/" v0)
= (v0
"\/" v1) by
A7;
hence thesis by
A66;
end;
A117: for v1, v0 holds (v0
"\/" (v0
"/\" v1))
= (v0
"/\" (v0
"\/" v1))
proof
let v1, v0;
(v1
"\/" v0)
= (v0
"\/" v1) by
A7;
hence thesis by
A109;
end;
A119: for v2, v1, v0 holds (v0
"/\" ((v0
"/\" (v0
"\/" v1))
"/\" (v1
"\/" v2)))
= (v0
"/\" (v1
"\/" v2))
proof
let v2, v1, v0;
(v0
"\/" (v0
"/\" v1))
= (v0
"/\" (v0
"\/" v1)) by
A117;
hence thesis by
A87;
end;
A121: for v2, v1, v0 holds (v0
"/\" (v0
"/\" ((v0
"\/" v1)
"/\" (v1
"\/" v2))))
= (v0
"/\" (v1
"\/" v2))
proof
let v2, v1, v0;
((v0
"/\" (v0
"\/" v1))
"/\" (v1
"\/" v2))
= (v0
"/\" ((v0
"\/" v1)
"/\" (v1
"\/" v2))) by
A2;
hence thesis by
A119;
end;
A123: for v2, v1, v0 holds (v0
"/\" ((v0
"\/" v1)
"/\" (v1
"\/" v2)))
= (v0
"/\" (v1
"\/" v2))
proof
let v2, v1, v0;
(v0
"/\" (v0
"/\" ((v0
"\/" v1)
"/\" (v1
"\/" v2))))
= (v0
"/\" ((v0
"\/" v1)
"/\" (v1
"\/" v2))) by
A17;
hence thesis by
A121;
end;
A126: for v102, v1, v100 holds ((v100
"/\" (v100
"\/" v1))
"\/" v102)
= (v100
"\/" ((v1
"/\" v100)
"\/" v102))
proof
let v102, v1, v100;
(v100
"\/" (v1
"/\" v100))
= (v100
"/\" (v100
"\/" v1)) by
A115;
hence thesis by
A6;
end;
A130: for v101, v102 holds ((v102
"\/" v101)
"/\" (v101
"\/" v102))
= (v101
"\/" (v102
"/\" v102))
proof
let v101, v102;
(v102
"/\" v102)
= v102 by
A1;
hence thesis by
A55;
end;
A133: for v1, v0 holds ((v0
"\/" v1)
"/\" (v1
"\/" v0))
= (v1
"\/" v0)
proof
let v1, v0;
(v0
"/\" v0)
= v0 by
A1;
hence thesis by
A130;
end;
A136: for v101, v100, v1 holds ((v100
"\/" v101)
"/\" (v101
"\/" (v1
"/\" v100)))
= (v101
"\/" (v100
"/\" (v1
"/\" v100)))
proof
let v101, v100, v1;
(v100
"/\" (v1
"/\" v100))
= (v1
"/\" v100) by
A25;
hence thesis by
A55;
end;
A139: for v1, v0, v2 holds ((v0
"\/" v1)
"/\" (v1
"\/" (v2
"/\" v0)))
= (v1
"\/" (v2
"/\" v0))
proof
let v1, v0, v2;
(v0
"/\" (v2
"/\" v0))
= (v2
"/\" v0) by
A25;
hence thesis by
A136;
end;
A142: for v102, v1, v100 holds ((v100
"/\" (v100
"\/" v1))
"/\" ((v100
"/\" v1)
"\/" (v100
"/\" v102)))
= ((v100
"/\" v1)
"\/" (v100
"/\" v102))
proof
let v102, v1, v100;
(v100
"\/" (v100
"/\" v1))
= (v100
"/\" (v100
"\/" v1)) by
A117;
hence thesis by
A55;
end;
A145: for v2, v1, v0 holds (v0
"/\" ((v0
"\/" v1)
"/\" ((v0
"/\" v1)
"\/" (v0
"/\" v2))))
= ((v0
"/\" v1)
"\/" (v0
"/\" v2))
proof
let v2, v1, v0;
((v0
"/\" (v0
"\/" v1))
"/\" ((v0
"/\" v1)
"\/" (v0
"/\" v2)))
= (v0
"/\" ((v0
"\/" v1)
"/\" ((v0
"/\" v1)
"\/" (v0
"/\" v2)))) by
A2;
hence thesis by
A142;
end;
A148: for v100, v102, v101 holds ((v100
"\/" v101)
"/\" ((v101
"/\" v102)
"\/" v100))
= ((v100
"\/" (v101
"/\" v102))
"/\" ((v101
"/\" v102)
"\/" v100))
proof
let v100, v102, v101;
((v100
"\/" (v101
"/\" v102))
"/\" ((v101
"/\" v102)
"\/" v100))
= ((v101
"/\" v102)
"\/" v100) by
A133;
hence thesis by
A51;
end;
A151: for v0, v2, v1 holds ((v0
"\/" v1)
"/\" ((v1
"/\" v2)
"\/" v0))
= ((v1
"/\" v2)
"\/" v0)
proof
let v0, v2, v1;
((v0
"\/" (v1
"/\" v2))
"/\" ((v1
"/\" v2)
"\/" v0))
= ((v1
"/\" v2)
"\/" v0) by
A133;
hence thesis by
A148;
end;
A154: for v101, v102 holds ((v102
"/\" v101)
"\/" (v101
"/\" v102))
= (v101
"/\" (v102
"\/" v102))
proof
let v101, v102;
(v102
"\/" v102)
= v102 by
A5;
hence thesis by
A74;
end;
A157: for v1, v0 holds ((v0
"/\" v1)
"\/" (v1
"/\" v0))
= (v1
"/\" v0)
proof
let v1, v0;
(v0
"\/" v0)
= v0 by
A5;
hence thesis by
A154;
end;
A159: for v1, v0, v2 holds ((v0
"/\" v1)
"\/" (v1
"/\" (v2
"\/" v0)))
= (v1
"/\" (v0
"\/" v2))
proof
let v1, v0, v2;
(v0
"\/" v2)
= (v2
"\/" v0) by
A7;
hence thesis by
A74;
end;
A162: for v102, v0, v1 holds ((v1
"/\" v0)
"\/" v102)
= ((v0
"/\" v1)
"\/" ((v1
"/\" v0)
"\/" v102))
proof
let v102, v0, v1;
((v0
"/\" v1)
"\/" (v1
"/\" v0))
= (v1
"/\" v0) by
A157;
hence thesis by
A6;
end;
A168: for v101, v2, v1 holds (((v101
"\/" v1)
"/\" v101)
"\/" ((v101
"\/" v1)
"/\" ((v1
"/\" v2)
"\/" v101)))
= (v101
"\/" (v1
"/\" v2))
proof
let v101, v2, v1;
((v101
"\/" v1)
"/\" (v101
"\/" (v1
"/\" v2)))
= (v101
"\/" (v1
"/\" v2)) by
A10;
hence thesis by
A77;
end;
A171: for v0, v2, v1 holds ((v0
"/\" (v0
"\/" v1))
"\/" ((v0
"\/" v1)
"/\" ((v1
"/\" v2)
"\/" v0)))
= (v0
"\/" (v1
"/\" v2))
proof
let v0, v2, v1;
