bvfunc14.miz
begin
reserve Y for non
empty
set,
G for
Subset of (
PARTITIONS Y),
A,B,C,D,E,F for
a_partition of Y;
theorem ::
BVFUNC14:1
Th1: for z be
Element of Y, PA,PB be
a_partition of Y holds (
EqClass (z,(PA
'/\' PB)))
= ((
EqClass (z,PA))
/\ (
EqClass (z,PB)))
proof
let z be
Element of Y, PA,PB be
a_partition of Y;
A1: ((
EqClass (z,PA))
/\ (
EqClass (z,PB)))
c= (
EqClass (z,(PA
'/\' PB)))
proof
set Z = (
EqClass (z,(PA
'/\' PB)));
let x be
object;
assume
A2: x
in ((
EqClass (z,PA))
/\ (
EqClass (z,PB)));
then
reconsider x as
Element of Y;
A3: x
in (
EqClass (x,PA)) by
EQREL_1:def 6;
x
in (
EqClass (z,PA)) by
A2,
XBOOLE_0:def 4;
then
A4: (
EqClass (x,PA))
meets (
EqClass (z,PA)) by
A3,
XBOOLE_0: 3;
A5: x
in (
EqClass (x,PB)) by
EQREL_1:def 6;
(PA
'/\' PB)
= ((
INTERSECTION (PA,PB))
\
{
{} }) by
PARTIT1:def 4;
then Z
in (
INTERSECTION (PA,PB)) by
XBOOLE_0:def 5;
then
consider X,Y be
set such that
A6: X
in PA and
A7: Y
in PB and
A8: Z
= (X
/\ Y) by
SETFAM_1:def 5;
A9: z
in (X
/\ Y) by
A8,
EQREL_1:def 6;
then z
in (
EqClass (z,PB)) & z
in Y by
EQREL_1:def 6,
XBOOLE_0:def 4;
then Y
meets (
EqClass (z,PB)) by
XBOOLE_0: 3;
then
A10: Y
= (
EqClass (z,PB)) by
A7,
EQREL_1:def 4;
x
in (
EqClass (z,PB)) by
A2,
XBOOLE_0:def 4;
then
A11: (
EqClass (x,PB))
meets (
EqClass (z,PB)) by
A5,
XBOOLE_0: 3;
z
in (
EqClass (z,PA)) & z
in X by
A9,
EQREL_1:def 6,
XBOOLE_0:def 4;
then X
meets (
EqClass (z,PA)) by
XBOOLE_0: 3;
then X
= (
EqClass (z,PA)) by
A6,
EQREL_1:def 4;
then
A12: X
= (
EqClass (x,PA)) by
A4,
EQREL_1: 41;
x
in ((
EqClass (x,PA))
/\ (
EqClass (x,PB))) by
A3,
A5,
XBOOLE_0:def 4;
hence thesis by
A11,
A8,
A10,
A12,
EQREL_1: 41;
end;
(
EqClass (z,(PA
'/\' PB)))
c= ((
EqClass (z,PA))
/\ (
EqClass (z,PB)))
proof
let x be
object;
A13: (
EqClass (z,(PA
'/\' PB)))
c= (
EqClass (z,PA)) & (
EqClass (z,(PA
'/\' PB)))
c= (
EqClass (z,PB)) by
BVFUNC11: 3;
assume x
in (
EqClass (z,(PA
'/\' PB)));
hence thesis by
A13,
XBOOLE_0:def 4;
end;
hence thesis by
A1,
XBOOLE_0:def 10;
end;
theorem ::
BVFUNC14:2
G
=
{A, B} & A
<> B implies (
'/\' G)
= (A
'/\' B)
proof
assume that
A1: G
=
{A, B} and
A2: A
<> B;
A3: (A
'/\' B)
c= (
'/\' G)
proof
let x be
object;
reconsider xx = x as
set by
TARSKI: 1;
assume
A4: x
in (A
'/\' B);
then
A5: x
<>
{} by
EQREL_1:def 4;
x
in ((
INTERSECTION (A,B))
\
{
{} }) by
A4,
PARTIT1:def 4;
then
consider a,b be
set such that
A6: a
in A and
A7: b
in B and
A8: x
= (a
/\ b) by
SETFAM_1:def 5;
set h0 = ((A,B)
--> (a,b));
A9: (
rng ((A,B)
--> (a,b)))
=
{a, b} by
A2,
FUNCT_4: 64;
(
rng h0)
c= (
bool Y)
proof
let y be
object;
assume
A10: y
in (
rng h0);
now
per cases by
A9,
A10,
TARSKI:def 2;
case y
= a;
hence thesis by
A6;
end;
case y
= b;
hence thesis by
A7;
end;
end;
hence thesis;
end;
then
reconsider F = (
rng h0) as
Subset-Family of Y;
A11: xx
c= (
Intersect F)
proof
let u be
object;
assume
A12: u
in xx;
for y be
set holds y
in F implies u
in y
proof
let y be
set;
assume
A13: y
in F;
now
per cases by
A9,
A13,
TARSKI:def 2;
case y
= a;
hence thesis by
A8,
A12,
XBOOLE_0:def 4;
end;
case y
= b;
hence thesis by
A8,
A12,
XBOOLE_0:def 4;
end;
end;
hence thesis;
end;
then u
in (
meet F) by
A9,
SETFAM_1:def 1;
hence thesis by
A9,
SETFAM_1:def 9;
end;
A14: for d be
set st d
in G holds (h0
. d)
in d
proof
let d be
set;
assume
A15: d
in G;
now
per cases by
A1,
A15,
TARSKI:def 2;
case d
= A;
hence thesis by
A2,
A6,
FUNCT_4: 63;
end;
case d
= B;
hence thesis by
A7,
FUNCT_4: 63;
end;
end;
hence thesis;
end;
A16: (
rng h0)
=
{a, b} by
A2,
FUNCT_4: 64;
(
Intersect F)
c= xx
proof
let u be
object;
assume
A17: u
in (
Intersect F);
A18: a
in
{a, b} by
TARSKI:def 2;
then a
in F by
A2,
FUNCT_4: 64;
then
A19: (
Intersect F)
= (
meet F) by
SETFAM_1:def 9;
b
in
{a, b} by
TARSKI:def 2;
then
A20: u
in b by
A16,
A17,
A19,
SETFAM_1:def 1;
u
in a by
A16,
A17,
A18,
A19,
SETFAM_1:def 1;
hence thesis by
A8,
A20,
XBOOLE_0:def 4;
end;
then (
dom ((A,B)
--> (a,b)))
=
{A, B} & x
= (
Intersect F) by
A11,
FUNCT_4: 62,
XBOOLE_0:def 10;
hence thesis by
A1,
A14,
A5,
BVFUNC_2:def 1;
end;
(
'/\' G)
c= (A
'/\' B)
proof
let x be
object;
reconsider xx = x as
set by
TARSKI: 1;
assume x
in (
'/\' G);
then
consider h be
Function, F be
Subset-Family of Y such that
A21: (
dom h)
= G and
A22: (
rng h)
= F and
A23: for d be
set st d
in G holds (h
. d)
in d and
A24: x
= (
Intersect F) and
A25: x
<>
{} by
BVFUNC_2:def 1;
A26: not x
in
{
{} } by
A25,
TARSKI:def 1;
A
in (
dom h) by
A1,
A21,
TARSKI:def 2;
then
A27: (h
. A)
in (
rng h) by
FUNCT_1:def 3;
A28: ((h
. A)
/\ (h
. B))
c= xx
proof
let m be
object;
assume
A29: m
in ((h
. A)
/\ (h
. B));
A30: (
rng h)
c=
{(h
. A), (h
. B)}
proof
let u be
object;
assume u
in (
rng h);
then
consider x1 be
object such that
A31: x1
in (
dom h) and
A32: u
= (h
. x1) by
FUNCT_1:def 3;
now
per cases by
A1,
A21,
A31,
TARSKI:def 2;
case x1
= A;
hence thesis by
A32,
TARSKI:def 2;
end;
case x1
= B;
hence thesis by
A32,
TARSKI:def 2;
end;
end;
hence thesis;
end;
for y be
set holds y
in (
rng h) implies m
in y
proof
let y be
set;
assume
A33: y
in (
rng h);
now
per cases by
A30,
A33,
TARSKI:def 2;
case y
= (h
. A);
hence thesis by
A29,
XBOOLE_0:def 4;
end;
case y
= (h
. B);
hence thesis by
A29,
XBOOLE_0:def 4;
end;
end;
hence thesis;
end;
then m
in (
meet (
rng h)) by
A27,
SETFAM_1:def 1;
hence thesis by
A22,
A24,
A27,
SETFAM_1:def 9;
end;
B
in G by
A1,
TARSKI:def 2;
then
A34: (h
. B)
in B by
A23;
A
in G by
A1,
TARSKI:def 2;
then
A35: (h
. A)
in A by
A23;
B
in (
dom h) by
A1,
A21,
TARSKI:def 2;
then
A36: (h
. B)
in (
rng h) by
FUNCT_1:def 3;
xx
c= ((h
. A)
/\ (h
. B))
proof
let m be
object;
assume m
in xx;
then m
in (
meet (
rng h)) by
A22,
A24,
A27,
SETFAM_1:def 9;
then m
in (h
. A) & m
in (h
. B) by
A27,
A36,
SETFAM_1:def 1;
hence thesis by
XBOOLE_0:def 4;
end;
then ((h
. A)
/\ (h
. B))
= x by
A28,
XBOOLE_0:def 10;
then x
in (
INTERSECTION (A,B)) by
A35,
A34,
SETFAM_1:def 5;
then x
in ((
INTERSECTION (A,B))
\
{
{} }) by
A26,
XBOOLE_0:def 5;
hence thesis by
PARTIT1:def 4;
end;
hence thesis by
A3,
XBOOLE_0:def 10;
end;
Lm1: for f be
Function, C,D,c,d be
object st C
<> D holds (((f
+* (C
.--> c))
+* (D
.--> d))
. C)
= c
proof
let f be
Function;
let C,D,c,d be
object;
set h = ((f
+* (C
.--> c))
+* (D
.--> d));
assume C
<> D;
then not C
in (
dom (D
.--> d)) by
TARSKI:def 1;
then
A2: (h
. C)
= ((f
+* (C
.--> c))
. C) by
FUNCT_4: 11;
C
in (
dom (C
.--> c)) by
TARSKI:def 1;
hence (h
. C)
= ((C
.--> c)
. C) by
A2,
FUNCT_4: 13
.= c by
FUNCOP_1: 72;
end;
Lm2: for B,C,D,b,c,d be
object, h be
Function st h
= ((B,C,D)
--> (b,c,d)) holds (
rng h)
=
{(h
. B), (h
. C), (h
. D)}
proof
let B,C,D,b,c,d be
object, h be
Function;
assume h
= ((B,C,D)
--> (b,c,d));
then
A1: (
dom h)
=
{B, C, D} by
FUNCT_4: 128;
then
A2: B
in (
dom h) by
ENUMSET1:def 1;
A3: (
rng h)
c=
{(h
. B), (h
. C), (h
. D)}
proof
let t be
object;
assume t
in (
rng h);
then
consider x1 be
object such that
A4: x1
in (
dom h) and
A5: t
= (h
. x1) by
FUNCT_1:def 3;
now
per cases by
A1,
A4,
ENUMSET1:def 1;
case x1
= D;
hence thesis by
A5,
ENUMSET1:def 1;
end;
case x1
= B;
hence thesis by
A5,
ENUMSET1:def 1;
end;
case x1
= C;
hence thesis by
A5,
ENUMSET1:def 1;
end;
end;
hence thesis;
end;
A6: C
in (
dom h) by
A1,
ENUMSET1:def 1;
A7: D
in (
dom h) by
A1,
ENUMSET1:def 1;
{(h
. B), (h
. C), (h
. D)}
c= (
rng h)
proof
let t be
object;
assume
A8: t
in
{(h
. B), (h
. C), (h
. D)};
now
per cases by
A8,
ENUMSET1:def 1;
case t
= (h
. D);
hence thesis by
A7,
FUNCT_1:def 3;
end;
case t
= (h
. B);
hence thesis by
A2,
FUNCT_1:def 3;
end;
case t
= (h
. C);
hence thesis by
A6,
FUNCT_1:def 3;
end;
end;
hence thesis;
end;
hence thesis by
A3,
XBOOLE_0:def 10;
end;
theorem ::
BVFUNC14:3
G
=
{B, C, D} & B
<> C & C
<> D & D
<> B implies (
'/\' G)
= ((B
'/\' C)
'/\' D)
proof
assume that
A1: G
=
{B, C, D} and
A2: B
<> C and
A3: C
<> D and
A4: D
<> B;
A5: ((B
'/\' C)
'/\' D)
c= (
'/\' G)
proof
let x be
object;
reconsider xx = x as
set by
TARSKI: 1;
assume
A6: x
in ((B
'/\' C)
'/\' D);
then
A7: x
<>
{} by
EQREL_1:def 4;
x
in ((
INTERSECTION ((B
'/\' C),D))
\
{
{} }) by
A6,
PARTIT1:def 4;
then
consider a,d be
set such that
A8: a
in (B
'/\' C) and
A9: d
in D and
A10: x
= (a
/\ d) by
SETFAM_1:def 5;
a
in ((
INTERSECTION (B,C))
\
{
{} }) by
A8,
PARTIT1:def 4;
then
consider b,c be
set such that
A11: b
in B and
A12: c
in C and
A13: a
= (b
/\ c) by
SETFAM_1:def 5;
set h = ((B,C,D)
--> (b,c,d));
A14: (
rng h)
=
{(h
. B), (h
. C), (h
. D)} by
Lm2
.=
{(h
. D), (h
. B), (h
. C)} by
ENUMSET1: 59;
A15: (h
. D)
= d by
FUNCT_7: 94;
(
rng h)
c= (
bool Y)
proof
let t be
object;
assume
A16: t
in (
rng h);
now
per cases by
A14,
A16,
ENUMSET1:def 1;
case t
= (h
. D);
hence thesis by
A9,
A15;
end;
case t
= (h
. B);
then t
= b by
A2,
A4,
FUNCT_4: 134;
hence thesis by
A11;
end;
case t
= (h
. C);
then t
= c by
A3,
Lm1;
hence thesis by
A12;
end;
end;
hence thesis;
end;
then
reconsider F = (
rng h) as
Subset-Family of Y;
A17: (h
. C)
= c by
A3,
Lm1;
A18: for p be
set st p
in G holds (h
. p)
in p
proof
let p be
set;
assume
A19: p
in G;
now
per cases by
A1,
A19,
ENUMSET1:def 1;
case p
= D;
hence thesis by
A9,
FUNCT_7: 94;
end;
case p
= B;
hence thesis by
A2,
A4,
A11,
FUNCT_4: 134;
end;
case p
= C;
hence thesis by
A3,
A12,
Lm1;
end;
end;
hence thesis;
end;
A20: (h
. B)
= b by
A2,
A4,
FUNCT_4: 134;
A21: xx
c= (
Intersect F)
proof
let u be
object;
assume
A22: u
in xx;
for y be
set holds y
in F implies u
in y
proof
let y be
set;
assume
A23: y
in F;
now
per cases by
A14,
A23,
ENUMSET1:def 1;
case y
= (h
. D);
hence thesis by
A10,
A15,
A22,
XBOOLE_0:def 4;
end;
case
A24: y
= (h
. B);
u
in (b
/\ (c
/\ d)) by
A10,
A13,
A22,
XBOOLE_1: 16;
hence thesis by
A20,
A24,
XBOOLE_0:def 4;
end;
case
A25: y
= (h
. C);
u
in (c
/\ (b
/\ d)) by
A10,
A13,
A22,
XBOOLE_1: 16;
hence thesis by
A17,
A25,
XBOOLE_0:def 4;
end;
end;
hence thesis;
end;
then u
in (
meet F) by
A14,
SETFAM_1:def 1;
hence thesis by
A14,
SETFAM_1:def 9;
end;
A26: (
dom h)
=
{B, C, D} by
FUNCT_4: 128;
then D
in (
dom h) by
ENUMSET1:def 1;
then
A27: (
rng h)
<>
{} by
FUNCT_1: 3;
(
Intersect F)
c= xx
proof
let t be
object;
assume t
in (
Intersect F);
then
A28: t
in (
meet (
rng h)) by
A27,
SETFAM_1:def 9;
(h
. C)
in
{(h
. D), (h
. B), (h
. C)} by
ENUMSET1:def 1;
then t
in (h
. C) by
A14,
A28,
SETFAM_1:def 1;
then
A29: t
in c by
A3,
Lm1;
(h
. B)
in
{(h
. D), (h
. B), (h
. C)} by
ENUMSET1:def 1;
then t
in (h
. B) by
A14,
A28,
SETFAM_1:def 1;
then t
in b by
A2,
A4,
FUNCT_4: 134;
then
A30: t
in (b
/\ c) by
A29,
XBOOLE_0:def 4;
(h
. D)
in
{(h
. D), (h
. B), (h
. C)} by
ENUMSET1:def 1;
then t
in (h
. D) by
A14,
A28,
SETFAM_1:def 1;
hence thesis by
A10,
A13,
A15,
A30,
XBOOLE_0:def 4;
end;
then x
= (
Intersect F) by
A21,
XBOOLE_0:def 10;
hence thesis by
A1,
A26,
A18,
A7,
BVFUNC_2:def 1;
end;
(
'/\' G)
c= ((B
'/\' C)
'/\' D)
proof
let x be
object;
reconsider xx = x as
set by
TARSKI: 1;
assume x
in (
'/\' G);
then
consider h be
Function, F be
Subset-Family of Y such that
A31: (
dom h)
= G and
A32: (
rng h)
= F and
A33: for d be
set st d
in G holds (h
. d)
in d and
A34: x
= (
Intersect F) and
A35: x
<>
{} by
BVFUNC_2:def 1;
D
in (
dom h) by
A1,
A31,
ENUMSET1:def 1;
then
A36: (h
. D)
in (
rng h) by
FUNCT_1:def 3;
set m = ((h
. B)
/\ (h
. C));
B
in (
dom h) by
A1,
A31,
ENUMSET1:def 1;
then
A37: (h
. B)
in (
rng h) by
FUNCT_1:def 3;
C
in (
dom h) by
A1,
A31,
ENUMSET1:def 1;
then
A38: (h
. C)
in (
rng h) by
FUNCT_1:def 3;
A39: xx
c= (((h
. B)
/\ (h
. C))
/\ (h
. D))
proof
let m be
object;
assume m
in xx;
then
A40: m
in (
meet (
rng h)) by
A32,
A34,
A37,
SETFAM_1:def 9;
then m
in (h
. B) & m
in (h
. C) by
A37,
A38,
SETFAM_1:def 1;
then
A41: m
in ((h
. B)
/\ (h
. C)) by
XBOOLE_0:def 4;
m
in (h
. D) by
A36,
A40,
SETFAM_1:def 1;
hence thesis by
A41,
XBOOLE_0:def 4;
end;
then m
<>
{} by
A35;
then
A42: not m
in
{
{} } by
TARSKI:def 1;
D
in G by
A1,
ENUMSET1:def 1;
then
A43: (h
. D)
in D by
A33;
A44: not x
in
{
{} } by
A35,
TARSKI:def 1;
C
in G by
A1,
ENUMSET1:def 1;
then
A45: (h
. C)
in C by
A33;
B
in G by
A1,
ENUMSET1:def 1;
then (h
. B)
in B by
A33;
then m
in (
INTERSECTION (B,C)) by
A45,
SETFAM_1:def 5;
then m
in ((
INTERSECTION (B,C))
\
{
{} }) by
A42,
XBOOLE_0:def 5;
then
A46: m
in (B
'/\' C) by
PARTIT1:def 4;
(((h
. B)
/\ (h
. C))
/\ (h
. D))
c= xx
proof
let m be
object;
assume
A47: m
in (((h
. B)
/\ (h
. C))
/\ (h
. D));
then
A48: m
in ((h
. B)
/\ (h
. C)) by
XBOOLE_0:def 4;
A49: (
rng h)
c=
{(h
. B), (h
. C), (h
. D)}
proof
let u be
object;
assume u
in (
rng h);
then
consider x1 be
object such that
A50: x1
in (
dom h) and
A51: u
= (h
. x1) by