((v0
"\/" v1)
"/\" v0)
= (v0
"/\" (v0
"\/" v1)) by
A3;
hence thesis by
A168;
end;
A173: for v0, v2, v1 holds ((v0
"/\" (v0
"\/" v1))
"\/" ((v1
"/\" v2)
"\/" v0))
= (v0
"\/" (v1
"/\" v2))
proof
let v0, v2, v1;
((v0
"\/" v1)
"/\" ((v1
"/\" v2)
"\/" v0))
= ((v1
"/\" v2)
"\/" v0) by
A151;
hence thesis by
A171;
end;
A175: for v0, v2, v1 holds (v0
"\/" ((v1
"/\" v0)
"\/" ((v1
"/\" v2)
"\/" v0)))
= (v0
"\/" (v1
"/\" v2))
proof
let v0, v2, v1;
((v0
"/\" (v0
"\/" v1))
"\/" ((v1
"/\" v2)
"\/" v0))
= (v0
"\/" ((v1
"/\" v0)
"\/" ((v1
"/\" v2)
"\/" v0))) by
A126;
hence thesis by
A173;
end;
A177: for v0, v2, v1 holds ((v1
"/\" v0)
"\/" ((v1
"/\" v2)
"\/" v0))
= (v0
"\/" (v1
"/\" v2))
proof
let v0, v2, v1;
(v0
"\/" ((v1
"/\" v0)
"\/" ((v1
"/\" v2)
"\/" v0)))
= ((v1
"/\" v0)
"\/" ((v1
"/\" v2)
"\/" v0)) by
A107;
hence thesis by
A175;
end;
A180: for v2, v3, v1, v0 holds ((v0
"/\" v1)
"\/" (v2
"/\" (v3
"/\" (v0
"\/" v1))))
= ((v0
"\/" v1)
"/\" ((v0
"/\" v1)
"\/" (v2
"/\" v3)))
proof
let v2, v3, v1, v0;
((v2
"/\" v3)
"\/" (v0
"/\" v1))
= ((v0
"/\" v1)
"\/" (v2
"/\" v3)) by
A7;
hence thesis by
A69;
end;
A183: for v102, v1, v100 holds (v100
"/\" ((v100
"/\" (v100
"\/" v1))
"/\" ((v100
"/\" v1)
"\/" v102)))
= (v100
"/\" ((v100
"/\" v1)
"\/" v102))
proof
let v102, v1, v100;
(v100
"\/" (v100
"/\" v1))
= (v100
"/\" (v100
"\/" v1)) by
A117;
hence thesis by
A123;
end;
A186: for v2, v1, v0 holds (v0
"/\" (v0
"/\" ((v0
"\/" v1)
"/\" ((v0
"/\" v1)
"\/" v2))))
= (v0
"/\" ((v0
"/\" v1)
"\/" v2))
proof
let v2, v1, v0;
((v0
"/\" (v0
"\/" v1))
"/\" ((v0
"/\" v1)
"\/" v2))
= (v0
"/\" ((v0
"\/" v1)
"/\" ((v0
"/\" v1)
"\/" v2))) by
A2;
hence thesis by
A183;
end;
A188: for v2, v1, v0 holds (v0
"/\" ((v0
"\/" v1)
"/\" ((v0
"/\" v1)
"\/" v2)))
= (v0
"/\" ((v0
"/\" v1)
"\/" v2))
proof
let v2, v1, v0;
(v0
"/\" (v0
"/\" ((v0
"\/" v1)
"/\" ((v0
"/\" v1)
"\/" v2))))
= (v0
"/\" ((v0
"\/" v1)
"/\" ((v0
"/\" v1)
"\/" v2))) by
A17;
hence thesis by
A186;
end;
A190: for v2, v1, v0 holds (v0
"/\" ((v0
"/\" v1)
"\/" (v0
"/\" v2)))
= ((v0
"/\" v1)
"\/" (v0
"/\" v2))
proof
let v2, v1, v0;
(v0
"/\" ((v0
"\/" v1)
"/\" ((v0
"/\" v1)
"\/" (v0
"/\" v2))))
= (v0
"/\" ((v0
"/\" v1)
"\/" (v0
"/\" v2))) by
A188;
hence thesis by
A145;
end;
A193: for v102, v1, v100 holds ((v100
"/\" v1)
"\/" (v102
"/\" (v100
"\/" (v100
"/\" v1))))
= ((v100
"\/" (v100
"/\" v1))
"/\" ((v100
"/\" (v100
"/\" v1))
"\/" v102))
proof
let v102, v1, v100;
(v100
"/\" (v100
"/\" v1))
= (v100
"/\" v1) by
A17;
hence thesis by
A72;
end;
A196: for v2, v1, v0 holds ((v0
"/\" v1)
"\/" (v2
"/\" (v0
"/\" (v0
"\/" v1))))
= ((v0
"\/" (v0
"/\" v1))
"/\" ((v0
"/\" (v0
"/\" v1))
"\/" v2))
proof
let v2, v1, v0;
(v0
"\/" (v0
"/\" v1))
= (v0
"/\" (v0
"\/" v1)) by
A117;
hence thesis by
A193;
end;
A198: for v2, v1, v0 holds ((v0
"\/" v1)
"/\" ((v0
"/\" v1)
"\/" (v2
"/\" v0)))
= ((v0
"\/" (v0
"/\" v1))
"/\" ((v0
"/\" (v0
"/\" v1))
"\/" v2))
proof
let v2, v1, v0;
((v0
"/\" v1)
"\/" (v2
"/\" (v0
"/\" (v0
"\/" v1))))
= ((v0
"\/" v1)
"/\" ((v0
"/\" v1)
"\/" (v2
"/\" v0))) by
A180;
hence thesis by
A196;
end;
A200: for v2, v1, v0 holds ((v0
"\/" v1)
"/\" ((v0
"/\" v1)
"\/" (v2
"/\" v0)))
= ((v0
"/\" (v0
"\/" v1))
"/\" ((v0
"/\" (v0
"/\" v1))
"\/" v2))
proof
let v2, v1, v0;
(v0
"\/" (v0
"/\" v1))
= (v0
"/\" (v0
"\/" v1)) by
A117;
hence thesis by
A198;
end;
A202: for v2, v1, v0 holds ((v0
"\/" v1)
"/\" ((v0
"/\" v1)
"\/" (v2
"/\" v0)))
= ((v0
"/\" (v0
"\/" v1))
"/\" ((v0
"/\" v1)
"\/" v2))
proof
let v2, v1, v0;
(v0
"/\" (v0
"/\" v1))
= (v0
"/\" v1) by
A17;
hence thesis by
A200;
end;
A204: for v2, v1, v0 holds ((v0
"\/" v1)
"/\" ((v0
"/\" v1)
"\/" (v2
"/\" v0)))
= (v0
"/\" ((v0
"\/" v1)
"/\" ((v0
"/\" v1)
"\/" v2)))
proof
let v2, v1, v0;
((v0
"/\" (v0
"\/" v1))
"/\" ((v0
"/\" v1)
"\/" v2))
= (v0
"/\" ((v0
"\/" v1)
"/\" ((v0
"/\" v1)
"\/" v2))) by
A2;
hence thesis by
A202;
end;
A206: for v2, v1, v0 holds ((v0
"\/" v1)
"/\" ((v0
"/\" v1)
"\/" (v2
"/\" v0)))
= (v0
"/\" ((v0
"/\" v1)
"\/" v2))
proof
let v2, v1, v0;
(v0
"/\" ((v0
"\/" v1)
"/\" ((v0
"/\" v1)
"\/" v2)))
= (v0
"/\" ((v0
"/\" v1)
"\/" v2)) by
A188;
hence thesis by
A204;
end;
A209: for v102, v1, v100 holds ((v100
"/\" (v100
"\/" v1))
"/\" ((v100
"/\" v1)
"\/" (v102
"/\" v100)))
= ((v100
"/\" v1)
"\/" (v102
"/\" v100))
proof
let v102, v1, v100;
(v100
"\/" (v100
"/\" v1))
= (v100
"/\" (v100
"\/" v1)) by
A117;
hence thesis by
A139;
end;
A212: for v2, v1, v0 holds (v0
"/\" ((v0
"\/" v1)
"/\" ((v0
"/\" v1)
"\/" (v2
"/\" v0))))
= ((v0
"/\" v1)
"\/" (v2
"/\" v0))
proof
let v2, v1, v0;
((v0
"/\" (v0
"\/" v1))
"/\" ((v0
"/\" v1)
"\/" (v2
"/\" v0)))
= (v0
"/\" ((v0
"\/" v1)
"/\" ((v0
"/\" v1)
"\/" (v2
"/\" v0)))) by
A2;
hence thesis by
A209;
end;
A214: for v2, v1, v0 holds (v0
"/\" (v0
"/\" ((v0
"/\" v1)
"\/" v2)))
= ((v0
"/\" v1)
"\/" (v2
"/\" v0))
proof
let v2, v1, v0;
((v0
"\/" v1)
"/\" ((v0
"/\" v1)
"\/" (v2
"/\" v0)))
= (v0
"/\" ((v0
"/\" v1)
"\/" v2)) by
A206;
hence thesis by
A212;
end;
A216: for v2, v1, v0 holds (v0
"/\" ((v0
"/\" v1)
"\/" v2))
= ((v0
"/\" v1)
"\/" (v2
"/\" v0))
proof
let v2, v1, v0;
(v0
"/\" (v0
"/\" ((v0
"/\" v1)
"\/" v2)))
= (v0
"/\" ((v0
"/\" v1)
"\/" v2)) by
A17;
hence thesis by
A214;
end;
A220: for v100, v0, v2 holds ((v100
"/\" (v0
"\/" v100))
"\/" (v100
"\/" (v2
"/\" v0)))
= ((v0
"\/" v100)
"/\" (v100
"\/" (v2
"/\" v0)))
proof
let v100, v0, v2;
((v0
"\/" v100)
"/\" (v100
"\/" (v2
"/\" v0)))
= (v100
"\/" (v2
"/\" v0)) by
A139;
hence thesis by
A74;
end;
A223: for v2, v0, v1 holds (v0
"\/" ((v0
"/\" (v1
"\/" v0))
"\/" (v2
"/\" v1)))
= ((v1
"\/" v0)
"/\" (v0
"\/" (v2
"/\" v1)))
proof
let v2, v0, v1;
((v0
"/\" (v1
"\/" v0))
"\/" (v0
"\/" (v2
"/\" v1)))
= (v0
"\/" ((v0
"/\" (v1
"\/" v0))
"\/" (v2
"/\" v1))) by
A48;
hence thesis by
A220;
end;
A225: for v2, v0, v1 holds (v0
"\/" (v0
"\/" ((v1
"/\" v0)
"\/" (v2
"/\" v1))))
= ((v1
"\/" v0)
"/\" (v0
"\/" (v2
"/\" v1)))
proof
let v2, v0, v1;
((v0
"/\" (v1
"\/" v0))
"\/" (v2
"/\" v1))
= (v0
"\/" ((v1
"/\" v0)
"\/" (v2
"/\" v1))) by
A112;
hence thesis by
A223;
end;
A227: for v2, v0, v1 holds (v0
"\/" (v0
"\/" (v1
"/\" ((v1
"/\" v0)
"\/" v2))))
= ((v1
"\/" v0)
"/\" (v0
"\/" (v2
"/\" v1)))
proof
let v2, v0, v1;
((v1
"/\" v0)
"\/" (v2
"/\" v1))
= (v1
"/\" ((v1
"/\" v0)
"\/" v2)) by
A216;
hence thesis by
A225;
end;
A229: for v2, v0, v1 holds (v0
"\/" (v1
"/\" ((v1
"/\" v0)
"\/" v2)))
= ((v1
"\/" v0)
"/\" (v0
"\/" (v2
"/\" v1)))
proof
let v2, v0, v1;
(v0
"\/" (v0
"\/" (v1
"/\" ((v1
"/\" v0)
"\/" v2))))
= (v0
"\/" (v1
"/\" ((v1
"/\" v0)
"\/" v2))) by
A34;
hence thesis by
A227;
end;
A231: for v2, v0, v1 holds (v0
"\/" (v1
"/\" ((v1
"/\" v0)
"\/" v2)))
= (v0
"\/" (v2
"/\" v1))
proof
let v2, v0, v1;
((v1
"\/" v0)
"/\" (v0
"\/" (v2
"/\" v1)))
= (v0
"\/" (v2
"/\" v1)) by
A139;
hence thesis by
A229;
end;
A234: for v1, v102, v100 holds ((v1
"/\" v100)
"\/" ((v100
"/\" v102)
"\/" (v1
"/\" v100)))
= ((v1
"/\" v100)
"\/" (v100
"/\" v102))
proof
let v1, v102, v100;
(v100
"/\" (v1
"/\" v100))
= (v1
"/\" v100) by
A25;
hence thesis by
A177;
end;
A237: for v0, v2, v1 holds ((v0
"/\" v1)
"\/" (v1
"/\" ((v1
"/\" v2)
"\/" v0)))
= ((v0
"/\" v1)
"\/" (v1
"/\" v2))
proof
let v0, v2, v1;
((v1
"/\" v2)
"\/" (v0
"/\" v1))
= (v1
"/\" ((v1
"/\" v2)
"\/" v0)) by
A216;
hence thesis by
A234;
end;
A239: for v0, v2, v1 holds (v1
"/\" (v0
"\/" (v1
"/\" v2)))
= ((v0
"/\" v1)
"\/" (v1
"/\" v2))
proof
let v0, v2, v1;
((v0
"/\" v1)
"\/" (v1
"/\" ((v1
"/\" v2)
"\/" v0)))
= (v1
"/\" (v0
"\/" (v1
"/\" v2))) by
A159;
hence thesis by
A237;
end;
A245: for v103, v102, v101 holds ((v102
"/\" v101)
"\/" ((v101
"/\" v102)
"\/" (((v102
"/\" v101)
"\/" v101)
"/\" ((v101
"/\" v102)
"\/" v103))))
= (((v102
"/\" v101)
"\/" v101)
"/\" ((v102
"/\" v101)
"\/" ((v101
"/\" v102)
"\/" v103)))
proof
let v103, v102, v101;
((v102
"/\" v101)
"\/" ((v101
"/\" v102)
"\/" v103))
= ((v101
"/\" v102)
"\/" v103) by
A162;
hence thesis by
A97;
end;
A248: for v2, v0, v1 holds ((v0
"/\" v1)
"\/" ((v1
"/\" v0)
"\/" ((v1
"\/" (v0
"/\" v1))
"/\" ((v1
"/\" v0)
"\/" v2))))
= (((v0
"/\" v1)
"\/" v1)
"/\" ((v0
"/\" v1)
"\/" ((v1
"/\" v0)
"\/" v2)))
proof
let v2, v0, v1;
((v0
"/\" v1)
"\/" v1)
= (v1
"\/" (v0
"/\" v1)) by
A7;
hence thesis by
A245;
end;
A250: for v2, v0, v1 holds ((v0
"/\" v1)
"\/" ((v1
"/\" v0)
"\/" ((v1
"/\" (v1
"\/" v0))
"/\" ((v1