FUNCT_1:def 3;
now
per cases by
A1,
A31,
A50,
ENUMSET1:def 1;
case x1
= B;
hence thesis by
A51,
ENUMSET1:def 1;
end;
case x1
= C;
hence thesis by
A51,
ENUMSET1:def 1;
end;
case x1
= D;
hence thesis by
A51,
ENUMSET1:def 1;
end;
end;
hence thesis;
end;
for y be
set holds y
in (
rng h) implies m
in y
proof
let y be
set;
assume
A52: y
in (
rng h);
now
per cases by
A49,
A52,
ENUMSET1:def 1;
case y
= (h
. B);
hence thesis by
A48,
XBOOLE_0:def 4;
end;
case y
= (h
. C);
hence thesis by
A48,
XBOOLE_0:def 4;
end;
case y
= (h
. D);
hence thesis by
A47,
XBOOLE_0:def 4;
end;
end;
hence thesis;
end;
then m
in (
meet (
rng h)) by
A37,
SETFAM_1:def 1;
hence thesis by
A32,
A34,
A37,
SETFAM_1:def 9;
end;
then (((h
. B)
/\ (h
. C))
/\ (h
. D))
= x by
A39,
XBOOLE_0:def 10;
then x
in (
INTERSECTION ((B
'/\' C),D)) by
A43,
A46,
SETFAM_1:def 5;
then x
in ((
INTERSECTION ((B
'/\' C),D))
\
{
{} }) by
A44,
XBOOLE_0:def 5;
hence thesis by
PARTIT1:def 4;
end;
hence thesis by
A5,
XBOOLE_0:def 10;
end;
theorem ::
BVFUNC14:4
Th4: G
=
{A, B, C} & A
<> B & C
<> A implies (
CompF (A,G))
= (B
'/\' C)
proof
assume that
A1: G
=
{A, B, C} and
A2: A
<> B and
A3: C
<> A;
per cases ;
suppose
A4: B
= C;
G
=
{B, C, A} by
A1,
ENUMSET1: 59
.=
{B, A} by
A4,
ENUMSET1: 30;
hence (
CompF (A,G))
= B by
A2,
BVFUNC11: 7
.= (B
'/\' C) by
A4,
PARTIT1: 13;
end;
suppose
A5: B
<> C;
A6: (G
\
{A})
= ((
{A}
\/
{B, C})
\
{A}) by
A1,
ENUMSET1: 2
.= ((
{A}
\
{A})
\/ (
{B, C}
\
{A})) by
XBOOLE_1: 42;
( not B
in
{A}) & not C
in
{A} by
A2,
A3,
TARSKI:def 1;
then
A7: (G
\
{A})
= ((
{A}
\
{A})
\/
{B, C}) by
A6,
ZFMISC_1: 63
.= (
{}
\/
{B, C}) by
XBOOLE_1: 37
.=
{B, C};
A8: (
'/\' (G
\
{A}))
c= (B
'/\' C)
proof
let x be
object;
reconsider xx = x as
set by
TARSKI: 1;
assume x
in (
'/\' (G
\
{A}));
then
consider h be
Function, F be
Subset-Family of Y such that
A9: (
dom h)
= (G
\
{A}) and
A10: (
rng h)
= F and
A11: for d be
set st d
in (G
\
{A}) holds (h
. d)
in d and
A12: x
= (
Intersect F) and
A13: x
<>
{} by
BVFUNC_2:def 1;
A14: not x
in
{
{} } by
A13,
TARSKI:def 1;
B
in (
dom h) by
A7,
A9,
TARSKI:def 2;
then
A15: (h
. B)
in (
rng h) by
FUNCT_1:def 3;
A16: ((h
. B)
/\ (h
. C))
c= xx
proof
let m be
object;
assume
A17: m
in ((h
. B)
/\ (h
. C));
A18: (
rng h)
c=
{(h
. B), (h
. C)}
proof
let u be
object;
assume u
in (
rng h);
then
consider x1 be
object such that
A19: x1
in (
dom h) and
A20: u
= (h
. x1) by
FUNCT_1:def 3;
now
per cases by
A7,
A9,
A19,
TARSKI:def 2;
case x1
= B;
hence thesis by
A20,
TARSKI:def 2;
end;
case x1
= C;
hence thesis by
A20,
TARSKI:def 2;
end;
end;
hence thesis;
end;
for y be
set holds y
in (
rng h) implies m
in y
proof
let y be
set;
assume
A21: y
in (
rng h);
now
per cases by
A18,
A21,
TARSKI:def 2;
case y
= (h
. B);
hence thesis by
A17,
XBOOLE_0:def 4;
end;
case y
= (h
. C);
hence thesis by
A17,
XBOOLE_0:def 4;
end;
end;
hence thesis;
end;
then m
in (
meet (
rng h)) by
A15,
SETFAM_1:def 1;
hence thesis by
A10,
A12,
A15,
SETFAM_1:def 9;
end;
C
in (G
\
{A}) by
A7,
TARSKI:def 2;
then
A22: (h
. C)
in C by
A11;
B
in (G
\
{A}) by
A7,
TARSKI:def 2;
then
A23: (h
. B)
in B by
A11;
C
in (
dom h) by
A7,
A9,
TARSKI:def 2;
then
A24: (h
. C)
in (
rng h) by
FUNCT_1:def 3;
xx
c= ((h
. B)
/\ (h
. C))
proof
let m be
object;
assume m
in xx;
then m
in (
meet (
rng h)) by
A10,
A12,
A15,
SETFAM_1:def 9;
then m
in (h
. B) & m
in (h
. C) by
A15,
A24,
SETFAM_1:def 1;
hence thesis by
XBOOLE_0:def 4;
end;
then ((h
. B)
/\ (h
. C))
= x by
A16,
XBOOLE_0:def 10;
then x
in (
INTERSECTION (B,C)) by
A23,
A22,
SETFAM_1:def 5;
then x
in ((
INTERSECTION (B,C))
\
{
{} }) by
A14,
XBOOLE_0:def 5;
hence thesis by
PARTIT1:def 4;
end;
A25: (B
'/\' C)
c= (
'/\' (G
\
{A}))
proof
let x be
object;
reconsider xx = x as
set by
TARSKI: 1;
assume
A26: x
in (B
'/\' C);
then
A27: x
<>
{} by
EQREL_1:def 4;
x
in ((
INTERSECTION (B,C))
\
{
{} }) by
A26,
PARTIT1:def 4;
then
consider a,b be
set such that
A28: a
in B and
A29: b
in C and
A30: x
= (a
/\ b) by
SETFAM_1:def 5;
set h0 = ((B,C)
--> (a,b));
A31: (
dom h0)
= (G
\
{A}) by
A7,
FUNCT_4: 62;
A32: (
rng h0)
=
{a, b} by
A5,
FUNCT_4: 64;
(
rng h0)
c= (
bool Y)
proof
let y be
object;
assume
A33: y
in (
rng h0);
now
per cases by
A32,
A33,
TARSKI:def 2;
case y
= a;
hence thesis by
A28;
end;
case y
= b;
hence thesis by
A29;
end;
end;
hence thesis;
end;
then
reconsider F = (
rng h0) as
Subset-Family of Y;
A34: xx
c= (
Intersect F)
proof
let u be
object;
assume
A35: u
in xx;
for y be
set holds y
in F implies u
in y
proof
let y be
set;
assume
A36: y
in F;
now
per cases by
A32,
A36,
TARSKI:def 2;
case y
= a;
hence thesis by
A30,
A35,
XBOOLE_0:def 4;
end;
case y
= b;
hence thesis by
A30,
A35,
XBOOLE_0:def 4;
end;
end;
hence thesis;
end;
then u
in (
meet F) by
A32,
SETFAM_1:def 1;
hence thesis by
A32,
SETFAM_1:def 9;
end;
A37: for d be
set st d
in (G
\
{A}) holds (h0
. d)
in d
proof
let d be
set;
assume
A38: d
in (G
\
{A});
now
per cases by
A7,
A38,
TARSKI:def 2;
case d
= B;
hence thesis by
A5,
A28,
FUNCT_4: 63;
end;
case d
= C;
hence thesis by
A29,
FUNCT_4: 63;
end;
end;
hence thesis;
end;
(
Intersect F)
c= xx
proof
let u be
object;
assume
A39: u
in (
Intersect F);
A40: (
Intersect F)
= (
meet F) by
A32,
SETFAM_1:def 9;
b
in F by
A32,
TARSKI:def 2;
then
A41: u
in b by
A39,
A40,
SETFAM_1:def 1;
a
in F by
A32,
TARSKI:def 2;
then u
in a by
A39,
A40,
SETFAM_1:def 1;
hence thesis by
A30,
A41,
XBOOLE_0:def 4;
end;
then x
= (
Intersect F) by
A34,
XBOOLE_0:def 10;
hence thesis by
A31,
A37,
A27,
BVFUNC_2:def 1;
end;
(
CompF (A,G))
= (
'/\' (G
\
{A})) by
BVFUNC_2:def 7;
hence thesis by
A25,
A8,
XBOOLE_0:def 10;
end;
end;
theorem ::
BVFUNC14:5
Th5: G
=
{A, B, C} & A
<> B & B
<> C implies (
CompF (B,G))
= (C
'/\' A)
proof
{A, B, C}
=
{B, C, A} by
ENUMSET1: 59;
hence thesis by
Th4;
end;
theorem ::
BVFUNC14:6
G
=
{A, B, C} & B
<> C & C
<> A implies (
CompF (C,G))
= (A
'/\' B)
proof
{A, B, C}
=
{C, A, B} by
ENUMSET1: 59;
hence thesis by
Th4;
end;
theorem ::
BVFUNC14:7
Th7: G
=
{A, B, C, D} & A
<> B & A
<> C & A
<> D implies (
CompF (A,G))
= ((B
'/\' C)
'/\' D)
proof
assume that
A1: G
=
{A, B, C, D} and
A2: A
<> B and
A3: A
<> C and
A4: A
<> D;
per cases ;
suppose
A5: B
= C;
then G
=
{B, B, A, D} by
A1,
ENUMSET1: 71
.=
{B, A, D} by
ENUMSET1: 31
.=
{A, B, D} by
ENUMSET1: 58;
hence (
CompF (A,G))
= (B
'/\' D) by
A2,
A4,
Th4
.= ((B
'/\' C)
'/\' D) by
A5,
PARTIT1: 13;
end;
suppose
A6: B
= D;
then G
=
{B, B, A, C} by
A1,
ENUMSET1: 69
.=
{B, A, C} by
ENUMSET1: 31
.=
{A, B, C} by
ENUMSET1: 58;
hence (
CompF (A,G))
= (B
'/\' C) by
A2,
A3,
Th4
.= ((B
'/\' D)
'/\' C) by
A6,
PARTIT1: 13
.= ((B
'/\' C)
'/\' D) by
PARTIT1: 14;
end;
suppose
A7: C
= D;
then G
=
{C, C, A, B} by
A1,
ENUMSET1: 73
.=
{C, A, B} by
ENUMSET1: 31
.=
{A, B, C} by
ENUMSET1: 59;
hence (
CompF (A,G))
= (B
'/\' C) by
A2,
A3,
Th4
.= (B
'/\' (C
'/\' D)) by
A7,
PARTIT1: 13
.= ((B
'/\' C)
'/\' D) by
PARTIT1: 14;
end;
suppose
A8: B
<> C & B
<> D & C
<> D;
(G
\
{A})
= ((
{A}
\/
{B, C, D})
\
{A}) by
A1,
ENUMSET1: 4;
then
A9: (G
\
{A})
= ((
{A}
\
{A})
\/ (
{B, C, D}
\
{A})) by
XBOOLE_1: 42;
A10: not B
in
{A} by
A2,
TARSKI:def 1;
A11: ( not C
in
{A}) & not D
in
{A} by
A3,
A4,
TARSKI:def 1;
(
{B, C, D}
\
{A})
= ((
{B}
\/
{C, D})
\
{A}) by
ENUMSET1: 2
.= ((
{B}
\
{A})
\/ (
{C, D}
\
{A})) by
XBOOLE_1: 42
.= ((
{B}
\
{A})
\/
{C, D}) by
A11,
ZFMISC_1: 63
.= (
{B}
\/
{C, D}) by
A10,
ZFMISC_1: 59
.=
{B, C, D} by
ENUMSET1: 2;
then
A12: (G
\
{A})
= (
{}
\/
{B, C, D}) by
A9,
XBOOLE_1: 37
.=
{B, C, D};
A13: ((B
'/\' C)
'/\' D)
c= (
'/\' (G
\
{A}))
proof
let x be
object;
reconsider xx = x as
set by
TARSKI: 1;
assume
A14: x
in ((B
'/\' C)
'/\' D);
then
A15: x
<>
{} by
EQREL_1:def 4;
x
in ((
INTERSECTION ((B
'/\' C),D))
\
{
{} }) by
A14,
PARTIT1:def 4;
then
consider a,d be
set such that
A16: a
in (B
'/\' C) and
A17: d
in D and
A18: x
= (a
/\ d) by
SETFAM_1:def 5;
a
in ((
INTERSECTION (B,C))
\
{
{} }) by
A16,
PARTIT1:def 4;
then
consider b,c be
set such that
A19: b
in B and
A20: c
in C and
A21: a
= (b
/\ c) by
SETFAM_1:def 5;
set h = ((B,C,D)
--> (b,c,d));
A22: (h
. D)
= d by
FUNCT_7: 94;
A23: (h
. C)
= c by
A8,
Lm1;
A24: (
rng h)
=
{(h
. B), (h
. C), (h
. D)} by
Lm2
.=
{(h
. D), (h
. B), (h
. C)} by
ENUMSET1: 59;
A25: (h
. B)
= b by
A8,
FUNCT_4: 134;
(
rng h)
c= (
bool Y)
proof
let t be
object;
assume
A26: t
in (
rng h);
now
per cases by
A24,
A26,
ENUMSET1:def 1;
case t
= (h
. D);
hence thesis by
A17,
A22;
end;
case t
= (h
. B);
hence thesis by
A19,
A25;
end;
case t
= (h
. C);
hence thesis by
A20,
A23;
end;
end;
hence thesis;
end;
then
reconsider F = (
rng h) as
Subset-Family of Y;
A27: xx
c= (
Intersect F)
proof
let u be
object;
assume
A28: u
in xx;
for y be
set holds y
in F implies u
in y
proof
let y be
set;
assume
A29: y
in F;
now
per cases by
A24,
A29,
ENUMSET1:def 1;
case y
= (h
. D);
hence thesis by
A18,
A22,
A28,
XBOOLE_0:def 4;
end;
case
A30: y
= (h
. B);
u
in (b
/\ (c
/\ d)) by
A18,
A21,
A28,
XBOOLE_1: 16;
hence thesis by
A25,
A30,
XBOOLE_0:def 4;
end;
case
A31: y
= (h
. C);
u
in (c
/\ (b
/\ d)) by
A18,
A21,
A28,
XBOOLE_1: 16;
hence thesis by
A23,
A31,
XBOOLE_0:def 4;
end;
end;
hence thesis;
end;
then u
in (
meet F) by
A24,
SETFAM_1:def 1;
hence thesis by
A24,
SETFAM_1:def 9;
end;
A32: for p be
set st p
in (G
\
{A}) holds (h
. p)
in p
proof
let p be
set;
assume
A33: p
in (G
\
{A});
now
per cases by
A12,
A33,
ENUMSET1:def 1;
case p
= D;
hence thesis by
A17,
FUNCT_7: 94;
end;
case p
= B;
hence thesis by
A8,
A19,
FUNCT_4: 134;
end;
case p
= C;
hence thesis by
A8,
A20,
Lm1;
end;
end;
hence thesis;
end;
A34: (
dom h)
=
{B, C, D} by
FUNCT_4: 128;
then D
in (
dom h) by
ENUMSET1:def 1;
then
A35: (
rng h)
<>
{} by
FUNCT_1: 3;
(
Intersect F)
c= xx
proof
let t be
object;
assume t
in (
Intersect F);
then
A36: t
in (
meet (
rng h)) by
A35,
SETFAM_1:def 9;
(h
. D)
in (
rng h) by
A24,
ENUMSET1:def 1;
then
A37: t
in (h
. D) by
A36,
SETFAM_1:def 1;
(h
. C)
in (
rng h) by
A24,
ENUMSET1:def 1;
then
A38: t
in (h
. C) by
A36,
SETFAM_1:def 1;
(h
. B)
in (
rng h) by
A24,
ENUMSET1:def 1;
then t
in (h
. B) by
A36,
SETFAM_1:def 1;
then t
in (b
/\ c) by
A25,
A23,
A38,
XBOOLE_0:def 4;
hence thesis by
A18,
A21,
A22,
A37,
XBOOLE_0:def 4;
end;
then x
= (
Intersect F) by
A27,
XBOOLE_0:def 10;
hence thesis by
A12,
A34,
A32,
A15,
BVFUNC_2:def 1;
end;
(
'/\' (G
\
{A}))
c= ((B
'/\' C)
'/\' D)
proof
let x be
object;
reconsider xx = x as
set by
TARSKI: 1;
assume x
in (
'/\' (G
\
{A}));
then
consider h be
Function, F be
Subset-Family of Y such that
A39: (
dom h)
= (G
\
{A}) and
A40: (
rng h)
= F and
A41: for d be
set st d
in (G
\
{A}) holds (h
. d)
in d and
A42: x
= (
Intersect F) and
A43: x
<>
{} by
BVFUNC_2:def 1;
D
in (
dom h) by
A12,
A39,
ENUMSET1:def 1;
then
A44: (h
. D)
in (
rng h) by
FUNCT_1:def 3;
set m = ((h
. B)
/\ (h
. C));
B
in (
dom h) by
A12,
A39,
ENUMSET1:def 1;
then
A45: (h
. B)
in (
rng h) by
FUNCT_1:def 3;
C
in (
dom h) by
A12,
A39,
ENUMSET1:def 1;
then
A46: (h
. C)
in (
rng h) by
FUNCT_1:def 3;
A47: xx
c= (((h
. B)
/\ (h
. C))
/\ (h
. D))
proof
let m be
object;
assume m
in xx;
then
A48: m
in (
meet (
rng h)) by
A40,
A42,
A45,
SETFAM_1:def 9;
then m
in (h
. B) & m
in (h
. C) by
A45,
A46,
SETFAM_1:def 1;
then
A49: m
in ((h
. B)
/\ (h
. C)) by
XBOOLE_0:def 4;
m
in (h
. D) by
A44,
A48,
SETFAM_1:def 1;
hence thesis by
A49,
XBOOLE_0:def 4;
end;
then m
<>
{} by
A43;
then
A50: not m
in
{
{} } by
TARSKI:def 1;
D
in (G
\
{A}) by
A12,
ENUMSET1:def 1;
then
A51: (h
. D)
in D by
A41;
A52: not x
in
{
{} } by
A43,
TARSKI:def 1;
C
in (G
\
{A}) by
A12,
ENUMSET1:def 1;
then
A53: (h
. C)
in C by
A41;
B
in (G
\
{A}) by
A12,
ENUMSET1:def 1;
then (h
. B)
in B by
A41;
then m
in (
INTERSECTION (B,C)) by
A53,
SETFAM_1:def 5;
then m
in ((
INTERSECTION (B,C))
\
{
{} }) by
A50,
XBOOLE_0:def 5;
then
A54: m
in (B
'/\' C) by
PARTIT1:def 4;
(((h
. B)
/\ (h
. C))
/\ (h
. D))
c= xx
proof
let m be
object;
assume
A55: m
in (((h
. B)
/\ (h
. C))
/\ (h
. D));
then
A56: m
in ((h
. B)
/\ (h
. C)) by
XBOOLE_0:def 4;
A57: (
rng h)
c=
{(h
. B), (h
. C), (h
. D)}
proof
let u be
object;
assume u
in (
rng h);
then
consider x1 be
object such that
A58: x1
in (
dom h) and
A59: u
= (h
. x1) by
FUNCT_1:def 3;
now
per cases by
A12,
A39,
A58,
ENUMSET1:def 1;
case x1
= B;
hence thesis by
A59,
ENUMSET1:def 1;
end;
case x1
= C;
hence thesis by
A59,
ENUMSET1:def 1;
end;
case x1
= D;
hence thesis by
A59,
ENUMSET1:def 1;
end;
end;
hence thesis;
end;
for y be
set holds y
in (
rng h) implies m
in y
proof
let y be
set;
assume
A60: y
in (
rng h);
now
per cases by
A57,
A60,
ENUMSET1:def 1;
case y
= (h
. B);
hence thesis by
A56,
XBOOLE_0:def 4;
end;
case y
= (h
. C);
hence thesis by
A56,
XBOOLE_0:def 4;
end;
case y
= (h
. D);
hence thesis by
A55,
XBOOLE_0:def 4;
end;
end;
hence thesis;
end;
then m
in (
meet (
rng h)) by
A45,
SETFAM_1:def 1;
hence thesis by
A40,
A42,
A45,
SETFAM_1:def 9;
end;
then (((h
. B)
/\ (h
. C))
/\ (h
. D))
= x by
A47,
XBOOLE_0:def 10;
then x
in (
INTERSECTION ((B
'/\' C),D)) by
A51,
A54,
SETFAM_1:def 5;
then x
in ((
INTERSECTION ((B
'/\' C),D))
\
{
{} }) by
A52,
XBOOLE_0:def 5;
hence thesis by
PARTIT1:def 4;
end;
then (
'/\' (G
\
{A}))
= ((B
'/\' C)
'/\' D) by
A13,
XBOOLE_0:def 10;
hence thesis by
BVFUNC_2:def 7;
end;
end;
theorem ::
BVFUNC14:8
Th8: G
=
{A, B, C, D} & A
<> B & B
<> C & B
<> D implies (
CompF (B,G))
= ((A
'/\' C)
'/\' D)
proof
{A, B, C, D}
=
{B, A, C, D} by
ENUMSET1: 65;
hence thesis by
Th7;
end;
theorem ::
BVFUNC14:9
G
=
{A, B, C, D} & A
<> C & B
<> C & C
<> D implies (
CompF (C,G))
= ((A
'/\' B)
'/\' D)
proof
{A, B, C, D}
=
{C, A, B, D} by
ENUMSET1: 67;
hence thesis by
Th7;
end;
theorem ::
BVFUNC14:10
G
=
{A, B, C, D} & A
<> D & B
<> D & C
<> D implies (
CompF (D,G))
= ((A
'/\' C)
'/\' B)
proof
{A, B, C, D}
=
{D, A, C, B} by
ENUMSET1: 70;
hence thesis by
Th7;
end;
theorem ::
BVFUNC14:11
for B,C,D,b,c,d be
object holds (
dom ((B,C,D)
--> (b,c,d)))
=
{B, C, D} by
FUNCT_4: 128;
theorem ::
BVFUNC14:12
for f be
Function, C,D,c,d be
object st C
<> D holds (((f
+* (C
.--> c))
+* (D
.--> d))
. C)
= c by
Lm1;
theorem ::
BVFUNC14:13
for B,C,D,b,c,d be
object st B
<> C & D
<> B holds (((B,C,D)
--> (b,c,d))
. B)
= b by
FUNCT_4: 134;
theorem ::
BVFUNC14:14
for B,C,D,b,c,d be
object, h be
Function st h
= ((B,C,D)
--> (b,c,d)) holds (
rng h)
=
{(h
. B), (h
. C), (h
. D)} by
Lm2;
theorem ::
BVFUNC14:15
Th15: for h be
Function, A9,B9,C9,D9 be
object st A
<> B & A
<> C & A
<> D & B
<> C & B
<> D & C
<> D & h
= ((((B
.--> B9)
+* (C
.--> C9))
+* (D
.--> D9))
+* (A
.--> A9)) holds (h
. B)
= B9 & (h
. C)
= C9 & (h
. D)
= D9
proof
let h be
Function;
let A9,B9,C9,D9 be
object;
assume that
A1: A
<> B and
A2: A
<> C and
A3: A
<> D and
A4: B
<> C and
A5: B
<> D and
A6: C
<> D and
A7: h
= ((((B
.--> B9)
+* (C
.--> C9))
+* (D
.--> D9))
+* (A
.--> A9));
not D
in (
dom (A
.--> A9)) by
A3,
TARSKI:def 1;
then
A9: (h
. D)
= ((((B
.--> B9)
+* (C
.--> C9))
+* (D
.--> D9))
. D) by
A7,
FUNCT_4: 11;
not C
in (
dom (A
.--> A9)) by
A2,
TARSKI:def 1;
then
A10: (h
. C)
= ((((B
.--> B9)
+* (C
.--> C9))
+* (D
.--> D9))
. C) by
A7,
FUNCT_4: 11;
not C
in (
dom (D
.--> D9)) by
A6,
TARSKI:def 1;
then
A12: (h
. C)
= (((B
.--> B9)
+* (C
.--> C9))
. C) by
A10,
FUNCT_4: 11;
not B
in (
dom (A
.--> A9)) by
A1,
TARSKI:def 1;
then
A13: (h
. B)
= ((((B
.--> B9)
+* (C
.--> C9))
+* (D
.--> D9))
. B) by
A7,
FUNCT_4: 11;
not B
in (
dom (D
.--> D9)) by
A5,
TARSKI:def 1;
then
A14: (h
. B)
= (((B
.--> B9)
+* (C
.--> C9))
. B) by
A13,
FUNCT_4: 11;
not B
in (
dom (C
.--> C9)) by
A4,
TARSKI:def 1;
then (h
. B)
= ((B
.--> B9)
. B) by
A14,
FUNCT_4: 11;
hence (h
. B)
= B9 by
FUNCOP_1: 72;
C
in (
dom (C
.--> C9)) by
TARSKI:def 1;
then (h
. C)
= ((C
.--> C9)
. C) by
A12,
FUNCT_4: 13;
hence (h
. C)
= C9 by
FUNCOP_1: 72;
D
in (
dom (D
.--> D9)) by
TARSKI:def 1;
then (h
. D)
= ((D
.--> D9)
. D) by
A9,
FUNCT_4: 13;
hence thesis by
FUNCOP_1: 72;
end;
theorem ::
BVFUNC14:16
Th16: for A,B,C,D be
object, h be
Function, A9,B9,C9,D9 be
object st h
= ((((B
.--> B9)
+* (C
.--> C9))
+* (D
.--> D9))
+* (A
.--> A9)) holds (
dom h)
=
{A, B, C, D}
proof
let A,B,C,D be
object;
let h be
Function;
let A9,B9,C9,D9 be
object;
assume
A1: h
= ((((B
.--> B9)
+* (C
.--> C9))
+* (D
.--> D9))
+* (A
.--> A9));
(
dom ((B
.--> B9)
+* (C
.--> C9)))
= ((
dom (B
.--> B9))
\/ (
dom (C
.--> C9))) by
FUNCT_4:def 1;
then
A2: (
dom (((B
.--> B9)
+* (C
.--> C9))
+* (D
.--> D9)))
= (((
dom (B
.--> B9))
\/ (
dom (C
.--> C9)))
\/ (
dom (D
.--> D9))) by
FUNCT_4:def 1;
(
dom ((((B
.--> B9)
+* (C
.--> C9))
+* (D
.--> D9))
+* (A
.--> A9)))
= (((
{B}
\/ (
dom (C
.--> C9)))
\/ (
dom (D
.--> D9)))
\/ (
dom (A
.--> A9))) by
A2,
FUNCT_4:def 1
.= (((
{B}
\/
{C})
\/ (
dom (D
.--> D9)))
\/ (
dom (A
.--> A9)))
.= (((
{B}
\/
{C})
\/
{D})
\/ (
dom (A
.--> A9)))
.= (
{A}
\/ ((
{B}
\/
{C})
\/
{D}))
.= (
{A}
\/ (
{B, C}
\/
{D})) by
ENUMSET1: 1
.= (
{A}
\/
{B, C, D}) by
ENUMSET1: 3
.=
{A, B, C, D} by
ENUMSET1: 4;
hence thesis by
A1;
end;
theorem ::
BVFUNC14:17
Th17: for h be
Function, A9,B9,C9,D9 be
object st G
=
{A, B, C, D} & h
= ((((B
.--> B9)
+* (C
.--> C9))
+* (D
.--> D9))
+* (A
.--> A9)) holds (
rng h)
=
{(h
. A), (h
. B), (h
. C), (h
. D)}
proof
let h be
Function;
let A9,B9,C9,D9 be
object;
assume that
A1: G
=
{A, B, C, D} and
A2: h
= ((((B
.--> B9)
+* (C
.--> C9))
+* (D
.--> D9))
+* (A
.--> A9));
A3: (
dom h)
= G by
A1,
A2,
Th16;
then
A4: B
in (
dom h) by
A1,
ENUMSET1:def 2;
A5: (
rng h)
c=
{(h
. A), (h
. B), (h
. C), (h
. D)}
proof
let t be
object;
assume t
in (
rng h);
then
consider x1 be
object such that
A6: x1
in (
dom h) and
A7: t
= (h
. x1) by
FUNCT_1:def 3;
now
per cases by
A1,
A3,
A6,
ENUMSET1:def 2;
case x1
= A;
hence thesis by
A7,
ENUMSET1:def 2;
end;
case x1
= B;
hence thesis by
A7,
ENUMSET1:def 2;
end;
case x1
= C;
hence thesis by
A7,
ENUMSET1:def 2;
end;
case x1
= D;
hence thesis by
A7,
ENUMSET1:def 2;
end;
end;
hence thesis;
end;
A8: D
in (
dom h) by
A1,
A3,
ENUMSET1:def 2;
A9: C
in (
dom h) by
A1,
A3,
ENUMSET1:def 2;
A10: A
in (
dom h) by
A1,
A3,
ENUMSET1:def 2;
{(h
. A), (h
. B), (h
. C), (h
. D)}
c= (
rng h)
proof
let t be
object;
assume
A11: t
in
{(h
. A), (h
. B), (h
. C), (h
. D)};
per cases by
A11,
ENUMSET1:def 2;
suppose t
= (h
. A);
hence thesis by
A10,
FUNCT_1:def 3;
end;
suppose t
= (h
. B);
hence thesis by
A4,
FUNCT_1:def 3;
end;
suppose t
= (h
. C);
hence thesis by
A9,
FUNCT_1:def 3;
end;
suppose t
= (h
. D);
hence thesis by
A8,
FUNCT_1:def 3;
end;
end;
hence thesis by
A5,
XBOOLE_0:def 10;
end;
theorem ::
BVFUNC14:18
for z,u be
Element of Y, h be
Function st G is
independent & G
=
{A, B, C, D} & A
<> B & A
<> C & A
<> D & B
<> C & B
<> D & C
<> D holds (
EqClass (u,((B
'/\' C)
'/\' D)))
meets (
EqClass (z,A))
proof
let z,u be
Element of Y;
let h be
Function;
assume that
A1: G is
independent and
A2: G
=
{A, B, C, D} and
A3: A
<> B & A
<> C & A
<> D & B
<> C & B
<> D & C
<> D;
set h = ((((B
.--> (
EqClass (u,B)))
+* (C
.--> (
EqClass (u,C))))
+* (D
.--> (
EqClass (u,D))))
+* (A
.--> (
EqClass (z,A))));
A4: (h
. B)
= (
EqClass (u,B)) by
A3,
Th15;
A5: (h
. D)
= (
EqClass (u,D)) by
A3,
Th15;
A6: (h
. C)
= (
EqClass (u,C)) by
A3,
Th15;
A7: (
rng h)
=
{(h
. A), (h
. B), (h
. C), (h
. D)} by
A2,
Th17;
(
rng h)
c= (
bool Y)
proof
let t be
object;
assume
A8: t
in (
rng h);
per cases by
A7,
A8,
ENUMSET1:def 2;
suppose t
= (h
. A);
then t
= (
EqClass (z,A)) by
FUNCT_7: 94;
hence thesis;
end;
suppose t
= (h
. B);
hence thesis by
A4;
end;
suppose t
= (h
. C);
hence thesis by
A6;
end;
suppose t
= (h
. D);
hence thesis by
A5;
end;
end;
then
reconsider FF = (
rng h) as
Subset-Family of Y;
A9: (
dom h)
= G by
A2,
Th16;
for d be
set st d
in G holds (h
. d)
in d
proof
let d be
set;
assume
A10: d
in G;
per cases by
A2,
A10,
ENUMSET1:def 2;
suppose
A11: d
= A;
(h
. A)
= (
EqClass (z,A)) by
FUNCT_7: 94;
hence thesis by
A11;
end;
suppose
A12: d
= B;
(h
. B)
= (
EqClass (u,B)) by
A3,
Th15;
hence thesis by
A12;
end;
suppose
A13: d
= C;
(h
. C)
= (
EqClass (u,C)) by
A3,
Th15;
hence thesis by
A13;
end;
suppose
A14: d
= D;
(h
. D)
= (
EqClass (u,D)) by
A3,
Th15;
hence thesis by
A14;
end;
end;
then (
Intersect FF)
<>
{} by
A1,
A9,
BVFUNC_2:def 5;
then
consider m be
object such that
A15: m
in (
Intersect FF) by
XBOOLE_0:def 1;
A
in (
dom h) by
A2,
A9,
ENUMSET1:def 2;
then
A16: (h
. A)
in (
rng h) by
FUNCT_1:def 3;
then
A17: m
in (
meet FF) by
A15,
SETFAM_1:def 9;
D
in (
dom h) by
A2,
A9,
ENUMSET1:def 2;
then (h
. D)
in (
rng h) by
FUNCT_1:def 3;
then
A18: m
in (h
. D) by
A17,
SETFAM_1:def 1;
C
in (
dom h) by
A2,
A9,
ENUMSET1:def 2;
then (h
. C)
in (
rng h) by
FUNCT_1:def 3;
then
A19: m
in (h
. C) by
A17,
SETFAM_1:def 1;
B
in (
dom h) by
A2,
A9,
ENUMSET1:def 2;
then (h
. B)
in (
rng h) by
FUNCT_1:def 3;
then m
in (h
. B) by
A17,
SETFAM_1:def 1;
then m
in ((
EqClass (u,B))
/\ (
EqClass (u,C))) by
A4,
A6,
A19,
XBOOLE_0:def 4;
then
A20: m
in (((
EqClass (u,B))
/\ (
EqClass (u,C)))
/\ (
EqClass (u,D))) by
A5,
A18,
XBOOLE_0:def 4;
(h
. A)
= (
EqClass (z,A)) & m
in (h
. A) by
A16,
A17,
FUNCT_7: 94,
SETFAM_1:def 1;
then m
in ((((
EqClass (u,B))
/\ (
EqClass (u,C)))
/\ (
EqClass (u,D)))
/\ (
EqClass (z,A))) by
A20,
XBOOLE_0:def 4;
then
A21: (((
EqClass (u,B))
/\ (
EqClass (u,C)))
/\ (
EqClass (u,D)))
meets (
EqClass (z,A)) by
XBOOLE_0: 4;
(
EqClass (u,((B
'/\' C)
'/\' D)))
= ((
EqClass (u,(B
'/\' C)))
/\ (
EqClass (u,D))) by
Th1;
hence thesis by
A21,
Th1;
end;
theorem ::
BVFUNC14:19
for z,u be
Element of Y st G is
independent & G
=
{A, B, C, D} & A
<> B & A
<> C & A
<> D & B
<> C & B
<> D & C
<> D & (
EqClass (z,(C
'/\' D)))
= (
EqClass (u,(C
'/\' D))) holds (
EqClass (u,(
CompF (A,G))))
meets (
EqClass (z,(
CompF (B,G))))
proof
let z,u be
Element of Y;
assume that
A1: G is
independent and
A2: G
=
{A, B, C, D} and
A3: A
<> B and
A4: A
<> C & A
<> D and
A5: B
<> C & B
<> D and
A6: C
<> D and
A7: (
EqClass (z,(C
'/\' D)))
= (
EqClass (u,(C
'/\' D)));
set h = ((((B
.--> (
EqClass (u,B)))
+* (C
.--> (
EqClass (u,C))))
+* (D
.--> (
EqClass (u,D))))
+* (A
.--> (
EqClass (z,A))));
set H = (
EqClass (z,(
CompF (B,G))));
A8: (A
'/\' (C
'/\' D))
= ((A
'/\' C)
'/\' D) by
PARTIT1: 14;
A9: (
rng h)
=
{(h
. A), (h
. B), (h
. C), (h
. D)} by
A2,
Th17;
(
rng h)
c= (
bool Y)
proof
let t be
object;
assume
A10: t
in (
rng h);
per cases by
A9,
A10,
ENUMSET1:def 2;
suppose t
= (h
. A);
then t
= (
EqClass (z,A)) by
FUNCT_7: 94;
hence thesis;
end;
suppose t
= (h
. B);
then t
= (
EqClass (u,B)) by
A3,
A4,
A5,
A6,
Th15;
hence thesis;
end;
suppose t
= (h
. C);
then t
= (
EqClass (u,C)) by
A3,
A4,
A5,
A6,
Th15;
hence thesis;
end;
suppose t
= (h
. D);
then t
= (
EqClass (u,D)) by
A3,
A4,
A5,
A6,
Th15;
hence thesis;
end;
end;
then
reconsider FF = (
rng h) as
Subset-Family of Y;
set I = (
EqClass (z,A)), GG = (
EqClass (u,((B
'/\' C)
'/\' D)));
A11: GG
= ((
EqClass (u,(B
'/\' C)))
/\ (
EqClass (u,D))) by
Th1;
A12: for d be
set st d
in G holds (h
. d)
in d
proof
let d be
set;
assume
A13: d
in G;
per cases by
A2,
A13,
ENUMSET1:def 2;
suppose
A14: d
= A;
(h
. A)
= (
EqClass (z,A)) by
FUNCT_7: 94;
hence thesis by
A14;
end;
suppose
A15: d
= B;
(h
. B)
= (
EqClass (u,B)) by
A3,
A4,
A5,
A6,
Th15;
hence thesis by
A15;
end;
suppose
A16: d
= C;
(h
. C)
= (
EqClass (u,C)) by
A3,
A4,
A5,
A6,
Th15;
hence thesis by
A16;
end;
suppose
A17: d
= D;
(h
. D)
= (
EqClass (u,D)) by
A3,
A4,
A5,
A6,
Th15;
hence thesis by
A17;
end;
end;
(
dom h)
= G by
A2,
Th16;
then (
Intersect FF)
<>
{} by
A1,
A12,
BVFUNC_2:def 5;
then
consider m be
object such that
A18: m
in (
Intersect FF) by
XBOOLE_0:def 1;
A19: (
dom h)
= G by
A2,
Th16;
then A
in (
dom h) by
A2,
ENUMSET1:def 2;
then
A20: (h
. A)
in (
rng h) by
FUNCT_1:def 3;
then
A21: m
in (
meet FF) by
A18,
SETFAM_1:def 9;
then
A22: (h
. A)
= (
EqClass (z,A)) & m
in (h
. A) by
A20,
FUNCT_7: 94,
SETFAM_1:def 1;
D
in (
dom h) by
A2,
A19,
ENUMSET1:def 2;
then (h
. D)
in (
rng h) by
FUNCT_1:def 3;
then
A23: m
in (h
. D) by
A21,
SETFAM_1:def 1;
C
in (
dom h) by
A2,
A19,
ENUMSET1:def 2;
then (h
. C)
in (
rng h) by
FUNCT_1:def 3;
then
A24: m
in (h
. C) by
A21,
SETFAM_1:def 1;
B
in (
dom h) by
A2,
A19,
ENUMSET1:def 2;
then (h
. B)
in (
rng h) by
FUNCT_1:def 3;
then
A25: m
in (h
. B) by
A21,
SETFAM_1:def 1;
(h
. B)
= (
EqClass (u,B)) & (h
. C)
= (
EqClass (u,C)) by
A3,
A4,
A5,
A6,
Th15;
then
A26: m
in ((
EqClass (u,B))
/\ (
EqClass (u,C))) by
A25,
A24,
XBOOLE_0:def 4;
(h
. D)
= (
EqClass (u,D)) by
A3,
A4,
A5,
A6,
Th15;
then m
in (((
EqClass (u,B))
/\ (
EqClass (u,C)))
/\ (
EqClass (u,D))) by
A23,
A26,
XBOOLE_0:def 4;
then m
in ((((
EqClass (u,B))
/\ (
EqClass (u,C)))
/\ (
EqClass (u,D)))
/\ (
EqClass (z,A))) by
A22,
XBOOLE_0:def 4;
then (GG
/\ I)
<>
{} by
A11,
Th1;
then
consider p be
object such that
A27: p
in (GG
/\ I) by
XBOOLE_0:def 1;
reconsider p as
Element of Y by
A27;
set K = (
EqClass (p,(C
'/\' D)));
A28: p
in GG by
A27,
XBOOLE_0:def 4;
set L = (
EqClass (z,(C
'/\' D)));
A29: z
in I by
EQREL_1:def 6;
GG
= (
EqClass (u,(B
'/\' (C
'/\' D)))) by
PARTIT1: 14;
then
A30: GG
c= (
EqClass (u,(C
'/\' D))) by
BVFUNC11: 3;
p
in (
EqClass (p,(C
'/\' D))) by
EQREL_1:def 6;
then K
meets L by
A7,
A30,
A28,
XBOOLE_0: 3;
then K
= L by
EQREL_1: 41;
then z
in K by
EQREL_1:def 6;
then
A31: z
in (I