"/\" v0)
"\/" v2))))
= (((v0
"/\" v1)
"\/" v1)
"/\" ((v0
"/\" v1)
"\/" ((v1
"/\" v0)
"\/" v2)))
proof
let v2, v0, v1;
(v1
"\/" (v0
"/\" v1))
= (v1
"/\" (v1
"\/" v0)) by
A115;
hence thesis by
A248;
end;
A252: for v2, v0, v1 holds ((v0
"/\" v1)
"\/" ((v1
"/\" v0)
"\/" (v1
"/\" ((v1
"\/" v0)
"/\" ((v1
"/\" v0)
"\/" v2)))))
= (((v0
"/\" v1)
"\/" v1)
"/\" ((v0
"/\" v1)
"\/" ((v1
"/\" v0)
"\/" v2)))
proof
let v2, v0, v1;
((v1
"/\" (v1
"\/" v0))
"/\" ((v1
"/\" v0)
"\/" v2))
= (v1
"/\" ((v1
"\/" v0)
"/\" ((v1
"/\" v0)
"\/" v2))) by
A2;
hence thesis by
A250;
end;
A254: for v2, v0, v1 holds ((v0
"/\" v1)
"\/" ((v1
"/\" v0)
"\/" (v1
"/\" ((v1
"/\" v0)
"\/" v2))))
= (((v0
"/\" v1)
"\/" v1)
"/\" ((v0
"/\" v1)
"\/" ((v1
"/\" v0)
"\/" v2)))
proof
let v2, v0, v1;
(v1
"/\" ((v1
"\/" v0)
"/\" ((v1
"/\" v0)
"\/" v2)))
= (v1
"/\" ((v1
"/\" v0)
"\/" v2)) by
A188;
hence thesis by
A252;
end;
A256: for v2, v0, v1 holds ((v0
"/\" v1)
"\/" (v1
"/\" ((v1
"/\" v0)
"\/" v2)))
= (((v0
"/\" v1)
"\/" v1)
"/\" ((v0
"/\" v1)
"\/" ((v1
"/\" v0)
"\/" v2)))
proof
let v2, v0, v1;
((v1
"/\" v0)
"\/" (v1
"/\" ((v1
"/\" v0)
"\/" v2)))
= (v1
"/\" ((v1
"/\" v0)
"\/" v2)) by
A100;
hence thesis by
A254;
end;
A258: for v2, v0, v1 holds (v1
"/\" (v0
"\/" (v1
"/\" ((v1
"/\" v0)
"\/" v2))))
= (((v0
"/\" v1)
"\/" v1)
"/\" ((v0
"/\" v1)
"\/" ((v1
"/\" v0)
"\/" v2)))
proof
let v2, v0, v1;
((v0
"/\" v1)
"\/" (v1
"/\" ((v1
"/\" v0)
"\/" v2)))
= (v1
"/\" (v0
"\/" (v1
"/\" ((v1
"/\" v0)
"\/" v2)))) by
A239;
hence thesis by
A256;
end;
A261: for v1, v0, v2 holds (v0
"/\" (v1
"\/" (v2
"/\" v0)))
= (((v1
"/\" v0)
"\/" v0)
"/\" ((v1
"/\" v0)
"\/" ((v0
"/\" v1)
"\/" v2)))
proof
let v1, v0, v2;
(v1
"\/" (v0
"/\" ((v0
"/\" v1)
"\/" v2)))
= (v1
"\/" (v2
"/\" v0)) by
A231;
hence thesis by
A258;
end;
A263: for v1, v0, v2 holds (v0
"/\" (v1
"\/" (v2
"/\" v0)))
= ((v0
"\/" (v1
"/\" v0))
"/\" ((v1
"/\" v0)
"\/" ((v0
"/\" v1)
"\/" v2)))
proof
let v1, v0, v2;
((v1
"/\" v0)
"\/" v0)
= (v0
"\/" (v1
"/\" v0)) by
A7;
hence thesis by
A261;
end;
A265: for v1, v0, v2 holds (v0
"/\" (v1
"\/" (v2
"/\" v0)))
= ((v0
"/\" (v0
"\/" v1))
"/\" ((v1
"/\" v0)
"\/" ((v0
"/\" v1)
"\/" v2)))
proof
let v1, v0, v2;
(v0
"\/" (v1
"/\" v0))
= (v0
"/\" (v0
"\/" v1)) by
A115;
hence thesis by
A263;
end;
A267: for v1, v0, v2 holds (v0
"/\" (v1
"\/" (v2
"/\" v0)))
= ((v0
"/\" (v0
"\/" v1))
"/\" ((v0
"/\" v1)
"\/" v2))
proof
let v1, v0, v2;
((v1
"/\" v0)
"\/" ((v0
"/\" v1)
"\/" v2))
= ((v0
"/\" v1)
"\/" v2) by
A162;
hence thesis by
A265;
end;
A269: for v1, v0, v2 holds (v0
"/\" (v1
"\/" (v2
"/\" v0)))
= (v0
"/\" ((v0
"\/" v1)
"/\" ((v0
"/\" v1)
"\/" v2)))
proof
let v1, v0, v2;
((v0
"/\" (v0
"\/" v1))
"/\" ((v0
"/\" v1)
"\/" v2))
= (v0
"/\" ((v0
"\/" v1)
"/\" ((v0
"/\" v1)
"\/" v2))) by
A2;
hence thesis by
A267;
end;
A271: for v1, v0, v2 holds (v0
"/\" (v1
"\/" (v2
"/\" v0)))
= (v0
"/\" ((v0
"/\" v1)
"\/" v2))
proof
let v1, v0, v2;
(v0
"/\" ((v0
"\/" v1)
"/\" ((v0
"/\" v1)
"\/" v2)))
= (v0
"/\" ((v0
"/\" v1)
"\/" v2)) by
A188;
hence thesis by
A269;
end;
A274: for v1, v2, v0 holds (v0
"/\" (v1
"\/" ((v0
"/\" v2)
"/\" v0)))
= ((v0
"/\" v1)
"\/" (v0
"/\" v2))
proof
let v1, v2, v0;
(v0
"/\" ((v0
"/\" v1)
"\/" (v0
"/\" v2)))
= (v0
"/\" (v1
"\/" ((v0
"/\" v2)
"/\" v0))) by
A271;
hence thesis by
A190;
end;
A276: for v1, v2, v0 holds (v0
"/\" (v1
"\/" (v0
"/\" (v0
"/\" v2))))
= ((v0
"/\" v1)
"\/" (v0
"/\" v2))
proof
let v1, v2, v0;
((v0
"/\" v2)
"/\" v0)
= (v0
"/\" (v0
"/\" v2)) by
A3;
hence thesis by
A274;
end;
for v1, v2, v0 holds (v0
"/\" (v1
"\/" (v0
"/\" v2)))
= ((v0
"/\" v1)
"\/" (v0
"/\" v2))
proof
let v1, v2, v0;
(v0
"/\" (v0
"/\" v2))
= (v0
"/\" v2) by
A17;
hence thesis by
A276;
end;
hence thesis;
end;
Cluster5: L is
meet-idempotent
meet-associative
meet-commutative
satisfying_QLT1
join-idempotent
join-associative
join-commutative
satisfying_QLT2
QLT-selfmodular implies L is
modular
proof
assume
A1: L is
meet-idempotent
meet-associative
meet-commutative
satisfying_QLT1
join-idempotent
join-associative
join-commutative
satisfying_QLT2
QLT-selfmodular;
then (for v0 holds (v0
"/\" v0)
= v0) & (for v2, v1, v0 holds ((v0
"/\" v1)
"/\" v2)
= (v0