/\ K) by
A29,
XBOOLE_0:def 4;
A32: p
in K & p
in I by
A27,
EQREL_1:def 6,
XBOOLE_0:def 4;
then p
in (I
/\ K) by
XBOOLE_0:def 4;
then (I
/\ K)
in (
INTERSECTION (A,(C
'/\' D))) & not (I
/\ K)
in
{
{} } by
SETFAM_1:def 5,
TARSKI:def 1;
then
A33: (I
/\ K)
in ((
INTERSECTION (A,(C
'/\' D)))
\
{
{} }) by
XBOOLE_0:def 5;
(
CompF (B,G))
= ((A
'/\' C)
'/\' D) by
A2,
A3,
A5,
Th8;
then (I
/\ K)
in (
CompF (B,G)) by
A33,
A8,
PARTIT1:def 4;
then
A34: (I
/\ K)
= H or (I
/\ K)
misses H by
EQREL_1:def 4;
z
in H by
EQREL_1:def 6;
then p
in H by
A32,
A31,
A34,
XBOOLE_0: 3,
XBOOLE_0:def 4;
then p
in (GG
/\ H) by
A28,
XBOOLE_0:def 4;
then GG
meets H by
XBOOLE_0: 4;
hence thesis by
A2,
A3,
A4,
Th7;
end;
theorem ::
BVFUNC14:20
for z,u be
Element of Y st G is
independent & G
=
{A, B, C} & A
<> B & B
<> C & C
<> A & (
EqClass (z,C))
= (
EqClass (u,C)) holds (
EqClass (u,(
CompF (A,G))))
meets (
EqClass (z,(
CompF (B,G))))
proof
let z,u be
Element of Y;
assume that
A1: G is
independent and
A2: G
=
{A, B, C} and
A3: A
<> B and
A4: B
<> C and
A5: C
<> A and
A6: (
EqClass (z,C))
= (
EqClass (u,C));
set h = (((B
.--> (
EqClass (u,B)))
+* (C
.--> (
EqClass (u,C))))
+* (A
.--> (
EqClass (z,A))));
A
in (
dom (A
.--> (
EqClass (z,A)))) by
TARSKI:def 1;
then (h
. A)
= ((A
.--> (
EqClass (z,A)))
. A) by
FUNCT_4: 13;
then
A8: (h
. A)
= (
EqClass (z,A)) by
FUNCOP_1: 72;
set H = (
EqClass (z,(
CompF (B,G)))), I = (
EqClass (z,A)), GG = (
EqClass (u,(B
'/\' C)));
A9: (GG
/\ I)
= (((
EqClass (u,B))
/\ (
EqClass (u,C)))
/\ (
EqClass (z,A))) by
Th1;
(
dom ((B
.--> (
EqClass (u,B)))
+* (C
.--> (
EqClass (u,C)))))
= ((
dom (B
.--> (
EqClass (u,B))))
\/ (
dom (C
.--> (
EqClass (u,C))))) by
FUNCT_4:def 1;
then
A10: (
dom (((B
.--> (
EqClass (u,B)))
+* (C
.--> (
EqClass (u,C))))
+* (A
.--> (
EqClass (z,A)))))
= (((
dom (B
.--> (
EqClass (u,B))))
\/ (
dom (C
.--> (
EqClass (u,C)))))
\/ (
dom (A
.--> (
EqClass (z,A))))) by
FUNCT_4:def 1;
A12: C
in (
dom (C
.--> (
EqClass (u,C)))) by
TARSKI:def 1;
not B
in (
dom (A
.--> (
EqClass (z,A)))) by
A3,
TARSKI:def 1;
then
A13: (h
. B)
= (((B
.--> (
EqClass (u,B)))
+* (C
.--> (
EqClass (u,C))))
. B) by
FUNCT_4: 11;
not B
in (
dom (C
.--> (
EqClass (u,C)))) by
A4,
TARSKI:def 1;
then (h
. B)
= ((B
.--> (
EqClass (u,B)))
. B) by
A13,
FUNCT_4: 11;
then
A14: (h
. B)
= (
EqClass (u,B)) by
FUNCOP_1: 72;
not C
in (
dom (A
.--> (
EqClass (z,A)))) by
A5,
TARSKI:def 1;
then (h
. C)
= (((B
.--> (
EqClass (u,B)))
+* (C
.--> (
EqClass (u,C))))
. C) by
FUNCT_4: 11;
then (h
. C)
= ((C
.--> (
EqClass (u,C)))
. C) by
A12,
FUNCT_4: 13;
then
A15: (h
. C)
= (
EqClass (u,C)) by
FUNCOP_1: 72;
A16: (
dom (((B
.--> (
EqClass (u,B)))
+* (C
.--> (
EqClass (u,C))))
+* (A
.--> (
EqClass (z,A)))))
= ((
{A}
\/
{B})
\/
{C}) by
A10,
XBOOLE_1: 4
.= (
{A, B}
\/
{C}) by
ENUMSET1: 1
.=
{A, B, C} by
ENUMSET1: 3;
A17: (
rng h)
c=
{(h
. A), (h
. B), (h
. C)}
proof
let t be
object;
assume t
in (
rng h);
then
consider x1 be
object such that
A18: x1
in (
dom h) and
A19: t
= (h
. x1) by
FUNCT_1:def 3;
now
per cases by
A16,
A18,
ENUMSET1:def 1;
case x1
= A;
hence thesis by
A19,
ENUMSET1:def 1;
end;
case x1
= B;
hence thesis by
A19,
ENUMSET1:def 1;
end;
case x1
= C;
hence thesis by
A19,
ENUMSET1:def 1;
end;
end;
hence thesis;
end;
(
rng h)
c= (
bool Y)
proof
let t be
object;
assume
A20: t
in (
rng h);
now
per cases by
A17,
A20,
ENUMSET1:def 1;
case t
= (h
. A);
hence thesis by
A8;
end;
case t
= (h
. B);
hence thesis by
A14;
end;
case t
= (h
. C);
hence thesis by
A15;
end;
end;
hence thesis;
end;
then
reconsider FF = (
rng h) as
Subset-Family of Y;
A21: z
in H by
EQREL_1:def 6;
for d be
set st d
in G holds (h
. d)
in d
proof
let d be
set;
assume
A22: d
in G;
now
per cases by
A2,
A22,
ENUMSET1:def 1;
case d
= A;
hence thesis by
A8;
end;
case d
= B;
hence thesis by
A14;
end;
case d
= C;
hence thesis by
A15;
end;
end;
hence thesis;
end;
then (
Intersect FF)
<>
{} by
A1,
A2,
A16,
BVFUNC_2:def 5;
then
consider m be
object such that
A23: m
in (
Intersect FF) by
XBOOLE_0:def 1;
A
in (
dom h) by
A16,
ENUMSET1:def 1;
then
A24: (h
. A)
in (
rng h) by
FUNCT_1:def 3;
then
A25: (
Intersect FF)
= (
meet (
rng h)) by
SETFAM_1:def 9;
C
in (
dom h) by
A16,
ENUMSET1:def 1;
then (h
. C)
in (
rng h) by
FUNCT_1:def 3;
then
A26: m
in (h
. C) by
A25,
A23,
SETFAM_1:def 1;
B
in (
dom h) by
A16,
ENUMSET1:def 1;
then (h
. B)
in (
rng h) by
FUNCT_1:def 3;
then m
in (h
. B) by
A25,
A23,
SETFAM_1:def 1;
then
A27: m
in ((
EqClass (u,B))
/\ (
EqClass (u,C))) by
A14,
A15,
A26,
XBOOLE_0:def 4;
m
in (h
. A) by
A24,
A25,
A23,
SETFAM_1:def 1;
then (GG
/\ I)
<>
{} by
A8,
A9,
A27,
XBOOLE_0:def 4;
then
consider p be
object such that
A28: p
in (GG
/\ I) by
XBOOLE_0:def 1;
reconsider p as
Element of Y by
A28;
set K = (
EqClass (p,C));
A29: (I
/\ K)
in (
INTERSECTION (A,C)) by
SETFAM_1:def 5;
set L = (
EqClass (z,C));
A30: p
in (
EqClass (p,C)) by
EQREL_1:def 6;
A31: p
in GG by
A28,
XBOOLE_0:def 4;
p
in K & p
in I by
A28,
EQREL_1:def 6,
XBOOLE_0:def 4;
then
A32: p
in (I
/\ K) by
XBOOLE_0:def 4;
then not (I
/\ K)
in
{
{} } by
TARSKI:def 1;
then (I
/\ K)
in ((
INTERSECTION (A,C))
\
{
{} }) by
A29,
XBOOLE_0:def 5;
then
A33: (I
/\ K)
in (A
'/\' C) by
PARTIT1:def 4;
GG
c= L by
A6,
BVFUNC11: 3;
then K
meets L by
A31,
A30,
XBOOLE_0: 3;
then K
= L by
EQREL_1: 41;
then
A34: z
in K by
EQREL_1:def 6;
z
in I by
EQREL_1:def 6;
then
A35: z
in (I
/\ K) by
A34,
XBOOLE_0:def 4;
(
CompF (B,G))
= (A
'/\' C) by
A2,
A3,
A4,
Th5;
then
A36: (I
/\ K)
= H or (I
/\ K)
misses H by
A33,
EQREL_1:def 4;
GG
= (
EqClass (u,(
CompF (A,G)))) by
A2,
A3,
A5,
Th4;
hence thesis by
A32,
A31,
A35,
A21,
A36,
XBOOLE_0: 3;
end;
theorem ::
BVFUNC14:21
Th21: G
=
{A, B, C, D, E} & A
<> B & A
<> C & A
<> D & A
<> E implies (
CompF (A,G))
= (((B
'/\' C)
'/\' D)
'/\' E)
proof
assume that
A1: G
=
{A, B, C, D, E} and
A2: A
<> B and
A3: A
<> C and
A4: A
<> D and
A5: A
<> E;
per cases ;
suppose
A6: B
= C;
then G
= (
{A, B, B, D}
\/
{E}) by
A1,
ENUMSET1: 10
.= (
{B, B, A, D}
\/
{E}) by
ENUMSET1: 67
.=
{B, B, A, D, E} by
ENUMSET1: 10
.=
{B, A, D, E} by
ENUMSET1: 32
.=
{A, B, D, E} by
ENUMSET1: 65;
hence (
CompF (A,G))
= ((B
'/\' D)
'/\' E) by
A2,
A4,
A5,
Th7
.= (((B
'/\' C)
'/\' D)
'/\' E) by
A6,
PARTIT1: 13;
end;
suppose
A7: B
= D;
then G
= (
{A, B, C, B}
\/
{E}) by
A1,
ENUMSET1: 10
.= (
{B, B, A, C}
\/
{E}) by
ENUMSET1: 69
.=
{B, B, A, C, E} by
ENUMSET1: 10
.=
{B, A, C, E} by
ENUMSET1: 32
.=
{A, B, C, E} by
ENUMSET1: 65;
hence (
CompF (A,G))
= ((B
'/\' C)
'/\' E) by
A2,
A3,
A5,
Th7
.= (((B
'/\' D)
'/\' C)
'/\' E) by
A7,
PARTIT1: 13
.= (((B
'/\' C)
'/\' D)
'/\' E) by
PARTIT1: 14;
end;
suppose
A8: B
= E;
then G
= (
{A}
\/
{B, C, D, B}) by
A1,
ENUMSET1: 7
.= (
{A}
\/
{B, B, C, D}) by
ENUMSET1: 63
.= (
{A}
\/
{B, C, D}) by
ENUMSET1: 31
.=
{A, B, C, D} by
ENUMSET1: 4;
hence (
CompF (A,G))
= ((B
'/\' C)
'/\' D) by
A2,
A3,
A4,
Th7
.= (((B
'/\' E)
'/\' C)
'/\' D) by
A8,
PARTIT1: 13
.= ((B
'/\' E)
'/\' (C
'/\' D)) by
PARTIT1: 14
.= ((B
'/\' (C
'/\' D))
'/\' E) by
PARTIT1: 14
.= (((B
'/\' C)
'/\' D)
'/\' E) by
PARTIT1: 14;
end;
suppose
A9: C
= D;
then G
= (
{A, B, C, C}
\/
{E}) by
A1,
ENUMSET1: 10
.= (
{C, C, A, B}
\/
{E}) by
ENUMSET1: 73
.= (
{C, A, B}
\/
{E}) by
ENUMSET1: 31
.=
{C, A, B, E} by
ENUMSET1: 6
.=
{A, B, C, E} by
ENUMSET1: 67;
hence (
CompF (A,G))
= ((B
'/\' C)
'/\' E) by
A2,
A3,
A5,
Th7
.= ((B
'/\' (C
'/\' D))
'/\' E) by
A9,
PARTIT1: 13
.= (((B
'/\' C)
'/\' D)
'/\' E) by
PARTIT1: 14;
end;
suppose
A10: C
= E;
then G
= (
{A}
\/
{B, C, D, C}) by
A1,
ENUMSET1: 7
.= (
{A}
\/
{C, C, B, D}) by
ENUMSET1: 72
.= (
{A}
\/
{C, B, D}) by
ENUMSET1: 31
.=
{A, C, B, D} by
ENUMSET1: 4
.=
{A, B, C, D} by
ENUMSET1: 62;
hence (
CompF (A,G))
= ((B
'/\' C)
'/\' D) by
A2,
A3,
A4,
Th7
.= ((B
'/\' (C
'/\' E))
'/\' D) by
A10,
PARTIT1: 13
.= (B
'/\' ((C
'/\' E)
'/\' D)) by
PARTIT1: 14
.= (B
'/\' ((C
'/\' D)
'/\' E)) by
PARTIT1: 14
.= ((B
'/\' (C
'/\' D))
'/\' E) by
PARTIT1: 14
.= (((B
'/\' C)
'/\' D)
'/\' E) by
PARTIT1: 14;
end;
suppose
A11: D
= E;
then G
= (
{A}
\/
{B, C, D, D}) by
A1,
ENUMSET1: 7
.= (
{A}
\/
{D, D, B, C}) by
ENUMSET1: 73
.= (
{A}
\/
{D, B, C}) by
ENUMSET1: 31
.=
{A, D, B, C} by
ENUMSET1: 4
.=
{A, B, C, D} by
ENUMSET1: 63;
hence (
CompF (A,G))
= ((B
'/\' C)
'/\' D) by
A2,
A3,
A4,
Th7
.= ((B
'/\' C)
'/\' (D
'/\' E)) by
A11,
PARTIT1: 13
.= (B
'/\' (C
'/\' (D
'/\' E))) by
PARTIT1: 14
.= (B
'/\' ((C
'/\' D)
'/\' E)) by
PARTIT1: 14
.= ((B
'/\' (C
'/\' D))
'/\' E) by
PARTIT1: 14
.= (((B
'/\' C)
'/\' D)
'/\' E) by
PARTIT1: 14;
end;
suppose
A12: B
<> C & B
<> D & B
<> E & C
<> D & C
<> E & D
<> E;
A13: ( not D
in
{A}) & not E
in
{A} by
A4,
A5,
TARSKI:def 1;
A14: not B
in
{A} by
A2,
TARSKI:def 1;
(G
\
{A})
= ((
{A}
\/
{B, C, D, E})
\
{A}) by
A1,
ENUMSET1: 7;
then
A15: (G
\
{A})
= ((
{A}
\
{A})
\/ (
{B, C, D, E}
\
{A})) by
XBOOLE_1: 42;
A16: not C
in
{A} by
A3,
TARSKI:def 1;
A
in
{A} by
TARSKI:def 1;
then
A17: (
{A}
\
{A})
=
{} by
ZFMISC_1: 60;
(
{B, C, D, E}
\
{A})
= ((
{B}
\/
{C, D, E})
\
{A}) by
ENUMSET1: 4
.= ((
{B}
\
{A})
\/ (
{C, D, E}
\
{A})) by
XBOOLE_1: 42
.= (
{B}
\/ (
{C, D, E}
\
{A})) by
A14,
ZFMISC_1: 59
.= (
{B}
\/ ((
{C}
\/
{D, E})
\
{A})) by
ENUMSET1: 2
.= (
{B}
\/ ((
{C}
\
{A})
\/ (
{D, E}
\
{A}))) by
XBOOLE_1: 42
.= (
{B}
\/ ((
{C}
\
{A})
\/
{D, E})) by
A13,
ZFMISC_1: 63
.= (
{B}
\/ (
{C}
\/
{D, E})) by
A16,
ZFMISC_1: 59
.= (
{B}
\/
{C, D, E}) by
ENUMSET1: 2;
then
A18: (G
\
{A})
= ((
{A}
\
{A})
\/
{B, C, D, E}) by
A15,
ENUMSET1: 4;
A19: (((B
'/\' C)
'/\' D)
'/\' E)
c= (
'/\' (G
\
{A}))
proof
let x be
object;
reconsider xx = x as
set by
TARSKI: 1;
assume
A20: x
in (((B
'/\' C)
'/\' D)
'/\' E);
then
A21: x
<>
{} by
EQREL_1:def 4;
x
in ((
INTERSECTION (((B
'/\' C)
'/\' D),E))
\
{
{} }) by
A20,
PARTIT1:def 4;
then
consider bcd,e be
set such that
A22: bcd
in ((B
'/\' C)
'/\' D) and
A23: e
in E and
A24: x
= (bcd
/\ e) by
SETFAM_1:def 5;
bcd
in ((
INTERSECTION ((B
'/\' C),D))
\
{
{} }) by
A22,
PARTIT1:def 4;
then
consider bc,d be
set such that
A25: bc
in (B
'/\' C) and
A26: d
in D and
A27: bcd
= (bc
/\ d) by
SETFAM_1:def 5;
bc
in ((
INTERSECTION (B,C))
\
{
{} }) by
A25,
PARTIT1:def 4;
then
consider b,c be
set such that
A28: b
in B and
A29: c
in C and
A30: bc
= (b
/\ c) by
SETFAM_1:def 5;
set h = ((((B
.--> b)
+* (C
.--> c))
+* (D
.--> d))
+* (E
.--> e));
A32: C
in (
dom (C
.--> c)) by
TARSKI:def 1;
A34: D
in (
dom (D
.--> d)) by
TARSKI:def 1;
A35: not C
in (
dom (D
.--> d)) by
A12,
TARSKI:def 1;
E
in (
dom (E
.--> e)) by
TARSKI:def 1;
then
A37: (h
. E)
= ((E
.--> e)
. E) by
FUNCT_4: 13;
then
A38: (h
. E)
= e by
FUNCOP_1: 72;
not C
in (
dom (E
.--> e)) by
A12,
TARSKI:def 1;
then (h
. C)
= ((((B
.--> b)
+* (C
.--> c))
+* (D
.--> d))
. C) by
FUNCT_4: 11;
then (h
. C)
= (((B
.--> b)
+* (C
.--> c))
. C) by
A35,
FUNCT_4: 11;
then
A39: (h
. C)
= ((C
.--> c)
. C) by
A32,
FUNCT_4: 13;
then
A40: (h
. C)
= c by
FUNCOP_1: 72;
not D
in (
dom (E
.--> e)) by
A12,
TARSKI:def 1;
then (h
. D)
= ((((B
.--> b)
+* (C
.--> c))
+* (D
.--> d))
. D) by
FUNCT_4: 11;
then
A41: (h
. D)
= ((D
.--> d)
. D) by
A34,
FUNCT_4: 13;
then
A42: (h
. D)
= d by
FUNCOP_1: 72;
A43: not B
in (
dom (C
.--> c)) by
A12,
TARSKI:def 1;
A44: not B
in (
dom (D
.--> d)) by
A12,
TARSKI:def 1;
not B
in (
dom (E
.--> e)) by
A12,
TARSKI:def 1;
then (h
. B)
= ((((B
.--> b)
+* (C
.--> c))
+* (D
.--> d))
. B) by
FUNCT_4: 11;
then (h
. B)
= (((B
.--> b)
+* (C
.--> c))
. B) by
A44,
FUNCT_4: 11;
then
A45: (h
. B)
= ((B
.--> b)
. B) by
A43,
FUNCT_4: 11;
then
A46: (h
. B)
= b by
FUNCOP_1: 72;
A47: for p be
set st p
in (G
\
{A}) holds (h
. p)
in p
proof
let p be
set;
assume
A48: p
in (G
\
{A});
now
per cases by
A15,
A17,
A48,
ENUMSET1:def 2;
case p
= D;
hence thesis by
A26,
A41,
FUNCOP_1: 72;
end;
case p
= B;
hence thesis by
A28,
A45,
FUNCOP_1: 72;
end;
case p
= C;
hence thesis by
A29,
A39,
FUNCOP_1: 72;
end;
case p
= E;
hence thesis by
A23,
A37,
FUNCOP_1: 72;
end;
end;
hence thesis;
end;
(
dom ((B
.--> b)
+* (C
.--> c)))
= ((
dom (B
.--> b))
\/ (
dom (C
.--> c))) by
FUNCT_4:def 1;
then (
dom (((B
.--> b)
+* (C
.--> c))
+* (D
.--> d)))
= (((
dom (B
.--> b))
\/ (
dom (C
.--> c)))
\/ (
dom (D
.--> d))) by
FUNCT_4:def 1;
then
A49: (
dom ((((B
.--> b)
+* (C
.--> c))
+* (D
.--> d))
+* (E
.--> e)))
= ((((
dom (B
.--> b))
\/ (
dom (C
.--> c)))
\/ (
dom (D
.--> d)))
\/ (
dom (E
.--> e))) by
FUNCT_4:def 1;
A50: (
dom ((((B
.--> b)
+* (C
.--> c))
+* (D
.--> d))
+* (E
.--> e)))
= (((
{B}
\/
{C})
\/
{D})
\/
{E}) by
A49
.= ((
{B, C}
\/
{D})
\/
{E}) by
ENUMSET1: 1
.= (
{B, C, D}
\/
{E}) by
ENUMSET1: 3
.=
{B, C, D, E} by
ENUMSET1: 6;
then
A51: D
in (
dom h) by
ENUMSET1:def 2;
A52: (
rng h)
c=
{(h
. D), (h
. B), (h
. C), (h
. E)}
proof
let t be
object;
assume t
in (
rng h);
then
consider x1 be
object such that
A53: x1
in (
dom h) and
A54: t
= (h
. x1) by
FUNCT_1:def 3;
now
per cases by
A50,
A53,
ENUMSET1:def 2;
case x1
= D;
hence thesis by
A54,
ENUMSET1:def 2;
end;
case x1
= B;
hence thesis by
A54,
ENUMSET1:def 2;
end;
case x1
= C;
hence thesis by
A54,
ENUMSET1:def 2;
end;
case x1
= E;
hence thesis by
A54,
ENUMSET1:def 2;
end;
end;
hence thesis;
end;
(
rng h)
c= (
bool Y)
proof
let t be
object;
assume
A55: t
in (
rng h);
now
per cases by
A52,
A55,
ENUMSET1:def 2;
case t
= (h
. D);
hence thesis by
A26,
A42;
end;
case t
= (h
. B);
hence thesis by
A28,
A46;
end;
case t
= (h
. C);
hence thesis by
A29,
A40;
end;
case t
= (h
. E);
hence thesis by
A23,
A38;
end;
end;
hence thesis;
end;
then
reconsider F = (
rng h) as
Subset-Family of Y;
A56: C
in (
dom h) by
A50,
ENUMSET1:def 2;
A57: E
in (
dom h) by
A50,
ENUMSET1:def 2;
A58: B
in (
dom h) by
A50,
ENUMSET1:def 2;
A59:
{(h
. D), (h
. B), (h
. C), (h
. E)}
c= (
rng h)
proof
let t be
object;
assume
A60: t
in
{(h
. D), (h
. B), (h
. C), (h
. E)};
now
per cases by
A60,
ENUMSET1:def 2;
case t
= (h
. D);
hence thesis by
A51,
FUNCT_1:def 3;
end;
case t
= (h
. B);
hence thesis by
A58,
FUNCT_1:def 3;
end;
case t
= (h
. C);
hence thesis by
A56,
FUNCT_1:def 3;
end;
case t
= (h
. E);
hence thesis by
A57,
FUNCT_1:def 3;
end;
end;
hence thesis;
end;
then
A61:
{(h
. D), (h
. B), (h
. C), (h
. E)}
= (
rng h) by
A52,
XBOOLE_0:def 10;
reconsider h as
Function;
A62: xx
c= (
Intersect F)
proof
let u be
object;
A63: (h
. D)
in
{(h
. D), (h
. B), (h
. C), (h
. E)} by
ENUMSET1:def 2;
assume
A64: u
in xx;
for y be
set holds y
in F implies u
in y
proof
let y be
set;
assume
A65: y
in F;
now
per cases by
A52,
A65,
ENUMSET1:def 2;
case
A66: y
= (h
. D);
u
in (d
/\ ((b
/\ c)
/\ e)) by
A24,
A27,
A30,
A64,
XBOOLE_1: 16;
hence thesis by
A42,
A66,
XBOOLE_0:def 4;
end;
case
A67: y
= (h
. B);
u
in ((c
/\ (d
/\ b))
/\ e) by
A24,
A27,
A30,
A64,
XBOOLE_1: 16;
then u
in (c
/\ ((d
/\ b)
/\ e)) by
XBOOLE_1: 16;
then u
in (c
/\ ((d
/\ e)
/\ b)) by
XBOOLE_1: 16;
then u
in ((c
/\ (d
/\ e))
/\ b) by
XBOOLE_1: 16;
hence thesis by
A46,
A67,
XBOOLE_0:def 4;
end;
case
A68: y
= (h
. C);
u
in ((c
/\ (b
/\ d))
/\ e) by
A24,
A27,
A30,
A64,
XBOOLE_1: 16;
then u
in (c
/\ ((b
/\ d)
/\ e)) by
XBOOLE_1: 16;
hence thesis by
A40,
A68,
XBOOLE_0:def 4;
end;
case y
= (h
. E);
hence thesis by
A24,
A38,
A64,
XBOOLE_0:def 4;
end;
end;
hence thesis;
end;
then u
in (
meet F) by
A59,
A63,
SETFAM_1:def 1;
hence thesis by
A59,
A63,
SETFAM_1:def 9;
end;
A69: (
rng h)
<>
{} by
A51,
FUNCT_1: 3;
(
Intersect F)
c= xx
proof
let t be
object;
assume t
in (
Intersect F);
then
A70: t
in (
meet (
rng h)) by
A69,
SETFAM_1:def 9;
(h
. D)
in (
rng h) by
A61,
ENUMSET1:def 2;
then
A71: t
in (h
. D) by
A70,
SETFAM_1:def 1;
(h
. C)
in (
rng h) by
A61,
ENUMSET1:def 2;
then
A72: t
in (h
. C) by
A70,
SETFAM_1:def 1;
(h
. B)
in (
rng h) by
A61,
ENUMSET1:def 2;
then t
in (h
. B) by
A70,
SETFAM_1:def 1;
then t
in (b
/\ c) by
A46,
A40,
A72,
XBOOLE_0:def 4;
then
A73: t
in ((b
/\ c)
/\ d) by
A42,
A71,
XBOOLE_0:def 4;
(h
. E)
in (
rng h) by
A61,
ENUMSET1:def 2;
then t
in (h
. E) by
A70,
SETFAM_1:def 1;
hence thesis by
A24,
A27,
A30,
A38,
A73,
XBOOLE_0:def 4;
end;
then x
= (
Intersect F) by
A62,
XBOOLE_0:def 10;
hence thesis by
A18,
A17,
A50,
A47,
A21,
BVFUNC_2:def 1;
end;
(
'/\' (G
\
{A}))
c= (((B
'/\' C)
'/\' D)
'/\' E)
proof
let x be
object;
reconsider xx = x as
set by
TARSKI: 1;
assume x
in (
'/\' (G
\
{A}));
then
consider h be
Function, F be
Subset-Family of Y such that
A74: (
dom h)
= (G
\
{A}) and
A75: (
rng h)
= F and
A76: for d be
set st d
in (G
\
{A}) holds (h
. d)
in d and
A77: x
= (
Intersect F) and
A78: x
<>
{} by
BVFUNC_2:def 1;
D
in (
dom h) by
A18,
A17,
A74,
ENUMSET1:def 2;
then
A79: (h
. D)
in (
rng h) by
FUNCT_1:def 3;
set mbc = ((h
. B)
/\ (h
. C));
A80: not x
in
{
{} } by
A78,
TARSKI:def 1;
E
in (G
\
{A}) by
A18,
A17,
ENUMSET1:def 2;
then
A81: (h
. E)
in E by
A76;
D
in (G
\
{A}) by
A18,
A17,
ENUMSET1:def 2;
then
A82: (h
. D)
in D by
A76;
C
in (G
\
{A}) by
A18,
A17,
ENUMSET1:def 2;
then
A83: (h
. C)
in C by
A76;
E
in (
dom h) by
A18,
A17,
A74,
ENUMSET1:def 2;
then
A84: (h
. E)
in (
rng h) by
FUNCT_1:def 3;
set mbcd = (((h
. B)
/\ (h
. C))
/\ (h
. D));
B
in (
dom h) by
A18,
A17,
A74,
ENUMSET1:def 2;
then
A85: (h
. B)
in (
rng h) by
FUNCT_1:def 3;
C
in (
dom h) by
A18,
A17,
A74,
ENUMSET1:def 2;
then
A86: (h
. C)
in (
rng h) by
FUNCT_1:def 3;
A87: xx
c= ((((h
. B)
/\ (h
. C))
/\ (h
. D))
/\ (h
. E))
proof
let m be
object;
assume m
in xx;
then
A88: m
in (
meet (
rng h)) by
A75,
A77,
A85,
SETFAM_1:def 9;
then m
in (h
. B) & m
in (h
. C) by
A85,
A86,
SETFAM_1:def 1;
then
A89: m
in ((h
. B)
/\ (h
. C)) by
XBOOLE_0:def 4;
m
in (h
. D) by
A79,
A88,
SETFAM_1:def 1;
then
A90: m
in (((h
. B)
/\ (h
. C))
/\ (h
. D)) by
A89,
XBOOLE_0:def 4;
m
in (h
. E) by
A84,
A88,
SETFAM_1:def 1;
hence thesis by
A90,
XBOOLE_0:def 4;
end;
then mbcd
<>
{} by
A78;
then
A91: not mbcd
in
{
{} } by
TARSKI:def 1;
mbc
<>
{} by
A78,
A87;
then
A92: not mbc
in
{
{} } by
TARSKI:def 1;
B
in (G
\
{A}) by
A18,
A17,
ENUMSET1:def 2;
then (h
. B)
in B by
A76;
then mbc
in (
INTERSECTION (B,C)) by
A83,
SETFAM_1:def 5;
then mbc
in ((
INTERSECTION (B,C))
\
{
{} }) by
A92,
XBOOLE_0:def 5;
then mbc
in (B
'/\' C) by
PARTIT1:def 4;
then mbcd
in (
INTERSECTION ((B
'/\' C),D)) by
A82,
SETFAM_1:def 5;
then mbcd
in ((
INTERSECTION ((B
'/\' C),D))
\
{
{} }) by
A91,
XBOOLE_0:def 5;
then
A93: mbcd
in ((B
'/\' C)
'/\' D) by
PARTIT1:def 4;
((((h
. B)
/\ (h
. C))
/\ (h
. D))
/\ (h
. E))
c= xx
proof
let m be
object;
assume
A94: m
in ((((h
. B)
/\ (h
. C))
/\ (h
. D))
/\ (h
. E));
then
A95: m
in (((h
. B)
/\ (h
. C))
/\ (h
. D)) by
XBOOLE_0:def 4;
then
A96: m
in ((h
. B)
/\ (h
. C)) by
XBOOLE_0:def 4;
A97: (
rng h)
c=
{(h
. B), (h
. C), (h
. D), (h
. E)}
proof
let u be
object;
assume u
in (
rng h);
then
consider x1 be
object such that
A98: x1
in (
dom h) and
A99: u
= (h
. x1) by
FUNCT_1:def 3;
now
per cases by
A15,
A17,
A74,
A98,
ENUMSET1:def 2;
case x1
= B;
hence thesis by
A99,
ENUMSET1:def 2;
end;
case x1
= C;
hence thesis by
A99,
ENUMSET1:def 2;
end;
case x1
= D;
hence thesis by
A99,
ENUMSET1:def 2;
end;
case x1
= E;
hence thesis by
A99,
ENUMSET1:def 2;
end;
end;
hence thesis;
end;
for y be
set holds y
in (
rng h) implies m
in y
proof
let y be
set;
assume
A100: y
in (
rng h);
now
per cases by
A97,
A100,
ENUMSET1:def 2;
case y
= (h
. B);
hence thesis by
A96,
XBOOLE_0:def 4;
end;
case y
= (h
. C);
hence thesis by
A96,
XBOOLE_0:def 4;
end;
case y
= (h
. D);
hence thesis by
A95,
XBOOLE_0:def 4;
end;
case y
= (h
. E);
hence thesis by
A94,
XBOOLE_0:def 4;
end;
end;
hence thesis;
end;
then m
in (
meet (
rng h)) by
A85,
SETFAM_1:def 1;
hence thesis by
A75,
A77,
A85,
SETFAM_1:def 9;
end;
then ((((h
. B)
/\ (h
. C))
/\ (h
. D))
/\ (h
. E))
= x by
A87,
XBOOLE_0:def 10;
then x
in (
INTERSECTION (((B
'/\' C)
'/\' D),E)) by
A81,
A93,
SETFAM_1:def 5;
then x
in ((
INTERSECTION (((B
'/\' C)
'/\' D),E))
\
{
{} }) by
A80,
XBOOLE_0:def 5;
hence thesis by
PARTIT1:def 4;
end;
then (
'/\' (G
\
{A}))
= (((B
'/\' C)
'/\' D)
'/\' E) by
A19,
XBOOLE_0:def 10;
hence thesis by
BVFUNC_2:def 7;
end;
end;
theorem ::
BVFUNC14:22
Th22: G
=
{A, B, C, D, E} & A
<> B & B
<> C & B
<> D & B
<> E implies (
CompF (B,G))
= (((A
'/\' C)
'/\' D)
'/\' E)
proof
assume that
A1: G
=
{A, B, C, D, E} and
A2: A
<> B & B
<> C & B
<> D & B
<> E;
{A, B, C, D, E}
= (
{A, B}
\/
{C, D, E}) by
ENUMSET1: 8;
then G
=
{B, A, C, D, E} by
A1,
ENUMSET1: 8;
hence thesis by
A2,
Th21;
end;
theorem ::
BVFUNC14:23
Th23: G
=
{A, B, C, D, E} & A
<> C & B
<> C & C
<> D & C
<> E implies (
CompF (C,G))
= (((A
'/\' B)
'/\' D)
'/\' E)
proof
assume that
A1: G
=
{A, B, C, D, E} and
A2: A
<> C & B
<> C & C
<> D & C
<> E;
{A, B, C, D, E}
= (
{A, B, C}
\/
{D, E}) by
ENUMSET1: 9;
then
{A, B, C, D, E}
= ((
{A}
\/
{B, C})
\/
{D, E}) by
ENUMSET1: 2;
then
{A, B, C, D, E}
= (
{A, C, B}
\/
{D, E}) by
ENUMSET1: 2;
then
{A, B, C, D, E}
= ((
{A, C}
\/
{B})
\/
{D, E}) by
ENUMSET1: 3;
then
{A, B, C, D, E}
= (
{C, A, B}
\/
{D, E}) by
ENUMSET1: 3;
then G
=
{C, A, B, D, E} by
A1,
ENUMSET1: 9;
hence thesis by
A2,
Th21;
end;
theorem ::
BVFUNC14:24
Th24: G
=
{A, B, C, D, E} & A
<> D & B
<> D & C
<> D & D
<> E implies (
CompF (D,G))
= (((A
'/\' B)
'/\' C)
'/\' E)
proof
assume that
A1: G
=
{A, B, C, D, E} and
A2: A
<> D & B
<> D & C
<> D & D
<> E;
{A, B, C, D, E}
= (
{A, B}
\/
{C, D, E}) by
ENUMSET1: 8;
then
{A, B, C, D, E}
= (
{A, B}
\/ (
{C, D}
\/
{E})) by
ENUMSET1: 3;
then
{A, B, C, D, E}
= (
{A, B}
\/
{D, C, E}) by
ENUMSET1: 3;
then G
=
{A, B, D, C, E} by
A1,
ENUMSET1: 8;
hence thesis by
A2,
Th23;
end;
theorem ::
BVFUNC14:25
G
=
{A, B, C, D, E} & A
<> E & B
<> E & C
<> E & D
<> E implies (
CompF (E,G))
= (((A
'/\' B)
'/\' C)
'/\' D)
proof
assume that
A1: G
=
{A, B, C, D, E} and
A2: A
<> E & B
<> E & C
<> E & D
<> E;
{A, B, C, D, E}
= (
{A, B, C}
\/
{D, E}) by
ENUMSET1: 9;
then G
=
{A, B, C, E, D} by
A1,
ENUMSET1: 9;
hence thesis by
A2,
Th24;
end;
theorem ::
BVFUNC14:26
Th26: for A,B,C,D,E be
set, h be
Function, A9,B9,C9,D9,E9 be
set st A
<> B & A
<> C & A
<> D & A
<> E & B
<> C & B
<> D & B
<> E & C
<> D & C
<> E & D
<> E & h
= (((((B
.--> B9)
+* (C
.--> C9))
+* (D
.--> D9))
+* (E
.--> E9))
+* (A
.--> A9)) holds (h
. A)
= A9 & (h
. B)
= B9 & (h
. C)
= C9 & (h
. D)
= D9 & (h
. E)
= E9
proof
let A,B,C,D,E be
set;
let h be
Function;
let A9,B9,C9,D9,E9 be
set;
assume that
A1: A
<> B and
A2: A
<> C and
A3: A
<> D and
A4: A
<> E and
A5: B
<> C and
A6: B
<> D and
A7: B
<> E and
A8: C
<> D and
A9: C
<> E and
A10: D
<> E and
A11: h
= (((((B
.--> B9)
+* (C
.--> C9))
+* (D
.--> D9))
+* (E
.--> E9))
+* (A
.--> A9));
A
in (
dom (A
.--> A9)) by
TARSKI:def 1;
then
A13: (h
. A)
= ((A
.--> A9)
. A) by
A11,
FUNCT_4: 13;
not C
in (
dom (A
.--> A9)) by
A2,
TARSKI:def 1;
then
A14: (h
. C)
= (((((B
.--> B9)
+* (C
.--> C9))
+* (D
.--> D9))
+* (E
.--> E9))
. C) by
A11,
FUNCT_4: 11;
not B
in (
dom (D
.--> D9)) by
A6,
TARSKI:def 1;
then
A16: ((((B
.--> B9)
+* (C
.--> C9))
+* (D
.--> D9))
. B)
= (((B
.--> B9)
+* (C
.--> C9))
. B) by
FUNCT_4: 11;
not E
in (
dom (A
.--> A9)) by
A4,
TARSKI:def 1;
then
A17: (h
. E)
= (((((B
.--> B9)
+* (C
.--> C9))
+* (D
.--> D9))
+* (E
.--> E9))
. E) by
A11,
FUNCT_4: 11;
E
in (
dom (E
.--> E9)) by
TARSKI:def 1;
then
A19: (h
. E)
= ((E
.--> E9)
. E) by
A17,
FUNCT_4: 13;
not C
in (
dom (D
.--> D9)) by
A8,
TARSKI:def 1;
then
A20: ((((B
.--> B9)
+* (C
.--> C9))
+* (D
.--> D9))
. C)
= (((B
.--> B9)
+* (C
.--> C9))
. C) by
FUNCT_4: 11;
not C
in (
dom (E
.--> E9)) by
A9,
TARSKI:def 1;
then
A21: (h
. C)
= ((((B
.--> B9)
+* (C
.--> C9))
+* (D
.--> D9))
. C) by
A14,
FUNCT_4: 11;
C
in (
dom (C
.--> C9)) by
TARSKI:def 1;
then
A23: (h
. C)
= ((C
.--> C9)
. C) by
A21,
A20,
FUNCT_4: 13;
not D
in (
dom (A
.--> A9)) by
A3,
TARSKI:def 1;
then
A24: (h
. D)
= (((((B
.--> B9)
+* (C
.--> C9))
+* (D
.--> D9))
+* (E
.--> E9))
. D) by
A11,
FUNCT_4: 11;
not D
in (
dom (E
.--> E9)) by
A10,
TARSKI:def 1;
then
A25: (h
. D)
= ((((B
.--> B9)
+* (C
.--> C9))
+* (D
.--> D9))
. D) by
A24,
FUNCT_4: 11;
D
in (
dom (D
.--> D9)) by
TARSKI:def 1;
then
A26: (h
. D)
= ((D
.--> D9)
. D) by
A25,
FUNCT_4: 13;
not B
in (
dom (A
.--> A9)) by
A1,
TARSKI:def 1;
then
A27: (h
. B)
= (((((B
.--> B9)
+* (C
.--> C9))
+* (D
.--> D9))
+* (E
.--> E9))
. B) by
A11,
FUNCT_4: 11;
not B
in (
dom (E
.--> E9)) by
A7,
TARSKI:def 1;
then
A28: (h
. B)
= ((((B
.--> B9)
+* (C
.--> C9))
+* (D
.--> D9))
. B) by
A27,
FUNCT_4: 11;
not B
in (
dom (C
.--> C9)) by
A5,
TARSKI:def 1;
then (h
. B)
= ((B
.--> B9)
. B) by
A28,
A16,
FUNCT_4: 11;
hence thesis by
A13,
A23,
A26,
A19,
FUNCOP_1: 72;
end;
theorem ::
BVFUNC14:27
Th27: for A,B,C,D,E be
set, h be
Function, A9,B9,C9,D9,E9 be
set st h
= (((((B
.--> B9)
+* (C
.--> C9))
+* (D
.--> D9))
+* (E
.--> E9))
+* (A
.--> A9)) holds (
dom h)
=
{A, B, C, D, E}
proof
let A,B,C,D,E be
set;
let h be
Function;
let A9,B9,C9,D9,E9 be
set;
assume
A1: h
= (((((B
.--> B9)
+* (C
.--> C9))
+* (D
.--> D9))
+* (E
.--> E9))
+* (A
.--> A9));
(
dom ((B
.--> B9)
+* (C
.--> C9)))
= ((
dom (B
.--> B9))
\/ (
dom (C
.--> C9))) by
FUNCT_4:def 1;
then (
dom (((B
.--> B9)
+* (C
.--> C9))
+* (D
.--> D9)))
= (((
dom (B
.--> B9))
\/ (
dom (C
.--> C9)))
\/ (
dom (D
.--> D9))) by
FUNCT_4:def 1;
then (
dom ((((B
.--> B9)
+* (C
.--> C9))
+* (D
.--> D9))
+* (E
.--> E9)))
= ((((
dom (B
.--> B9))
\/ (
dom (C
.--> C9)))
\/ (
dom (D
.--> D9)))
\/ (