"/\" (v1
"/\" v2))) & (for v1, v0 holds (v0
"/\" v1)
= (v1
"/\" v0)) & (for v0, v2, v1 holds ((v0
"/\" (v1
"\/" v2))
"\/" (v0
"/\" v1))
= (v0
"/\" (v1
"\/" v2))) & (for v0 holds (v0
"\/" v0)
= v0) & (for v2, v1, v0 holds ((v0
"\/" v1)
"\/" v2)
= (v0
"\/" (v1
"\/" v2))) & (for v1, v0 holds (v0
"\/" v1)
= (v1
"\/" v0)) & (for v0, v2, v1 holds ((v0
"\/" (v1
"/\" v2))
"/\" (v0
"\/" v1))
= (v0
"\/" (v1
"/\" v2))) & for v2, v1, v0 holds ((v0
"/\" v1)
"\/" (v2
"/\" (v0
"\/" v1)))
= ((v0
"\/" v1)
"/\" (v2
"\/" (v0
"/\" v1))) by
LATTICES:def 4,
LATTICES:def 5,
LATTICES:def 6,
LATTICES:def 7,
SHEFFER1:def 9,
ROBBINS1:def 7;
then for v1, v2, v3 holds ((v1
"/\" v2)
"\/" (v1
"/\" v3))
= (v1
"/\" (v2
"\/" (v1
"/\" v3))) by
ThQLT5;
hence thesis by
ModRedef,
A1;
end;
registration
cluster
meet-idempotent
meet-associative
meet-commutative
satisfying_QLT1
join-idempotent
join-associative
join-commutative
satisfying_QLT2
QLT-selfmodular ->
modular for non
empty
LattStr;
coherence by
Cluster5;
end
begin
theorem ::
LATQUASI:13
ThQLT6: (for v0 holds (v0
"/\" v0)
= v0) & (for v2, v1, v0 holds ((v0
"/\" v1)
"/\" v2)
= (v0
"/\" (v1
"/\" v2))) & (for v1, v0 holds (v0
"/\" v1)
= (v1
"/\" v0)) & (for v0, v2, v1 holds ((v0
"/\" (v1
"\/" v2))
"\/" (v0
"/\" v1))
= (v0
"/\" (v1
"\/" v2))) & (for v0 holds (v0
"\/" v0)
= v0) & (for v2, v1, v0 holds ((v0
"\/" v1)
"\/" v2)
= (v0
"\/" (v1
"\/" v2))) & (for v1, v0 holds (v0
"\/" v1)
= (v1
"\/" v0)) & (for v0, v2, v1 holds ((v0
"\/" (v1
"/\" v2))
"/\" (v0
"\/" v1))
= (v0
"\/" (v1
"/\" v2))) & (for v2, v1, v0 holds (((v0
"\/" v1)
"/\" v2)
"\/" v1)
= (((v2
"\/" v1)
"/\" v0)
"\/" v1)) implies for v1, v2, v3 holds ((v1
"/\" v2)
"\/" (v1
"/\" v3))
= (v1
"/\" (v2
"\/" (v1
"/\" v3)))
proof
assume
A2: for v0 holds (v0
"/\" v0)
= v0;
assume
A3: for v2, v1, v0 holds ((v0
"/\" v1)
"/\" v2)
= (v0
"/\" (v1
"/\" v2));
assume
A4: for v1, v0 holds (v0
"/\" v1)
= (v1
"/\" v0);
assume
A5: for v0, v2, v1 holds ((v0
"/\" (v1
"\/" v2))
"\/" (v0
"/\" v1))
= (v0
"/\" (v1
"\/" v2));
assume
A6: for v0 holds (v0
"\/" v0)
= v0;
assume
A7: for v2, v1, v0 holds ((v0
"\/" v1)
"\/" v2)
= (v0
"\/" (v1
"\/" v2));
assume
A8: for v1, v0 holds (v0
"\/" v1)
= (v1
"\/" v0);
assume
A9: for v0, v2, v1 holds ((v0
"\/" (v1
"/\" v2))
"/\" (v0
"\/" v1))
= (v0
"\/" (v1
"/\" v2));
A11: for v0, v2, v1 holds ((v0
"\/" v1)
"/\" (v0
"\/" (v1
"/\" v2)))
= (v0
"\/" (v1
"/\" v2))
proof
let v0, v2, v1;
((v0
"\/" (v1
"/\" v2))
"/\" (v0
"\/" v1))
= ((v0
"\/" v1)
"/\" (v0
"\/" (v1
"/\" v2))) by
A4;
hence thesis by
A9;
end;
assume
A12: for v2, v1, v0 holds (((v0
"\/" v1)
"/\" v2)
"\/" v1)
= (((v2
"\/" v1)
"/\" v0)
"\/" v1);
A14: for v2, v1, v0 holds (v1
"\/" ((v0
"\/" v1)
"/\" v2))
= (((v2
"\/" v1)
"/\" v0)
"\/" v1)
proof
let v2, v1, v0;
(((v0
"\/" v1)
"/\" v2)
"\/" v1)
= (v1
"\/" ((v0
"\/" v1)
"/\" v2)) by
A8;
hence thesis by
A12;
end;
A17: for v2, v0, v1 holds (v0
"\/" ((v1
"\/" v0)
"/\" v2))
= (v0
"\/" ((v2
"\/" v0)
"/\" v1))
proof
let v2, v0, v1;
(((v2
"\/" v0)
"/\" v1)
"\/" v0)
= (v0
"\/" ((v2
"\/" v0)
"/\" v1)) by
A8;
hence thesis by
A14;
end;
A21: for v0, v2, v1 holds ((v0
"/\" v1)
"\/" (v0
"/\" (v1
"\/" v2)))
= (v0
"/\" (v1
"\/" v2))
proof
let v0, v2, v1;
((v0
"/\" (v1
"\/" v2))
"\/" (v0
"/\" v1))
= ((v0
"/\" v1)
"\/" (v0
"/\" (v1
"\/" v2))) by
A8;
hence thesis by
A5;
end;
A24: for v102, v101 holds (v101
"/\" v102)
= (v101
"/\" (v101
"/\" v102))
proof
let v102, v101;
(v101
"/\" v101)
= v101 by
A2;
hence thesis by
A3;
end;
A28: for v2, v0, v1 holds ((v1
"/\" v0)
"/\" v2)
= (v0
"/\" (v1
"/\" v2))
proof
let v2, v0, v1;
(v0
"/\" v1)
= (v1
"/\" v0) by
A4;
hence thesis by
A3;
end;
A31: for v0, v2, v1 holds (v0
"/\" (v1
"/\" v2))
= (v1
"/\" (v0
"/\" v2))
proof
let v0, v2, v1;
((v0
"/\" v1)
"/\" v2)
= (v0
"/\" (v1
"/\" v2)) by
A3;
hence thesis by
A28;
end;
A34: for v102, v100 holds ((v100
"\/" v102)
"\/" v102)
= (v100
"\/" v102)
proof
let v102, v100;
(v102
"\/" v102)
= v102 by
A6;
hence thesis by
A7;
end;
A37: for v1, v0 holds (v1
"\/" (v0
"\/" v1))
= (v0
"\/" v1)
proof
let v1, v0;
((v0
"\/" v1)
"\/" v1)
= (v1
"\/" (v0
"\/" v1)) by
A8;
hence thesis by
A34;
end;
A40: for v0, v1, v2 holds ((v0