dom (E
.--> E9))) by
FUNCT_4:def 1;
then
A3: (
dom (((((B
.--> B9)
+* (C
.--> C9))
+* (D
.--> D9))
+* (E
.--> E9))
+* (A
.--> A9)))
= (((((
dom (B
.--> B9))
\/ (
dom (C
.--> C9)))
\/ (
dom (D
.--> D9)))
\/ (
dom (E
.--> E9)))
\/ (
dom (A
.--> A9))) by
FUNCT_4:def 1;
(
dom (((((B
.--> B9)
+* (C
.--> C9))
+* (D
.--> D9))
+* (E
.--> E9))
+* (A
.--> A9)))
= (
{A}
\/ (((
{B}
\/
{C})
\/
{D})
\/
{E})) by
A3
.= (
{A}
\/ ((
{B, C}
\/
{D})
\/
{E})) by
ENUMSET1: 1
.= (
{A}
\/ (
{B, C, D}
\/
{E})) by
ENUMSET1: 3
.= (
{A}
\/
{B, C, D, E}) by
ENUMSET1: 6
.=
{A, B, C, D, E} by
ENUMSET1: 7;
hence thesis by
A1;
end;
theorem ::
BVFUNC14:28
Th28: for A,B,C,D,E be
set, h be
Function, A9,B9,C9,D9,E9 be
set st h
= (((((B
.--> B9)
+* (C
.--> C9))
+* (D
.--> D9))
+* (E
.--> E9))
+* (A
.--> A9)) holds (
rng h)
=
{(h
. A), (h
. B), (h
. C), (h
. D), (h
. E)}
proof
let A,B,C,D,E be
set;
let h be
Function;
let A9,B9,C9,D9,E9 be
set;
assume h
= (((((B
.--> B9)
+* (C
.--> C9))
+* (D
.--> D9))
+* (E
.--> E9))
+* (A
.--> A9));
then
A1: (
dom h)
=
{A, B, C, D, E} by
Th27;
then
A2: B
in (
dom h) by
ENUMSET1:def 3;
A3: D
in (
dom h) by
A1,
ENUMSET1:def 3;
A4: C
in (
dom h) by
A1,
ENUMSET1:def 3;
A5: (
rng h)
c=
{(h
. A), (h
. B), (h
. C), (h
. D), (h
. E)}
proof
let t be
object;
assume t
in (
rng h);
then
consider x1 be
object such that
A6: x1
in (
dom h) and
A7: t
= (h
. x1) by
FUNCT_1:def 3;
now
per cases by
A1,
A6,
ENUMSET1:def 3;
case x1
= A;
hence thesis by
A7,
ENUMSET1:def 3;
end;
case x1
= B;
hence thesis by
A7,
ENUMSET1:def 3;
end;
case x1
= C;
hence thesis by
A7,
ENUMSET1:def 3;
end;
case x1
= D;
hence thesis by
A7,
ENUMSET1:def 3;
end;
case x1
= E;
hence thesis by
A7,
ENUMSET1:def 3;
end;
end;
hence thesis;
end;
A8: E
in (
dom h) by
A1,
ENUMSET1:def 3;
A9: A
in (
dom h) by
A1,
ENUMSET1:def 3;
{(h
. A), (h
. B), (h
. C), (h
. D), (h
. E)}
c= (
rng h)
proof
let t be
object;
assume
A10: t
in
{(h
. A), (h
. B), (h
. C), (h
. D), (h
. E)};
now
per cases by
A10,
ENUMSET1:def 3;
case t
= (h
. A);
hence thesis by
A9,
FUNCT_1:def 3;
end;
case t
= (h
. B);
hence thesis by
A2,
FUNCT_1:def 3;
end;
case t
= (h
. C);
hence thesis by
A4,
FUNCT_1:def 3;
end;
case t
= (h
. D);
hence thesis by
A3,
FUNCT_1:def 3;
end;
case t
= (h
. E);
hence thesis by
A8,
FUNCT_1:def 3;
end;
end;
hence thesis;
end;
hence thesis by
A5,
XBOOLE_0:def 10;
end;
theorem ::
BVFUNC14:29
for G be
Subset of (
PARTITIONS Y), A,B,C,D,E be
a_partition of Y, z,u be
Element of Y, h be
Function st G is
independent & G
=
{A, B, C, D, E} & A
<> B & A
<> C & A
<> D & A
<> E & B
<> C & B
<> D & B
<> E & C
<> D & C
<> E & D
<> E holds (
EqClass (u,(((B
'/\' C)
'/\' D)
'/\' E)))
meets (
EqClass (z,A))
proof
let G be
Subset of (
PARTITIONS Y);
let A,B,C,D,E be
a_partition of Y;
let z,u be
Element of Y;
let h be
Function;
assume that
A1: G is
independent and
A2: G
=
{A, B, C, D, E} and
A3: A
<> B & A
<> C & A
<> D & A
<> E & B
<> C & B
<> D & B
<> E & C
<> D & C
<> E & D
<> E;
set h = (((((B
.--> (
EqClass (u,B)))
+* (C
.--> (
EqClass (u,C))))
+* (D
.--> (
EqClass (u,D))))
+* (E
.--> (
EqClass (u,E))))
+* (A
.--> (
EqClass (z,A))));
A4: (h
. B)
= (
EqClass (u,B)) by
A3,
Th26;
A5: (h
. D)
= (
EqClass (u,D)) by
A3,
Th26;
A6: (h
. C)
= (
EqClass (u,C)) by
A3,
Th26;
A7: (h
. E)
= (
EqClass (u,E)) by
A3,
Th26;
A8: (
rng h)
=
{(h
. A), (h
. B), (h
. C), (h
. D), (h
. E)} by
Th28;
(
rng h)
c= (
bool Y)
proof
let t be
object;
assume
A9: t
in (
rng h);
now
per cases by
A8,
A9,
ENUMSET1:def 3;
case t
= (h
. A);
then t
= (
EqClass (z,A)) by
A3,
Th26;
hence thesis;
end;
case t
= (h
. B);
hence thesis by
A4;
end;
case t
= (h
. C);
hence thesis by
A6;
end;
case t
= (h
. D);
hence thesis by
A5;
end;
case t
= (h
. E);
hence thesis by
A7;
end;
end;
hence thesis;
end;
then
reconsider FF = (
rng h) as
Subset-Family of Y;
A10: (
dom h)
= G by
A2,
Th27;
for d be
set st d
in G holds (h
. d)
in d
proof
let d be
set;
assume
A11: d
in G;
now
per cases by
A2,
A11,
ENUMSET1:def 3;
case
A12: d
= A;
(h
. A)
= (
EqClass (z,A)) by
A3,
Th26;
hence thesis by
A12;
end;
case
A13: d
= B;
(h
. B)
= (
EqClass (u,B)) by
A3,
Th26;
hence thesis by
A13;
end;
case
A14: d
= C;
(h
. C)
= (
EqClass (u,C)) by
A3,
Th26;
hence thesis by
A14;
end;
case
A15: d
= D;
(h
. D)
= (
EqClass (u,D)) by
A3,
Th26;
hence thesis by
A15;
end;
case
A16: d
= E;
(h
. E)
= (
EqClass (u,E)) by
A3,
Th26;
hence thesis by
A16;
end;
end;
hence thesis;
end;
then (
Intersect FF)
<>
{} by
A1,
A10,
BVFUNC_2:def 5;
then
consider m be
object such that
A17: m
in (
Intersect FF) by
XBOOLE_0:def 1;
A
in (
dom h) by
A2,
A10,
ENUMSET1:def 3;
then
A18: (h
. A)
in (
rng h) by
FUNCT_1:def 3;
then
A19: m
in (
meet FF) by
A17,
SETFAM_1:def 9;
then
A20: m
in (h
. A) by
A18,
SETFAM_1:def 1;
D
in (
dom h) by
A2,
A10,
ENUMSET1:def 3;
then (h
. D)
in (
rng h) by
FUNCT_1:def 3;
then
A21: m
in (h
. D) by
A19,
SETFAM_1:def 1;
C
in (
dom h) by
A2,
A10,
ENUMSET1:def 3;
then (h
. C)
in (
rng h) by
FUNCT_1:def 3;
then
A22: m
in (h
. C) by
A19,
SETFAM_1:def 1;
B
in (
dom h) by
A2,
A10,
ENUMSET1:def 3;
then (h
. B)
in (
rng h) by
FUNCT_1:def 3;
then m
in (h
. B) by
A19,
SETFAM_1:def 1;
then m
in ((
EqClass (u,B))
/\ (
EqClass (u,C))) by
A4,
A6,
A22,
XBOOLE_0:def 4;
then
A23: m
in (((
EqClass (u,B))
/\ (
EqClass (u,C)))
/\ (
EqClass (u,D))) by
A5,
A21,
XBOOLE_0:def 4;
E
in (
dom h) by
A2,
A10,
ENUMSET1:def 3;
then (h
. E)
in (
rng h) by
FUNCT_1:def 3;
then m
in (h
. E) by
A19,
SETFAM_1:def 1;
then
A24: m
in ((((
EqClass (u,B))
/\ (
EqClass (u,C)))
/\ (
EqClass (u,D)))
/\ (
EqClass (u,E))) by
A7,
A23,
XBOOLE_0:def 4;
set GG = (
EqClass (u,(((B
'/\' C)
'/\' D)
'/\' E)));
GG
= ((
EqClass (u,((B
'/\' C)
'/\' D)))
/\ (
EqClass (u,E))) by
Th1;
then
A25: GG
= (((
EqClass (u,(B
'/\' C)))
/\ (
EqClass (u,D)))
/\ (
EqClass (u,E))) by
Th1;
(h
. A)
= (
EqClass (z,A)) by
A3,
Th26;
then m
in (((((
EqClass (u,B))
/\ (
EqClass (u,C)))
/\ (
EqClass (u,D)))
/\ (
EqClass (u,E)))
/\ (
EqClass (z,A))) by
A20,
A24,
XBOOLE_0:def 4;
then ((((
EqClass (u,B))
/\ (
EqClass (u,C)))
/\ (
EqClass (u,D)))
/\ (
EqClass (u,E)))
meets (
EqClass (z,A)) by
XBOOLE_0: 4;
hence thesis by
A25,
Th1;
end;
theorem ::
BVFUNC14:30
for G be
Subset of (
PARTITIONS Y), A,B,C,D,E be
a_partition of Y, z,u be
Element of Y st G is
independent & G
=
{A, B, C, D, E} & A
<> B & A
<> C & A
<> D & A
<> E & B
<> C & B
<> D & B
<> E & C
<> D & C
<> E & D
<> E & (
EqClass (z,((C
'/\' D)
'/\' E)))
= (
EqClass (u,((C
'/\' D)
'/\' E))) holds (
EqClass (u,(
CompF (A,G))))
meets (
EqClass (z,(
CompF (B,G))))
proof
let G be
Subset of (
PARTITIONS Y);
let A,B,C,D,E be
a_partition of Y;
let z,u be
Element of Y;
assume that
A1: G is
independent and
A2: G
=
{A, B, C, D, E} and
A3: A
<> B and
A4: A
<> C & A
<> D & A
<> E and
A5: B
<> C & B
<> D & B
<> E and
A6: C
<> D & C
<> E & D
<> E and
A7: (
EqClass (z,((C
'/\' D)
'/\' E)))
= (
EqClass (u,((C
'/\' D)
'/\' E)));
set h = (((((B
.--> (
EqClass (u,B)))
+* (C
.--> (
EqClass (u,C))))
+* (D
.--> (
EqClass (u,D))))
+* (E
.--> (
EqClass (u,E))))
+* (A
.--> (
EqClass (z,A))));
A8: (h
. B)
= (
EqClass (u,B)) by
A3,
A4,
A5,
A6,
Th26;
A9: (h
. E)
= (
EqClass (u,E)) by
A3,
A4,
A5,
A6,
Th26;
A10: (h
. D)
= (
EqClass (u,D)) by
A3,
A4,
A5,
A6,
Th26;
A11: (h
. C)
= (
EqClass (u,C)) by
A3,
A4,
A5,
A6,
Th26;
A12: (
rng h)
=
{(h
. A), (h
. B), (h
. C), (h
. D), (h
. E)} by
Th28;
(
rng h)
c= (
bool Y)
proof
let t be
object;
assume
A13: t
in (
rng h);
now
per cases by
A12,
A13,
ENUMSET1:def 3;
case t
= (h
. A);
then t
= (
EqClass (z,A)) by
A3,
A4,
A5,
A6,
Th26;
hence thesis;
end;
case t
= (h
. B);
hence thesis by
A8;
end;
case t
= (h
. C);
hence thesis by
A11;
end;
case t
= (h
. D);
hence thesis by
A10;
end;
case t
= (h
. E);
hence thesis by
A9;
end;
end;
hence thesis;
end;
then
reconsider FF = (
rng h) as
Subset-Family of Y;
A14: (
dom h)
= G by
A2,
Th27;
for d be
set st d
in G holds (h
. d)
in d
proof
let d be
set;
assume
A15: d
in G;
now
per cases by
A2,
A15,
ENUMSET1:def 3;
case
A16: d
= A;
(h
. A)
= (
EqClass (z,A)) by
A3,
A4,
A5,
A6,
Th26;
hence thesis by
A16;
end;
case
A17: d
= B;
(h
. B)
= (
EqClass (u,B)) by
A3,
A4,
A5,
A6,
Th26;
hence thesis by
A17;
end;
case
A18: d
= C;
(h
. C)
= (
EqClass (u,C)) by
A3,
A4,
A5,
A6,
Th26;
hence thesis by
A18;
end;
case
A19: d
= D;
(h
. D)
= (
EqClass (u,D)) by
A3,
A4,
A5,
A6,
Th26;
hence thesis by
A19;
end;
case
A20: d
= E;
(h
. E)
= (
EqClass (u,E)) by
A3,
A4,
A5,
A6,
Th26;
hence thesis by
A20;
end;
end;
hence thesis;
end;
then (
Intersect FF)
<>
{} by
A1,
A14,
BVFUNC_2:def 5;
then
consider m be
object such that
A21: m
in (
Intersect FF) by
XBOOLE_0:def 1;
A
in (
dom h) by
A2,
A14,
ENUMSET1:def 3;
then
A22: (h
. A)
in (
rng h) by
FUNCT_1:def 3;
then
A23: m
in (
meet FF) by
A21,
SETFAM_1:def 9;
then
A24: m
in (h
. A) by
A22,
SETFAM_1:def 1;
D
in (
dom h) by
A2,
A14,
ENUMSET1:def 3;
then (h
. D)
in (
rng h) by
FUNCT_1:def 3;
then
A25: m
in (h
. D) by
A23,
SETFAM_1:def 1;
C
in (
dom h) by
A2,
A14,
ENUMSET1:def 3;
then (h
. C)
in (
rng h) by
FUNCT_1:def 3;
then
A26: m
in (h
. C) by
A23,
SETFAM_1:def 1;
B
in (
dom h) by
A2,
A14,
ENUMSET1:def 3;
then (h
. B)
in (
rng h) by
FUNCT_1:def 3;
then m
in (h
. B) by
A23,
SETFAM_1:def 1;
then m
in ((
EqClass (u,B))
/\ (
EqClass (u,C))) by
A8,
A11,
A26,
XBOOLE_0:def 4;
then
A27: m
in (((
EqClass (u,B))
/\ (
EqClass (u,C)))
/\ (
EqClass (u,D))) by
A10,
A25,
XBOOLE_0:def 4;
E
in (
dom h) by
A2,
A14,
ENUMSET1:def 3;
then (h
. E)
in (
rng h) by
FUNCT_1:def 3;
then m
in (h
. E) by
A23,
SETFAM_1:def 1;
then
A28: m
in ((((
EqClass (u,B))
/\ (
EqClass (u,C)))
/\ (
EqClass (u,D)))
/\ (
EqClass (u,E))) by
A9,
A27,
XBOOLE_0:def 4;
set GG = (
EqClass (u,(((B
'/\' C)
'/\' D)
'/\' E)));
set I = (
EqClass (z,A));
GG
= ((
EqClass (u,((B
'/\' C)
'/\' D)))
/\ (
EqClass (u,E))) by
Th1;
then
A29: GG
= (((
EqClass (u,(B
'/\' C)))
/\ (
EqClass (u,D)))
/\ (
EqClass (u,E))) by
Th1;
(h
. A)
= (
EqClass (z,A)) by
A3,
A4,
A5,
A6,
Th26;
then m
in (((((
EqClass (u,B))
/\ (
EqClass (u,C)))
/\ (
EqClass (u,D)))
/\ (
EqClass (u,E)))
/\ (
EqClass (z,A))) by
A24,
A28,
XBOOLE_0:def 4;
then (GG
/\ I)
<>
{} by
A29,
Th1;
then
consider p be
object such that
A30: p
in (GG
/\ I) by
XBOOLE_0:def 1;
reconsider p as
Element of Y by
A30;
set K = (
EqClass (p,((C
'/\' D)
'/\' E)));
A31: p
in GG by
A30,
XBOOLE_0:def 4;
A32: z
in I by
EQREL_1:def 6;
set L = (
EqClass (z,((C
'/\' D)
'/\' E)));
A33: p
in (
EqClass (p,((C
'/\' D)
'/\' E))) by
EQREL_1:def 6;
GG
= (
EqClass (u,((B
'/\' (C
'/\' D))
'/\' E))) by
PARTIT1: 14;
then GG
= (
EqClass (u,(B
'/\' ((C
'/\' D)
'/\' E)))) by
PARTIT1: 14;
then GG
c= L by
A7,
BVFUNC11: 3;
then K
meets L by
A31,
A33,
XBOOLE_0: 3;
then K
= L by
EQREL_1: 41;
then z
in K by
EQREL_1:def 6;
then
A34: z
in (I
/\ K) by
A32,
XBOOLE_0:def 4;
set H = (
EqClass (z,(
CompF (B,G))));
(A
'/\' ((C
'/\' D)
'/\' E))
= ((A
'/\' (C
'/\' D))
'/\' E) by
PARTIT1: 14;
then
A35: (A
'/\' ((C
'/\' D)
'/\' E))
= (((A
'/\' C)
'/\' D)
'/\' E) by
PARTIT1: 14;
A36: p
in K & p
in I by
A30,
EQREL_1:def 6,
XBOOLE_0:def 4;
then p
in (I
/\ K) by
XBOOLE_0:def 4;
then (I
/\ K)
in (
INTERSECTION (A,((C
'/\' D)
'/\' E))) & not (I
/\ K)
in
{
{} } by
SETFAM_1:def 5,
TARSKI:def 1;
then
A37: (I
/\ K)
in ((
INTERSECTION (A,((C
'/\' D)
'/\' E)))
\
{
{} }) by
XBOOLE_0:def 5;
(
CompF (B,G))
= (((A
'/\' C)
'/\' D)
'/\' E) by
A2,
A3,
A5,
Th22;
then (I
/\ K)
in (
CompF (B,G)) by
A37,
A35,
PARTIT1:def 4;
then
A38: (I
/\ K)
= H or (I
/\ K)
misses H by
EQREL_1:def 4;
z
in H by
EQREL_1:def 6;
then p
in H by
A36,
A34,
A38,
XBOOLE_0: 3,
XBOOLE_0:def 4;
then p
in (GG
/\ H) by
A31,
XBOOLE_0:def 4;
then GG
meets H by
XBOOLE_0: 4;
hence thesis by
A2,
A3,
A4,
Th21;
end;
theorem ::
BVFUNC14:31
Th31: G
=
{A, B, C, D, E, F} & A
<> B & A
<> C & A
<> D & A
<> E & A
<> F & B
<> C & B
<> D & B
<> E & B
<> F & C
<> D & C
<> E & C
<> F & D
<> E & D
<> F & E
<> F implies (
CompF (A,G))
= ((((B
'/\' C)
'/\' D)
'/\' E)
'/\' F)
proof
assume that
A1: G
=
{A, B, C, D, E, F} and
A2: A
<> B and
A3: A
<> C and
A4: A
<> D & A
<> E and
A5: A
<> F and
A6: B
<> C & B
<> D & B
<> E & B
<> F & C
<> D & C
<> E & C
<> F & D
<> E & D
<> F & E
<> F;
A7: (G
\
{A})
= ((
{A}
\/
{B, C, D, E, F})
\
{A}) by
A1,
ENUMSET1: 11
.= ((
{A}
\
{A})
\/ (
{B, C, D, E, F}
\
{A})) by
XBOOLE_1: 42;
A8: not F
in
{A} by
A5,
TARSKI:def 1;
A9: ( not D
in
{A}) & not E
in
{A} by
A4,
TARSKI:def 1;
A10: not C
in
{A} by
A3,
TARSKI:def 1;
A11: not B
in
{A} by
A2,
TARSKI:def 1;
A
in
{A} by
TARSKI:def 1;
then
A12: (
{A}
\
{A})
=
{} by
ZFMISC_1: 60;
A13: (
{B, C, D, E, F}
\
{A})
= ((
{B}
\/
{C, D, E, F})
\
{A}) by