"\/" v1)
"/\" (v0
"\/" (v2
"/\" v1)))
= (v0
"\/" (v1
"/\" v2))
proof
let v0, v1, v2;
(v1
"/\" v2)
= (v2
"/\" v1) by
A4;
hence thesis by
A11;
end;
A43: for v100, v101 holds (v100
"\/" (v101
"\/" v100))
= (v100
"\/" (((v101
"\/" v100)
"\/" v100)
"/\" v101))
proof
let v100, v101;
((v101
"\/" v100)
"/\" (v101
"\/" v100))
= (v101
"\/" v100) by
A2;
hence thesis by
A17;
end;
A46: for v0, v1 holds (v1
"\/" v0)
= (v0
"\/" (((v1
"\/" v0)
"\/" v0)
"/\" v1))
proof
let v0, v1;
(v0
"\/" (v1
"\/" v0))
= (v1
"\/" v0) by
A37;
hence thesis by
A43;
end;
A49: for v1, v0 holds (v0
"\/" v1)
= (v1
"\/" ((v1
"\/" (v0
"\/" v1))
"/\" v0))
proof
let v1, v0;
((v0
"\/" v1)
"\/" v1)
= (v1
"\/" (v0
"\/" v1)) by
A8;
hence thesis by
A46;
end;
A51: for v1, v0 holds (v0
"\/" v1)
= (v1
"\/" ((v0
"\/" v1)
"/\" v0))
proof
let v1, v0;
(v1
"\/" (v0
"\/" v1))
= (v0
"\/" v1) by
A37;
hence thesis by
A49;
end;
A53: for v1, v0 holds (v0
"\/" v1)
= (v1
"\/" (v0
"/\" (v0
"\/" v1)))
proof
let v1, v0;
((v0
"\/" v1)
"/\" v0)
= (v0
"/\" (v0
"\/" v1)) by
A4;
hence thesis by
A51;
end;
A57: for v2, v0, v1 holds (v0
"\/" (v2
"/\" (v1
"\/" v0)))
= (v0
"\/" ((v2
"\/" v0)
"/\" v1))
proof
let v2, v0, v1;
((v1
"\/" v0)
"/\" v2)
= (v2
"/\" (v1
"\/" v0)) by
A4;
hence thesis by
A17;
end;
A62: for v102, v100 holds (v100
"\/" (v100
"/\" v102))
= (v100
"\/" ((v102
"\/" v100)
"/\" v100))
proof
let v102, v100;
(v100
"\/" v100)
= v100 by
A6;
hence thesis by
A17;
end;
A65: for v1, v0 holds (v0
"\/" (v0
"/\" v1))
= (v0
"\/" (v0
"/\" (v1
"\/" v0)))
proof
let v1, v0;
((v1
"\/" v0)
"/\" v0)
= (v0
"/\" (v1
"\/" v0)) by
A4;
hence thesis by
A62;
end;
A69: for v102, v2, v100, v1 holds ((v100
"\/" ((v2
"\/" v100)
"/\" v1))
"/\" (v100
"\/" (((v1
"\/" v100)
"/\" v2)
"/\" v102)))
= (v100
"\/" (((v1
"\/" v100)
"/\" v2)
"/\" v102))
proof
let v102, v2, v100, v1;
(v100
"\/" ((v1
"\/" v100)
"/\" v2))
= (v100
"\/" ((v2
"\/" v100)
"/\" v1)) by
A17;
hence thesis by
A11;
end;
A72: for v3, v1, v0, v2 holds ((v0
"\/" (v1
"/\" (v2
"\/" v0)))
"/\" (v0
"\/" (((v2
"\/" v0)
"/\" v1)
"/\" v3)))
= (v0
"\/" (((v2
"\/" v0)
"/\" v1)
"/\" v3))
proof
let v3, v1, v0, v2;
(v0
"\/" ((v1
"\/" v0)
"/\" v2))
= (v0
"\/" (v1
"/\" (v2
"\/" v0))) by
A57;
hence thesis by
A69;
end;
A74: for v1, v3, v0, v2 holds ((v0
"\/" (v1
"/\" (v2
"\/" v0)))
"/\" (v0
"\/" ((v2
"\/" v0)
"/\" (v1
"/\" v3))))
= (v0
"\/" (((v2
"\/" v0)
"/\" v1)
"/\" v3))
proof
let v1, v3, v0, v2;
(((v2
"\/" v0)
"/\" v1)
"/\" v3)
= ((v2
"\/" v0)
"/\" (v1
"/\" v3)) by
A3;
hence thesis by
A72;
end;
A76: for v2, v0, v3, v1 holds ((v0
"\/" (v1
"/\" (v2
"\/" v0)))
"/\" (v0
"\/" (v2
"/\" ((v1
"/\" v3)
"\/" v0))))
= (v0
"\/" (((v2
"\/" v0)
"/\" v1)
"/\" v3))
proof
let v2, v0, v3, v1;
(v0
"\/" ((v2
"\/" v0)
"/\" (v1
"/\" v3)))
= (v0
"\/" (v2
"/\" ((v1
"/\" v3)
"\/" v0))) by
A57;
hence thesis by
A74;
end;
A78: for v2, v0, v3, v1 holds ((v0
"\/" (v1
"/\" (v2
"\/" v0)))
"/\" (v0
"\/" (v2
"/\" ((v1
"/\" v3)
"\/" v0))))
= (v0
"\/" ((v2
"\/" v0)
"/\" (v1
"/\" v3)))
proof
let v2, v0, v3, v1;
(((v2
"\/" v0)
"/\" v1)
"/\" v3)
= ((v2
"\/" v0)
"/\" (v1
"/\" v3)) by
A3;
hence thesis by
A76;
end;
A80: for v2, v0, v3, v1 holds ((v0
"\/" (v1
"/\" (v2
"\/" v0)))
"/\" (v0
"\/" (v2
"/\" ((v1
"/\" v3)
"\/" v0))))
= (v0
"\/" (v2
"/\" ((v1
"/\" v3)
"\/" v0)))
proof
let v2, v0, v3, v1;
(v0
"\/" ((v2
"\/" v0)
"/\" (v1
"/\" v3)))
= (v0
"\/" (v2
"/\" ((v1
"/\" v3)
"\/" v0))) by
A57;
hence thesis by
A78;
end;
A83: for v102, v101 holds (v101
"\/" (v101
"/\" (v101
"\/" v102)))
= (v101
"/\" (v101
"\/" v102))
proof
let v102, v101;
(v101
"/\" v101)
= v101 by
A2;
hence thesis by
A21;
end;
A87: for v100, v101, v1 holds ((v100
"/\" v101)
"\/" (v100
"/\" (v1
"\/" v101)))
= (v100
"/\" (v101
"\/" (v1
"\/" v101)))
proof
let v100, v101, v1;
(v101
"\/" (v1
"\/" v101))
= (v1
"\/" v101) by
A37;
hence thesis by
A21;
end;
A90: for v0, v1, v2 holds ((v0
"/\" v1)
"\/" (v0
"/\" (v2
"\/" v1)))
= (v0
"/\" (v2
"\/" v1))
proof
let v0, v1, v2;
(v1
"\/" (v2
"\/" v1))
= (v2
"\/" v1) by
A37;
hence thesis by
A87;
end;
A93: for v101, v1 holds ((v1
"\/" v101)
"\/" (v101
"/\" (v1
"\/" v101)))
= (v101
"\/" (v1
"\/" v101))
proof
let v101, v1;
(v101
"\/" (v1
"\/" v101))
= (v1
"\/" v101) by
A37;
hence thesis by
A53;
end;
A96: for v1, v0 holds (v0
"\/" (v1
"\/" (v1
"/\" (v0
"\/" v1))))
= (v1
"\/" (v0
"\/" v1))
proof
let v1, v0;
((v0
"\/" v1)
"\/" (v1
"/\" (v0
"\/" v1)))
= (v0
"\/" (v1
"\/" (v1
"/\" (v0
"\/" v1)))) by
A7;
hence thesis by
A93;
end;
A98: for v0, v1 holds (v0
"\/" (v1
"\/" (v1
"/\" v0)))
= (v1
"\/" (v0
"\/" v1))
proof
let v0, v1;
(v1
"\/" (v1
"/\" (v0
"\/" v1)))
= (v1
"\/" (v1
"/\" v0)) by
A65;
hence thesis by
A96;
end;
A100: for v0, v1 holds (v0
"\/" (v1
"\/" (v1
"/\" v0)))
= (v0
"\/" v1)
proof
let v0, v1;
(v1
"\/" (v0
"\/" v1))
= (v0
"\/" v1) by
A37;
hence thesis by
A98;
end;
A103: for v0, v100, v1 holds (v100
"\/" ((v0
"/\" v1)
"\/" (v0
"/\" (v1
"/\" v100))))
= (v100
"\/" (v0
"/\" v1))
proof
let v0, v100, v1;
((v0
"/\" v1)
"/\" v100)
= (v0
"/\" (v1
"/\" v100)) by
A3;
hence thesis by
A100;
end;
A106: for v1, v0 holds (v0
"\/" (v0
"/\" (v0
"\/" v1)))
= (v0
"\/" (v0
"/\" v1))
proof
let v1, v0;
(v1
"\/" v0)
= (v0
"\/" v1) by
A8;
hence thesis by
A65;
end;
A108: for v1, v0 holds (v0
"/\" (v0
"\/" v1))
= (v0
"\/" (v0
"/\" v1))
proof
let v1, v0;
(v0
"\/" (v0
"/\" (v0
"\/" v1)))
= (v0
"/\" (v0
"\/" v1)) by
A83;
hence thesis by
A106;
end;
A111: for v100, v1, v101 holds (v100
"/\" (v101
"\/" (v101
"/\" v1)))
= (v101
"/\" (v100
"/\" (v101
"\/" v1)))
proof
let v100, v1, v101;
(v101
"/\" (v101
"\/" v1))
= (v101
"\/" (v101
"/\" v1)) by
A108;
hence thesis by
A31;
end;
A115: for v1, v100, v101 holds ((v100
"\/" v101)
"/\" (v100
"\/" (v1
"/\" (v101
"\/" v100))))
= (v100
"\/" (v101
"/\" (v1
"\/" v100)))
proof
let v1, v100, v101;
(v100
"\/" ((v1
"\/" v100)
"/\" v101))
= (v100
"\/" (v1
"/\" (v101
"\/" v100))) by
A57;
hence thesis by
A40;
end;
A119: for v102, v1, v100 holds ((v100
"/\" v1)
"\/" (v100
"/\" (v102
"\/" (v100
"/\" v1))))
= (v100
"/\" (v102
"\/" (v100
"/\" v1)))
proof
let v102, v1, v100;
(v100
"/\" (v100
"/\" v1))
= (v100
"/\" v1) by
A24;
hence thesis by
A90;
end;
A123: for v100, v103, v101 holds ((v100
"\/" (v101
"/\" ((v101
"/\" v103)
"\/" v100)))
"/\" (v100
"\/" ((v101
"/\" v103)
"\/" ((v101
"/\" v103)
"/\" v100))))
= (v100
"\/" ((v101
"/\" v103)
"/\" ((v101
"/\" v103)
"\/" v100)))
proof
let v100, v103, v101;
((v101
"/\" v103)
"/\" ((v101
"/\" v103)
"\/" v100))
= ((v101
"/\" v103)
"\/" ((v101
"/\" v103)
"/\" v100)) by
A108;
hence thesis by
A80;
end;
A126: for v0, v2, v1 holds ((v0
"\/" (v1
"/\" ((v1
"/\" v2)
"\/" v0)))
"/\" (v0
"\/" ((v1
"/\" v2)
"\/" (v1
"/\" (v2
"/\" v0)))))
= (v0
"\/" ((v1
"/\" v2)
"/\" ((v1
"/\" v2)
"\/" v0)))
proof
let v0, v2, v1;
((v1
"/\" v2)
"/\" v0)
= (v1
"/\" (v2
"/\" v0)) by
A3;
hence thesis by
A123;
end;
A128: for v0, v2, v1 holds ((v0
"\/" (v1
"/\" ((v1
"/\" v2)
"\/" v0)))
"/\" (v0
"\/" (v1
"/\" v2)))
= (v0
"\/" ((v1
"/\" v2)
"/\" ((v1
"/\" v2)
"\/" v0)))
proof
let v0, v2, v1;
(v0
"\/" ((v1
"/\" v2)
"\/" (v1
"/\" (v2
"/\" v0))))
= (v0
"\/" (v1
"/\" v2)) by
A103;
hence thesis by
A126;
end;
A130: for v0, v2, v1 holds ((v0
"\/" (v1
"/\" v2))
"/\" (v0
"\/" (v1
"/\" ((v1
"/\" v2)
"\/" v0))))
= (v0
"\/" ((v1
"/\" v2)
"/\" ((v1
"/\" v2)
"\/" v0)))
proof
let v0, v2, v1;
((v0
"\/" (v1
"/\" ((v1
"/\" v2)
"\/" v0)))
"/\" (v0
"\/" (v1
"/\" v2)))
= ((v0
"\/" (v1
"/\" v2))
"/\" (v0
"\/" (v1
"/\" ((v1
"/\" v2)
"\/" v0)))) by
A4;
hence thesis by
A128;
end;
A132: for v0, v2, v1 holds (v0
"\/" ((v1
"/\" v2)
"/\" (v1
"\/" v0)))
= (v0
"\/" ((v1
"/\" v2)
"/\" ((v1
"/\" v2)
"\/" v0)))
proof
let v0, v2, v1;
((v0
"\/" (v1
"/\" v2))
"/\" (v0
"\/" (v1
"/\" ((v1
"/\" v2)
"\/" v0))))
= (v0
"\/" ((v1
"/\" v2)
"/\" (v1
"\/" v0))) by
A115;
hence thesis by
A130;
end;
A134: for v2, v0, v1 holds (v0
"\/" (v1
"/\" (v2
"/\" (v1
"\/" v0))))
= (v0
"\/" ((v1
"/\" v2)
"/\" ((v1
"/\" v2)
"\/" v0)))
proof
let v2, v0, v1;
((v1
"/\" v2)
"/\" (v1
"\/" v0))
= (v1
"/\" (v2
"/\" (v1
"\/" v0))) by
A3;
hence thesis by
A132;
end;
A136: for v2, v0, v1 holds (v0
"\/" (v1
"/\" (v2
"/\" (v1
"\/" v0))))
= (v0
"\/" ((v1
"/\" v2)
"\/" ((v1
"/\" v2)
"/\" v0)))
proof
let v2, v0, v1;
((v1
"/\" v2)
"/\" ((v1
"/\" v2)
"\/" v0))
= ((v1
"/\" v2)
"\/" ((v1
"/\" v2)
"/\" v0)) by
A108;
hence thesis by