ENUMSET1: 7
.= ((
{B}
\
{A})
\/ (
{C, D, E, F}
\
{A})) by
XBOOLE_1: 42
.= (
{B}
\/ (
{C, D, E, F}
\
{A})) by
A11,
ZFMISC_1: 59
.= (
{B}
\/ ((
{C}
\/
{D, E, F})
\
{A})) by
ENUMSET1: 4
.= (
{B}
\/ ((
{C}
\
{A})
\/ (
{D, E, F}
\
{A}))) by
XBOOLE_1: 42
.= (
{B}
\/ ((
{C}
\
{A})
\/ ((
{D, E}
\/
{F})
\
{A}))) by
ENUMSET1: 3
.= (
{B}
\/ ((
{C}
\
{A})
\/ ((
{D, E}
\
{A})
\/ (
{F}
\
{A})))) by
XBOOLE_1: 42
.= (
{B}
\/ ((
{C}
\
{A})
\/ (
{D, E}
\/ (
{F}
\
{A})))) by
A9,
ZFMISC_1: 63
.= (
{B}
\/ ((
{C}
\
{A})
\/ (
{D, E}
\/
{F}))) by
A8,
ZFMISC_1: 59
.= (
{B}
\/ (
{C}
\/ (
{D, E}
\/
{F}))) by
A10,
ZFMISC_1: 59
.= (
{B}
\/ (
{C}
\/
{D, E, F})) by
ENUMSET1: 3
.= (
{B}
\/
{C, D, E, F}) by
ENUMSET1: 4
.=
{B, C, D, E, F} by
ENUMSET1: 7;
A14: ((((B
'/\' C)
'/\' D)
'/\' E)
'/\' F)
c= (
'/\' (G
\
{A}))
proof
let x be
object;
reconsider xx = x as
set by
TARSKI: 1;
assume
A15: x
in ((((B
'/\' C)
'/\' D)
'/\' E)
'/\' F);
then
A16: x
<>
{} by
EQREL_1:def 4;
x
in ((
INTERSECTION ((((B
'/\' C)
'/\' D)
'/\' E),F))
\
{
{} }) by
A15,
PARTIT1:def 4;
then
consider bcde,f be
set such that
A17: bcde
in (((B
'/\' C)
'/\' D)
'/\' E) and
A18: f
in F and
A19: x
= (bcde
/\ f) by
SETFAM_1:def 5;
bcde
in ((
INTERSECTION (((B
'/\' C)
'/\' D),E))
\
{
{} }) by
A17,
PARTIT1:def 4;
then
consider bcd,e be
set such that
A20: bcd
in ((B
'/\' C)
'/\' D) and
A21: e
in E and
A22: bcde
= (bcd
/\ e) by
SETFAM_1:def 5;
bcd
in ((
INTERSECTION ((B
'/\' C),D))
\
{
{} }) by
A20,
PARTIT1:def 4;
then
consider bc,d be
set such that
A23: bc
in (B
'/\' C) and
A24: d
in D and
A25: bcd
= (bc
/\ d) by
SETFAM_1:def 5;
bc
in ((
INTERSECTION (B,C))
\
{
{} }) by
A23,
PARTIT1:def 4;
then
consider b,c be
set such that
A26: b
in B and
A27: c
in C and
A28: bc
= (b
/\ c) by
SETFAM_1:def 5;
set h = (((((B
.--> b)
+* (C
.--> c))
+* (D
.--> d))
+* (E
.--> e))
+* (F
.--> f));
A29: (h
. B)
= b by
A6,
Th26;
A30: (h
. E)
= e by
A6,
Th26;
A31: (h
. F)
= f by
A6,
Th26;
A32: (
dom (((((B
.--> b)
+* (C
.--> c))
+* (D
.--> d))
+* (E
.--> e))
+* (F
.--> f)))
=
{F, B, C, D, E} by
Th27
.= (
{F}
\/
{B, C, D, E}) by
ENUMSET1: 7
.=
{B, C, D, E, F} by
ENUMSET1: 10;
then
A33: C
in (
dom h) by
ENUMSET1:def 3;
A34: F
in (
dom h) by
A32,
ENUMSET1:def 3;
A35: E
in (
dom h) by
A32,
ENUMSET1:def 3;
A36: (h
. C)
= c by
A6,
Th26;
A37: (
rng h)
c=
{(h
. D), (h
. B), (h
. C), (h
. E), (h
. F)}
proof
let t be
object;
assume t
in (
rng h);
then
consider x1 be
object such that
A38: x1
in (
dom h) and
A39: t
= (h
. x1) by
FUNCT_1:def 3;
now
per cases by
A32,
A38,
ENUMSET1:def 3;
case x1
= D;
hence thesis by
A39,
ENUMSET1:def 3;
end;
case x1
= B;
hence thesis by
A39,
ENUMSET1:def 3;
end;
case x1
= C;
hence thesis by
A39,
ENUMSET1:def 3;
end;
case x1
= E;
hence thesis by
A39,
ENUMSET1:def 3;
end;
case x1
= F;
hence thesis by
A39,
ENUMSET1:def 3;
end;
end;
hence thesis;
end;
A40: (h
. D)
= d by
A6,
Th26;
(
rng h)
c= (
bool Y)
proof
let t be
object;
assume
A41: t
in (
rng h);
now
per cases by
A37,
A41,
ENUMSET1:def 3;
case t
= (h
. D);
hence thesis by
A24,
A40;
end;
case t
= (h
. B);
hence thesis by
A26,
A29;
end;
case t
= (h
. C);
hence thesis by
A27,
A36;
end;
case t
= (h
. E);
hence thesis by
A21,
A30;
end;
case t
= (h
. F);
hence thesis by
A18,
A31;
end;
end;
hence thesis;
end;
then
reconsider FF = (
rng h) as
Subset-Family of Y;
A42: D
in (
dom h) by
A32,
ENUMSET1:def 3;
then (h
. D)
in (
rng h) by
FUNCT_1:def 3;
then
A43: (
Intersect FF)
= (
meet (
rng h)) by
SETFAM_1:def 9;
A44: B
in (
dom h) by
A32,
ENUMSET1:def 3;
{(h
. D), (h
. B), (h
. C), (h
. E), (h
. F)}
c= (
rng h)
proof
let t be
object;
assume
A45: t
in
{(h
. D), (h
. B), (h
. C), (h
. E), (h
. F)};
now
per cases by
A45,
ENUMSET1:def 3;
case t
= (h
. D);
hence thesis by
A42,
FUNCT_1:def 3;
end;
case t
= (h
. B);
hence thesis by
A44,
FUNCT_1:def 3;
end;
case t
= (h
. C);
hence thesis by
A33,
FUNCT_1:def 3;
end;
case t
= (h
. E);
hence thesis by
A35,
FUNCT_1:def 3;
end;
case t
= (h
. F);
hence thesis by
A34,
FUNCT_1:def 3;
end;
end;
hence thesis;
end;
then
A46: (
rng h)
=
{(h
. D), (h
. B), (h
. C), (h
. E), (h
. F)} by
A37,
XBOOLE_0:def 10;
A47: xx
c= (
Intersect FF)
proof
let u be
object;
assume
A48: u
in xx;
for y be
set holds y
in FF implies u
in y
proof
let y be
set;
assume
A49: y
in FF;
now
per cases by
A37,
A49,
ENUMSET1:def 3;
case
A50: y
= (h
. D);
u
in ((d
/\ ((b
/\ c)
/\ e))
/\ f) by
A19,
A22,
A25,
A28,
A48,
XBOOLE_1: 16;
then
A51: u
in (d
/\ (((b
/\ c)
/\ e)
/\ f)) by
XBOOLE_1: 16;
y
= d by
A6,
A50,
Th26;
hence thesis by
A51,
XBOOLE_0:def 4;
end;
case
A52: y
= (h
. B);
u
in (((c
/\ (d
/\ b))
/\ e)
/\ f) by
A19,
A22,
A25,
A28,
A48,
XBOOLE_1: 16;
then u
in ((c
/\ ((d
/\ b)
/\ e))
/\ f) by
XBOOLE_1: 16;
then u
in ((c
/\ ((d
/\ e)
/\ b))
/\ f) by
XBOOLE_1: 16;
then u
in (c
/\ (((d
/\ e)
/\ b)
/\ f)) by
XBOOLE_1: 16;
then u
in (c
/\ ((d
/\ e)
/\ (f
/\ b))) by
XBOOLE_1: 16;
then u
in ((c
/\ (d
/\ e))
/\ (f
/\ b)) by
XBOOLE_1: 16;
then
A53: u
in (((c
/\ (d
/\ e))
/\ f)
/\ b) by
XBOOLE_1: 16;
y
= b by
A6,
A52,
Th26;
hence thesis by
A53,
XBOOLE_0:def 4;
end;
case
A54: y
= (h
. C);
u
in (((c
/\ (b
/\ d))
/\ e)
/\ f) by
A19,
A22,
A25,
A28,
A48,
XBOOLE_1: 16;
then u
in ((c
/\ ((b
/\ d)
/\ e))
/\ f) by
XBOOLE_1: 16;
then
A55: u
in (c
/\ (((b
/\ d)
/\ e)
/\ f)) by
XBOOLE_1: 16;
y
= c by
A6,
A54,
Th26;
hence thesis by
A55,
XBOOLE_0:def 4;
end;
case y
= (h
. E);
then
A56: y
= e by
A6,
Th26;
u
in ((((b
/\ c)
/\ d)
/\ f)
/\ e) by
A19,
A22,
A25,
A28,
A48,
XBOOLE_1: 16;
hence thesis by
A56,
XBOOLE_0:def 4;
end;
case y
= (h
. F);
hence thesis by
A19,
A31,
A48,
XBOOLE_0:def 4;
end;
end;
hence thesis;
end;
then u
in (
meet FF) by
A46,
SETFAM_1:def 1;
hence thesis by
A46,
SETFAM_1:def 9;
end;
A57: for p be
set st p
in (G
\
{A}) holds (h
. p)
in p
proof
let p be
set;
assume
A58: p
in (G
\
{A});
now
per cases by
A7,
A12,
A58,
ENUMSET1:def 3;
case p
= D;
hence thesis by
A6,
A24,
Th26;
end;
case p
= B;
hence thesis by
A6,
A26,
Th26;
end;
case p
= C;
hence thesis by
A6,
A27,
Th26;
end;
case p
= E;
hence thesis by
A6,
A21,
Th26;
end;
case p
= F;
hence thesis by
A6,
A18,
Th26;
end;
end;
hence thesis;
end;
(
Intersect FF)
c= xx
proof
let t be
object;
assume
A59: t
in (
Intersect FF);
(h
. C)
in (
rng h) by
A46,
ENUMSET1:def 3;
then
A60: t
in c by
A36,
A43,
A59,
SETFAM_1:def 1;
(h
. B)
in (
rng h) by
A46,
ENUMSET1:def 3;
then t
in b by
A29,
A43,
A59,
SETFAM_1:def 1;
then
A61: t
in (b
/\ c) by
A60,
XBOOLE_0:def 4;
(h
. D)
in (
rng h) by
A46,
ENUMSET1:def 3;
then t
in d by
A40,
A43,
A59,
SETFAM_1:def 1;
then
A62: t
in ((b
/\ c)
/\ d) by
A61,
XBOOLE_0:def 4;
(h
. E)
in (
rng h) by
A46,
ENUMSET1:def 3;
then t
in e by
A30,
A43,
A59,
SETFAM_1:def 1;
then
A63: t
in (((b
/\ c)
/\ d)
/\ e) by
A62,
XBOOLE_0:def 4;
(h
. F)
in (
rng h) by
A46,
ENUMSET1:def 3;
then t
in f by
A31,
A43,
A59,
SETFAM_1:def 1;
hence thesis by
A19,
A22,
A25,
A28,
A63,
XBOOLE_0:def 4;
end;
then x
= (
Intersect FF) by
A47,
XBOOLE_0:def 10;
hence thesis by
A7,
A13,
A12,
A32,
A57,
A16,
BVFUNC_2:def 1;
end;
A64: (
'/\' (G
\
{A}))
c= ((((B
'/\' C)
'/\' D)
'/\' E)
'/\' F)
proof
let x be
object;
reconsider xx = x as
set by
TARSKI: 1;
assume x
in (
'/\' (G
\
{A}));
then
consider h be
Function, FF be
Subset-Family of Y such that
A65: (
dom h)
= (G
\
{A}) and
A66: (
rng h)
= FF and
A67: for d be
set st d
in (G
\
{A}) holds (h
. d)
in d and
A68: x
= (
Intersect FF) and
A69: x
<>
{} by
BVFUNC_2:def 1;
A70: C
in (G
\
{A}) by
A7,
A13,
A12,
ENUMSET1:def 3;
then
A71: (h
. C)
in C by
A67;
set mbc = ((h
. B)
/\ (h
. C));
A72: B
in (G
\
{A}) by
A7,
A13,
A12,
ENUMSET1:def 3;
then (h
. B)
in B by
A67;
then
A73: mbc
in (
INTERSECTION (B,C)) by
A71,
SETFAM_1:def 5;
set mbcd = (((h
. B)
/\ (h
. C))
/\ (h
. D));
A74: E
in (G
\
{A}) by
A7,
A13,
A12,
ENUMSET1:def 3;
then
A75: (h
. E)
in (
rng h) by
A65,
FUNCT_1:def 3;
A76: (h
. B)
in (
rng h) by
A65,
A72,
FUNCT_1:def 3;
then
A77: (
Intersect FF)
= (
meet (
rng h)) by
A66,
SETFAM_1:def 9;
A78: (h
. C)
in (
rng h) by
A65,
A70,
FUNCT_1:def 3;
A79: F
in (G
\
{A}) by
A7,
A13,
A12,
ENUMSET1:def 3;
then
A80: (h
. F)
in (
rng h) by
A65,
FUNCT_1:def 3;
A81: D
in (G
\
{A}) by
A7,
A13,
A12,
ENUMSET1:def 3;
then
A82: (h
. D)
in (
rng h) by
A65,
FUNCT_1:def 3;
A83: xx
c= (((((h
. B)
/\ (h
. C))
/\ (h
. D))
/\ (h
. E))
/\ (h
. F))
proof
let m be
object;
assume
A84: m
in xx;
then m
in (h
. B) & m
in (h
. C) by
A68,
A76,
A78,
A77,
SETFAM_1:def 1;
then
A85: m
in ((h
. B)
/\ (h
. C)) by
XBOOLE_0:def 4;
m
in (h
. D) by
A68,
A82,
A77,
A84,
SETFAM_1:def 1;
then
A86: m
in (((h
. B)
/\ (h
. C))
/\ (h
. D)) by
A85,
XBOOLE_0:def 4;
m
in (h
. E) by
A68,
A75,
A77,
A84,
SETFAM_1:def 1;
then
A87: m
in ((((h
. B)
/\ (h
. C))
/\ (h
. D))
/\ (h
. E)) by
A86,
XBOOLE_0:def 4;
m
in (h
. F) by
A68,
A80,
A77,
A84,
SETFAM_1:def 1;
hence thesis by
A87,
XBOOLE_0:def 4;
end;
then mbcd
<>
{} by
A69;
then
A88: not mbcd
in
{
{} } by
TARSKI:def 1;
A89: (
rng h)
<>
{} by
A65,
A72,
FUNCT_1: 3;
(((((h
. B)
/\ (h
. C))
/\ (h
. D))
/\ (h
. E))
/\ (h
. F))
c= xx
proof
let m be
object;
assume
A90: m
in (((((h
. B)
/\ (h
. C))
/\ (h
. D))
/\ (h
. E))
/\ (h
. F));
then
A91: m
in ((((h
. B)
/\ (h
. C))
/\ (h
. D))
/\ (h
. E)) by
XBOOLE_0:def 4;
then
A92: m
in (((h
. B)
/\ (h
. C))
/\ (h
. D)) by
XBOOLE_0:def 4;
then
A93: m
in ((h
. B)
/\ (h
. C)) by
XBOOLE_0:def 4;
A94: (
rng h)
c=
{(h
. B), (h
. C), (h
. D), (h
. E), (h
. F)}
proof
let u be
object;
assume u
in (
rng h);
then
consider x1 be
object such that
A95: x1
in (
dom h) and
A96: u
= (h
. x1) by
FUNCT_1:def 3;
now
per cases by
A7,
A12,
A65,
A95,
ENUMSET1:def 3;
case x1
= B;
hence thesis by
A96,
ENUMSET1:def 3;
end;
case x1
= C;
hence thesis by
A96,
ENUMSET1:def 3;
end;
case x1
= D;
hence thesis by
A96,
ENUMSET1:def 3;
end;
case x1
= E;
hence thesis by
A96,
ENUMSET1:def 3;
end;
case x1
= F;
hence thesis by
A96,
ENUMSET1:def 3;
end;
end;
hence thesis;
end;
for y be
set holds y
in (
rng h) implies m
in y
proof
let y be
set;
assume
A97: y
in (
rng h);
now
per cases by
A94,
A97,
ENUMSET1:def 3;
case y
= (h
. B);
hence thesis by
A93,
XBOOLE_0:def 4;
end;
case y
= (h
. C);
hence thesis by
A93,
XBOOLE_0:def 4;
end;
case y
= (h
. D);
hence thesis by
A92,
XBOOLE_0:def 4;
end;
case y
= (h
. E);
hence thesis by
A91,
XBOOLE_0:def 4;
end;
case y
= (h
. F);
hence thesis by
A90,
XBOOLE_0:def 4;
end;
end;
hence thesis;
end;
hence thesis by
A68,
A89,
A77,
SETFAM_1:def 1;
end;
then
A98: (((((h
. B)
/\ (h
. C))
/\ (h
. D))
/\ (h
. E))
/\ (h
. F))
= x by
A83,
XBOOLE_0:def 10;
mbc
<>
{} by
A69,
A83;
then not mbc
in
{
{} } by
TARSKI:def 1;
then mbc
in ((
INTERSECTION (B,C))
\
{
{} }) by
A73,
XBOOLE_0:def 5;
then
A99: mbc
in (B
'/\' C) by
PARTIT1:def 4;
(h
. D)
in D by
A67,
A81;
then mbcd
in (
INTERSECTION ((B
'/\' C),D)) by
A99,
SETFAM_1:def 5;
then mbcd
in ((
INTERSECTION ((B
'/\' C),D))
\
{
{} }) by
A88,
XBOOLE_0:def 5;
then
A100: mbcd
in ((B
'/\' C)
'/\' D) by
PARTIT1:def 4;
set mbcde = ((((h
. B)
/\ (h
. C))
/\ (h
. D))
/\ (h
. E));
A101: not x
in
{
{} } by
A69,
TARSKI:def 1;
mbcde
<>
{} by
A69,
A83;
then
A102: not mbcde
in
{
{} } by
TARSKI:def 1;
(h
. E)
in E by
A67,
A74;
then mbcde
in (
INTERSECTION (((B
'/\' C)
'/\' D),E)) by
A100,
SETFAM_1:def 5;
then mbcde
in ((
INTERSECTION (((B
'/\' C)
'/\' D),E))
\
{
{} }) by
A102,
XBOOLE_0:def 5;
then
A103: mbcde
in (((B
'/\' C)
'/\' D)
'/\' E) by
PARTIT1:def 4;
(h
. F)
in F by
A67,
A79;
then x
in (
INTERSECTION ((((B
'/\' C)
'/\' D)
'/\' E),F)) by
A98,
A103,
SETFAM_1:def 5;
then x
in ((
INTERSECTION ((((B
'/\' C)
'/\' D)
'/\' E),F))
\
{
{} }) by
A101,
XBOOLE_0:def 5;
hence thesis by
PARTIT1:def 4;
end;
(
CompF (A,G))
= (
'/\' (G
\
{A})) by
BVFUNC_2:def 7;
hence thesis by
A14,
A64,
XBOOLE_0:def 10;
end;
theorem ::
BVFUNC14:32
Th32: G
=
{A, B, C, D, E, F} & A
<> B & A
<> C & A
<> D & A
<> E & A
<> F & B
<> C & B
<> D & B
<> E & B
<> F & C
<> D & C
<> E & C
<> F & D
<> E & D
<> F & E
<> F implies (
CompF (B,G))
= ((((A
'/\' C)
'/\' D)
'/\' E)
'/\' F)
proof
assume that
A1: G
=
{A, B, C, D, E, F} and
A2: A
<> B & A
<> C & A
<> D & A
<> E & A
<> F & B
<> C & B
<> D & B
<> E & B
<> F & C
<> D & C
<> E & C
<> F & D
<> E & D
<> F & E
<> F;
{A, B, C, D, E, F}
= (
{B, A}
\/
{C, D, E, F}) by
ENUMSET1: 12;
then G
=
{B, A, C, D, E, F} by
A1,
ENUMSET1: 12;
hence thesis by
A2,
Th31;
end;
theorem ::
BVFUNC14:33
Th33: G
=
{A, B, C, D, E, F} & A
<> B & A
<> C & A
<> D & A
<> E & A
<> F & B
<> C & B
<> D & B
<> E & B
<> F & C
<> D & C
<> E & C
<> F & D
<> E & D
<> F & E
<> F implies (
CompF (C,G))
= ((((A
'/\' B)
'/\' D)
'/\' E)
'/\' F)
proof
A1:
{A, B, C, D, E, F}
= (
{A, B, C}
\/
{D, E, F}) by
ENUMSET1: 13