A134;
end;
A138: for v2, v0, v1 holds (v0
"\/" (v1
"/\" (v2
"/\" (v1
"\/" v0))))
= (v0
"\/" ((v1
"/\" v2)
"\/" (v1
"/\" (v2
"/\" v0))))
proof
let v2, v0, v1;
((v1
"/\" v2)
"/\" v0)
= (v1
"/\" (v2
"/\" v0)) by
A3;
hence thesis by
A136;
end;
A140: for v2, v0, v1 holds (v0
"\/" (v1
"/\" (v2
"/\" (v1
"\/" v0))))
= (v0
"\/" (v1
"/\" v2))
proof
let v2, v0, v1;
(v0
"\/" ((v1
"/\" v2)
"\/" (v1
"/\" (v2
"/\" v0))))
= (v0
"\/" (v1
"/\" v2)) by
A103;
hence thesis by
A138;
end;
A143: for v101, v2, v102 holds ((v102
"/\" v2)
"\/" ((v101
"\/" (v102
"/\" v2))
"/\" v102))
= ((v102
"/\" v2)
"\/" (v102
"/\" (v101
"/\" (v102
"\/" v2))))
proof
let v101, v2, v102;
(v101
"/\" (v102
"\/" (v102
"/\" v2)))
= (v102
"/\" (v101
"/\" (v102
"\/" v2))) by
A111;
hence thesis by
A57;
end;
A146: for v2, v1, v0 holds ((v0
"/\" v1)
"\/" (v0
"/\" (v2
"\/" (v0
"/\" v1))))
= ((v0
"/\" v1)
"\/" (v0
"/\" (v2
"/\" (v0
"\/" v1))))
proof
let v2, v1, v0;
((v2
"\/" (v0
"/\" v1))
"/\" v0)
= (v0
"/\" (v2
"\/" (v0
"/\" v1))) by
A4;
hence thesis by
A143;
end;
A148: for v2, v1, v0 holds (v0
"/\" (v2
"\/" (v0
"/\" v1)))
= ((v0
"/\" v1)
"\/" (v0
"/\" (v2
"/\" (v0
"\/" v1))))
proof
let v2, v1, v0;
((v0
"/\" v1)
"\/" (v0
"/\" (v2
"\/" (v0
"/\" v1))))
= (v0
"/\" (v2
"\/" (v0
"/\" v1))) by
A119;
hence thesis by
A146;
end;
A154: for v102, v2, v101 holds ((v101
"/\" v2)
"\/" (v101
"/\" (v101
"/\" (v102
"/\" (v101
"\/" v2)))))
= ((v101
"/\" v2)
"\/" (v101
"/\" v102))
proof
let v102, v2, v101;
(v102
"/\" (v101
"\/" (v101
"/\" v2)))
= (v101
"/\" (v102
"/\" (v101
"\/" v2))) by
A111;
hence thesis by
A140;
end;
A157: for v2, v1, v0 holds ((v0
"/\" v1)
"\/" (v0
"/\" (v2
"/\" (v0
"\/" v1))))
= ((v0
"/\" v1)
"\/" (v0
"/\" v2))
proof
let v2, v1, v0;
(v0
"/\" (v0
"/\" (v2
"/\" (v0
"\/" v1))))
= (v0
"/\" (v2
"/\" (v0
"\/" v1))) by
A24;
hence thesis by
A154;
end;
A159: for v2, v1, v0 holds ((v0
"/\" v1)
"\/" (v0
"/\" v2))
= (v0
"/\" (v2
"\/" (v0
"/\" v1)))
proof
let v2, v1, v0;
((v0
"/\" v1)
"\/" (v0
"/\" (v2
"/\" (v0
"\/" v1))))
= ((v0
"/\" v1)
"\/" (v0
"/\" v2)) by
A157;
hence thesis by
A148;
end;
let v1, v2, v3;
(v1
"/\" (v2
"\/" (v1
"/\" v3)))
= ((v1
"/\" v3)
"\/" (v1
"/\" v2)) by
A159;
hence thesis by
A8;
end;
definition
let L;
::
LATQUASI:def15
attr L is
QLT-selfmodular' means for v2, v1, v0 holds (((v0
"\/" v1)
"/\" v2)
"\/" v1)
= (((v2
"\/" v1)
"/\" v0)
"\/" v1);
end
Cluster6: L is
meet-idempotent
meet-associative
meet-commutative
satisfying_QLT1
join-idempotent
join-associative
join-commutative
satisfying_QLT2
QLT-selfmodular' implies L is
modular
proof
assume
A1: L is
meet-idempotent
meet-associative
meet-commutative
satisfying_QLT1
join-idempotent
join-associative
join-commutative
satisfying_QLT2
QLT-selfmodular';
then (for v0 holds (v0
"/\" v0)
= v0) & (for v2, v1, v0 holds ((v0
"/\" v1)
"/\" v2)
= (v0
"/\" (v1
"/\" v2))) & (for v1, v0 holds (v0
"/\" v1)
= (v1
"/\" v0)) & (for v0, v2, v1 holds ((v0
"/\" (v1
"\/" v2))
"\/" (v0
"/\" v1))
= (v0
"/\" (v1
"\/" v2))) & (for v0 holds (v0
"\/" v0)
= v0) & (for v2, v1, v0 holds ((v0
"\/" v1)
"\/" v2)
= (v0
"\/" (v1
"\/" v2))) & (for v1, v0 holds (v0
"\/" v1)
= (v1
"\/" v0)) & (for v0, v2, v1 holds ((v0
"\/" (v1
"/\" v2))
"/\" (v0
"\/" v1))
= (v0
"\/" (v1
"/\" v2))) & (for v2, v1, v0 holds (((v0
"\/" v1)
"/\" v2)
"\/" v1)
= (((v2
"\/" v1)
"/\" v0)
"\/" v1)) by
LATTICES:def 4,
LATTICES:def 5,
LATTICES:def 6,
LATTICES:def 7,
SHEFFER1:def 9,
ROBBINS1:def 7;
then for v1, v2, v3 holds ((v1
"/\" v2)
"\/" (v1
"/\" v3))
= (v1
"/\" (v2
"\/" (v1
"/\" v3))) by
ThQLT6;
hence thesis by
ModRedef,
A1;
end;
registration
cluster
meet-idempotent
meet-associative
meet-commutative
satisfying_QLT1
join-idempotent
join-associative
join-commutative
satisfying_QLT2
QLT-selfmodular' ->
modular for non
empty
LattStr;
coherence by
Cluster6;
end
begin
theorem ::
LATQUASI:14
ex L1,L2 be
QuasiLattice st the
carrier of L1
= the
carrier of L2 & the
L_join of L1
= the
L_join of L2 & the
L_meet of L1
<> the
L_meet of L2
proof
take L1 =
QLTLattice1 ;
take L2 =
QLTLattice2 ;
thus the
carrier of L1
= the
carrier of L2;
thus the
L_join of L1
= the
L_join of L2;
thus thesis by
WazneQLT7;
end;