.= ((
{A}
\/
{B, C})
\/
{D, E, F}) by
ENUMSET1: 2
.= (
{A, C, B}
\/
{D, E, F}) by
ENUMSET1: 2
.=
{A, C, B, D, E, F} by
ENUMSET1: 13;
assume G
=
{A, B, C, D, E, F} & A
<> B & A
<> C & A
<> D & A
<> E & A
<> F & B
<> C & B
<> D & B
<> E & B
<> F & C
<> D & C
<> E & C
<> F & D
<> E & D
<> F & E
<> F;
hence thesis by
A1,
Th32;
end;
theorem ::
BVFUNC14:34
Th34: G
=
{A, B, C, D, E, F} & A
<> B & A
<> C & A
<> D & A
<> E & A
<> F & B
<> C & B
<> D & B
<> E & B
<> F & C
<> D & C
<> E & C
<> F & D
<> E & D
<> F & E
<> F implies (
CompF (D,G))
= ((((A
'/\' B)
'/\' C)
'/\' E)
'/\' F)
proof
A1:
{A, B, C, D, E, F}
= (
{A, B}
\/
{C, D, E, F}) by
ENUMSET1: 12
.= (
{A, B}
\/ (
{C, D}
\/
{E, F})) by
ENUMSET1: 5
.= (
{A, B}
\/
{D, C, E, F}) by
ENUMSET1: 5
.=
{A, B, D, C, E, F} by
ENUMSET1: 12;
assume G
=
{A, B, C, D, E, F} & A
<> B & A
<> C & A
<> D & A
<> E & A
<> F & B
<> C & B
<> D & B
<> E & B
<> F & C
<> D & C
<> E & C
<> F & D
<> E & D
<> F & E
<> F;
hence thesis by
A1,
Th33;
end;
theorem ::
BVFUNC14:35
Th35: G
=
{A, B, C, D, E, F} & A
<> B & A
<> C & A
<> D & A
<> E & A
<> F & B
<> C & B
<> D & B
<> E & B
<> F & C
<> D & C
<> E & C
<> F & D
<> E & D
<> F & E
<> F implies (
CompF (E,G))
= ((((A
'/\' B)
'/\' C)
'/\' D)
'/\' F)
proof
A1:
{A, B, C, D, E, F}
= (
{A, B, C}
\/
{D, E, F}) by
ENUMSET1: 13
.= (
{A, B, C}
\/ (
{D, E}
\/
{F})) by
ENUMSET1: 3
.= (
{A, B, C}
\/
{E, D, F}) by
ENUMSET1: 3
.=
{A, B, C, E, D, F} by
ENUMSET1: 13;
assume G
=
{A, B, C, D, E, F} & A
<> B & A
<> C & A
<> D & A
<> E & A
<> F & B
<> C & B
<> D & B
<> E & B
<> F & C
<> D & C
<> E & C
<> F & D
<> E & D
<> F & E
<> F;
hence thesis by
A1,
Th34;
end;
theorem ::
BVFUNC14:36
G
=
{A, B, C, D, E, F} & A
<> B & A
<> C & A
<> D & A
<> E & A
<> F & B
<> C & B
<> D & B
<> E & B
<> F & C
<> D & C
<> E & C
<> F & D
<> E & D
<> F & E
<> F implies (
CompF (F,G))
= ((((A
'/\' B)
'/\' C)
'/\' D)
'/\' E)
proof
A1:
{A, B, C, D, E, F}
= (
{A, B, C, D}
\/
{E, F}) by
ENUMSET1: 14
.=
{A, B, C, D, F, E} by
ENUMSET1: 14;
assume G
=
{A, B, C, D, E, F} & A
<> B & A
<> C & A
<> D & A
<> E & A
<> F & B
<> C & B
<> D & B
<> E & B
<> F & C
<> D & C
<> E & C
<> F & D
<> E & D
<> F & E
<> F;
hence thesis by
A1,
Th35;
end;
theorem ::
BVFUNC14:37
Th37: for A,B,C,D,E,F be
set, h be
Function, A9,B9,C9,D9,E9,F9 be
set st A
<> B & A
<> C & A
<> D & A
<> E & A
<> F & B
<> C & B
<> D & B
<> E & B
<> F & C
<> D & C
<> E & C
<> F & D
<> E & D
<> F & E
<> F & h
= ((((((B
.--> B9)
+* (C
.--> C9))
+* (D
.--> D9))
+* (E
.--> E9))
+* (F
.--> F9))
+* (A
.--> A9)) holds (h
. A)
= A9 & (h
. B)
= B9 & (h
. C)
= C9 & (h
. D)
= D9 & (h
. E)
= E9 & (h
. F)
= F9
proof
let A,B,C,D,E,F be
set;
let h be
Function;
let A9,B9,C9,D9,E9,F9 be
set;
assume that
A1: A
<> B and
A2: A
<> C and
A3: A
<> D and
A4: A
<> E and
A5: A
<> F and
A6: B
<> C & B
<> D & B
<> E & B
<> F & C
<> D & C
<> E & C
<> F & D
<> E & D
<> F & E
<> F and
A7: h
= ((((((B
.--> B9)
+* (C
.--> C9))
+* (D
.--> D9))
+* (E
.--> E9))
+* (F
.--> F9))
+* (A
.--> A9));
A
in (
dom (A
.--> A9)) by
TARSKI:def 1;
then
A9: (h
. A)
= ((A
.--> A9)
. A) by
A7,
FUNCT_4: 13;
not C
in (
dom (A
.--> A9)) by
A2,
TARSKI:def 1;
then
A10: (h
. C)
= ((((((B
.--> B9)
+* (C
.--> C9))
+* (D
.--> D9))
+* (E
.--> E9))
+* (F
.--> F9))
. C) by
A7,
FUNCT_4: 11;
not F
in (
dom (A
.--> A9)) by
A5,
TARSKI:def 1;
then
A11: (h
. F)
= ((((((B
.--> B9)
+* (C
.--> C9))
+* (D
.--> D9))
+* (E
.--> E9))
+* (F
.--> F9))
. F) by
A7,
FUNCT_4: 11
.= F9 by
A6,
Th26;
not E
in (
dom (A
.--> A9)) by
A4,
TARSKI:def 1;
then
A12: (h
. E)
= ((((((B
.--> B9)
+* (C
.--> C9))
+* (D
.--> D9))
+* (E
.--> E9))
+* (F
.--> F9))
. E) by
A7,
FUNCT_4: 11
.= E9 by
A6,
Th26;
not D
in (
dom (A
.--> A9)) by
A3,
TARSKI:def 1;
then
A13: (h
. D)
= ((((((B
.--> B9)
+* (C
.--> C9))
+* (D
.--> D9))
+* (E
.--> E9))
+* (F
.--> F9))
. D) by
A7,
FUNCT_4: 11
.= D9 by
A6,
Th26;
not B
in (
dom (A
.--> A9)) by
A1,
TARSKI:def 1;
then (h
. B)
= ((((((B
.--> B9)
+* (C
.--> C9))
+* (D
.--> D9))
+* (E
.--> E9))
+* (F
.--> F9))
. B) by
A7,
FUNCT_4: 11
.= B9 by
A6,
Th26;
hence thesis by
A6,
A9,
A10,
A13,
A12,
A11,
Th26,
FUNCOP_1: 72;
end;
theorem ::
BVFUNC14:38
Th38: for A,B,C,D,E,F be
set, h be
Function, A9,B9,C9,D9,E9,F9 be
set st h
= ((((((B
.--> B9)
+* (C
.--> C9))
+* (D
.--> D9))
+* (E
.--> E9))
+* (F
.--> F9))
+* (A
.--> A9)) holds (
dom h)
=
{A, B, C, D, E, F}
proof
let A,B,C,D,E,F be
set;
let h be
Function;
let A9,B9,C9,D9,E9,F9 be
set;
assume
A1: h
= ((((((B
.--> B9)
+* (C
.--> C9))
+* (D
.--> D9))
+* (E
.--> E9))
+* (F
.--> F9))
+* (A
.--> A9));
A2: (
dom (A
.--> A9))
=
{A};
(
dom (((((B
.--> B9)
+* (C
.--> C9))
+* (D
.--> D9))
+* (E
.--> E9))
+* (F
.--> F9)))
=
{F, B, C, D, E} by
Th27
.= (
{F}
\/
{B, C, D, E}) by
ENUMSET1: 7
.=
{B, C, D, E, F} by
ENUMSET1: 10;
then (
dom ((((((B
.--> B9)
+* (C
.--> C9))
+* (D
.--> D9))
+* (E
.--> E9))
+* (F
.--> F9))
+* (A
.--> A9)))
= (
{B, C, D, E, F}
\/
{A}) by
A2,
FUNCT_4:def 1
.=
{A, B, C, D, E, F} by
ENUMSET1: 11;
hence thesis by
A1;
end;
theorem ::
BVFUNC14:39
Th39: for A,B,C,D,E,F be
set, h be
Function, A9,B9,C9,D9,E9,F9 be
set st h
= ((((((B
.--> B9)
+* (C
.--> C9))
+* (D
.--> D9))
+* (E
.--> E9))
+* (F
.--> F9))
+* (A
.--> A9)) holds (
rng h)
=
{(h
. A), (h
. B), (h
. C), (h
. D), (h
. E), (h
. F)}
proof
let A,B,C,D,E,F be
set;
let h be
Function;
let A9,B9,C9,D9,E9,F9 be
set;
assume h
= ((((((B
.--> B9)
+* (C
.--> C9))
+* (D
.--> D9))
+* (E
.--> E9))
+* (F
.--> F9))
+* (A
.--> A9));
then
A1: (
dom h)
=
{A, B, C, D, E, F} by
Th38;
then
A2: B
in (
dom h) by
ENUMSET1:def 4;
A3: (
rng h)
c=
{(h
. A), (h
. B), (h
. C), (h
. D), (h
. E), (h
. F)}
proof
let t be
object;
assume t
in (
rng h);
then
consider x1 be
object such that
A4: x1
in (
dom h) and
A5: t
= (h
. x1) by
FUNCT_1:def 3;
now
per cases by
A1,
A4,
ENUMSET1:def 4;
case x1
= A;
hence thesis by
A5,
ENUMSET1:def 4;
end;
case x1
= B;
hence thesis by
A5,
ENUMSET1:def 4;
end;
case x1
= C;
hence thesis by
A5,
ENUMSET1:def 4;
end;
case x1
= D;
hence thesis by
A5,
ENUMSET1:def 4;
end;
case x1
= E;
hence thesis by
A5,
ENUMSET1:def 4;
end;
case x1
= F;
hence thesis by
A5,
ENUMSET1:def 4;
end;
end;
hence thesis;
end;
A6: D
in (
dom h) by
A1,
ENUMSET1:def 4;
A7: C
in (
dom h) by
A1,
ENUMSET1:def 4;
A8: F
in (
dom h) by
A1,
ENUMSET1:def 4;
A9: E
in (
dom h) by
A1,
ENUMSET1:def 4;
A10: A
in (
dom h) by
A1,
ENUMSET1:def 4;
{(h
. A), (h
. B), (h
. C), (h
. D), (h
. E), (h
. F)}
c= (
rng h)
proof
let t be
object;
assume
A11: t
in
{(h
. A), (h
. B), (h
. C), (h
. D), (h
. E), (h
. F)};
now
per cases by
A11,
ENUMSET1:def 4;
case t
= (h
. A);
hence thesis by
A10,
FUNCT_1:def 3;
end;
case t
= (h
. B);
hence thesis by
A2,
FUNCT_1:def 3;
end;
case t
= (h
. C);
hence thesis by
A7,
FUNCT_1:def 3;
end;
case t
= (h
. D);
hence thesis by
A6,
FUNCT_1:def 3;
end;
case t
= (h
. E);
hence thesis by
A9,
FUNCT_1:def 3;
end;
case t
= (h
. F);
hence thesis by
A8,
FUNCT_1:def 3;
end;
end;
hence thesis;
end;
hence thesis by
A3,
XBOOLE_0:def 10;
end;
theorem ::
BVFUNC14:40
for G be
Subset of (
PARTITIONS Y), A,B,C,D,E,F be
a_partition of Y, z,u be
Element of Y, h be
Function st G is
independent & G
=
{A, B, C, D, E, F} & A
<> B & A
<> C & A
<> D & A
<> E & A
<> F & B
<> C & B
<> D & B
<> E & B
<> F & C
<> D & C
<> E & C
<> F & D
<> E & D
<> F & E
<> F holds (
EqClass (u,((((B
'/\' C)
'/\' D)
'/\' E)
'/\' F)))
meets (
EqClass (z,A))
proof
let G be
Subset of (
PARTITIONS Y);
let A,B,C,D,E,F be
a_partition of Y;
let z,u be
Element of Y;
let h be
Function;
assume that
A1: G is
independent and
A2: G
=
{A, B, C, D, E, F} and
A3: A
<> B & A
<> C & A
<> D & A
<> E & A
<> F & B
<> C & B
<> D & B
<> E & B
<> F & C
<> D & C
<> E & C
<> F & D
<> E & D
<> F & E
<> F;
set h = ((((((B
.--> (
EqClass (u,B)))
+* (C
.--> (
EqClass (u,C))))
+* (D
.--> (
EqClass (u,D))))
+* (E
.--> (
EqClass (u,E))))
+* (F
.--> (
EqClass (u,F))))
+* (A
.--> (
EqClass (z,A))));
A4: (h
. A)
= (
EqClass (z,A)) by
A3,
Th37;
set GG = (
EqClass (u,((((B
'/\' C)
'/\' D)
'/\' E)
'/\' F)));
GG
= ((
EqClass (u,(((B
'/\' C)
'/\' D)
'/\' E)))
/\ (
EqClass (u,F))) by
Th1;
then GG
= (((
EqClass (u,((B
'/\' C)
'/\' D)))
/\ (
EqClass (u,E)))
/\ (
EqClass (u,F))) by
Th1;
then GG
= ((((
EqClass (u,(B
'/\' C)))
/\ (
EqClass (u,D)))
/\ (
EqClass (u,E)))
/\ (
EqClass (u,F))) by
Th1;
then
A5: (GG
/\ (
EqClass (z,A)))
= ((((((
EqClass (u,B))
/\ (
EqClass (u,C)))
/\ (
EqClass (u,D)))
/\ (
EqClass (u,E)))
/\ (
EqClass (u,F)))
/\ (
EqClass (z,A))) by
Th1;
A6: (h
. B)
= (
EqClass (u,B)) by
A3,
Th37;
A7: (h
. D)
= (
EqClass (u,D)) by
A3,
Th37;
A8: (h
. C)
= (
EqClass (u,C)) by
A3,
Th37;
A9: (h
. F)
= (
EqClass (u,F)) by
A3,
Th37;
A10: (h
. E)
= (
EqClass (u,E)) by
A3,
Th37;
A11: (
rng h)
=
{(h
. A), (h
. B), (h
. C), (h
. D), (h
. E), (h
. F)} by
Th39;
(
rng h)
c= (
bool Y)
proof
let t be
object;
assume
A12: t
in (
rng h);
now
per cases by
A11,
A12,
ENUMSET1:def 4;
case t
= (h
. A);
hence thesis by
A4;
end;
case t
= (h
. B);
hence thesis by
A6;
end;
case t
= (h
. C);
hence thesis by
A8;
end;
case t
= (h
. D);
hence thesis by
A7;
end;
case t
= (h
. E);
hence thesis by
A10;
end;
case t
= (h
. F);
hence thesis by
A9;
end;
end;
hence thesis;
end;
then
reconsider FF = (
rng h) as
Subset-Family of Y;
A13: (
dom h)
= G by
A2,
Th38;
then A
in (
dom h) by
A2,
ENUMSET1:def 4;
then
A14: (h
. A)
in (
rng h) by
FUNCT_1:def 3;
then
A15: (
Intersect FF)
= (
meet (
rng h)) by
SETFAM_1:def 9;
for d be
set st d
in G holds (h
. d)
in d
proof
let d be
set;
assume
A16: d
in G;
now
per cases by
A2,
A16,
ENUMSET1:def 4;
case d
= A;
hence thesis by
A4;
end;
case d
= B;
hence thesis by
A6;
end;
case d
= C;
hence thesis by
A8;
end;
case d
= D;
hence thesis by
A7;
end;
case d
= E;
hence thesis by
A10;
end;
case d
= F;
hence thesis by
A9;
end;
end;
hence thesis;
end;
then (
Intersect FF)
<>
{} by
A1,
A13,
BVFUNC_2:def 5;
then
consider m be
object such that
A17: m
in (
Intersect FF) by
XBOOLE_0:def 1;
C
in (
dom h) by
A2,
A13,
ENUMSET1:def 4;
then (h
. C)
in (
rng h) by
FUNCT_1:def 3;
then
A18: m
in (
EqClass (u,C)) by
A8,
A15,
A17,
SETFAM_1:def 1;
B
in (
dom h) by
A2,
A13,
ENUMSET1:def 4;
then (h
. B)
in (
rng h) by
FUNCT_1:def 3;
then m
in (
EqClass (u,B)) by
A6,
A15,
A17,
SETFAM_1:def 1;
then
A19: m
in ((
EqClass (u,B))
/\ (
EqClass (u,C))) by
A18,
XBOOLE_0:def 4;
D
in (
dom h) by
A2,
A13,
ENUMSET1:def 4;
then (h
. D)
in (
rng h) by
FUNCT_1:def 3;
then m
in (
EqClass (u,D)) by
A7,
A15,
A17,
SETFAM_1:def 1;
then
A20: m
in (((
EqClass (u,B))
/\ (
EqClass (u,C)))
/\ (
EqClass (u,D))) by
A19,
XBOOLE_0:def 4;
E
in (
dom h) by
A2,
A13,
ENUMSET1:def 4;
then (h
. E)
in (
rng h) by
FUNCT_1:def 3;
then m
in (
EqClass (u,E)) by
A10,
A15,
A17,
SETFAM_1:def 1;
then
A21: m
in ((((
EqClass (u,B))
/\ (
EqClass (u,C)))
/\ (
EqClass (u,D)))
/\ (
EqClass (u,E))) by
A20,
XBOOLE_0:def 4;
F
in (
dom h) by
A2,
A13,
ENUMSET1:def 4;
then (h
. F)
in (
rng h) by
FUNCT_1:def 3;
then m
in (
EqClass (u,F)) by
A9,
A15,
A17,
SETFAM_1:def 1;
then
A22: m
in (((((
EqClass (u,B))
/\ (
EqClass (u,C)))
/\ (
EqClass (u,D)))
/\ (
EqClass (u,E)))
/\ (
EqClass (u,F))) by
A21,
XBOOLE_0:def 4;
m
in (
EqClass (z,A)) by
A4,
A14,
A15,
A17,
SETFAM_1:def 1;
then m
in ((((((
EqClass (u,B))
/\ (
EqClass (u,C)))
/\ (
EqClass (u,D)))
/\ (
EqClass (u,E)))
/\ (
EqClass (u,F)))
/\ (
EqClass (z,A))) by
A22,
XBOOLE_0:def 4;
hence thesis by
A5,
XBOOLE_0:def 7;
end;
theorem ::
BVFUNC14:41
for G be
Subset of (
PARTITIONS Y), A,B,C,D,E,F be
a_partition of Y, z,u be
Element of Y, h be
Function st G is
independent & G
=
{A, B, C, D, E, F} & A
<> B & A
<> C & A
<> D & A
<> E & A
<> F & B
<> C & B
<> D & B
<> E & B
<> F & C
<> D & C
<> E & C
<> F & D
<> E & D
<> F & E
<> F & (
EqClass (z,(((C
'/\' D)
'/\' E)
'/\' F)))
= (
EqClass (u,(((C
'/\' D)
'/\' E)
'/\' F))) holds (
EqClass (u,(
CompF (A,G))))
meets (
EqClass (z,(
CompF (B,G))))
proof
let G be
Subset of (
PARTITIONS Y);
let A,B,C,D,E,F be
a_partition of Y;
let z,u be
Element of Y;
let h be
Function;
assume that
A1: G is
independent and
A2: G
=
{A, B, C, D, E, F} and
A3: A
<> B & A
<> C & A
<> D & A
<> E & A
<> F & B
<> C & B
<> D & B
<> E & B
<> F & C
<> D & C
<> E & C
<> F & D
<> E & D
<> F & E
<> F and
A4: (
EqClass (z,(((C
'/\' D)
'/\' E)
'/\' F)))
= (
EqClass (u,(((C
'/\' D)
'/\' E)
'/\' F)));
set h = ((((((B
.--> (
EqClass (u,B)))
+* (C
.--> (
EqClass (u,C))))
+* (D
.--> (
EqClass (u,D))))
+* (E
.--> (
EqClass (u,E))))
+* (F
.--> (
EqClass (u,F))))
+* (A
.--> (
EqClass (z,A))));
A5: (h
. A)
= (
EqClass (z,A)) by
A3,
Th37;
set I = (
EqClass (z,A)), GG = (
EqClass (u,((((B
'/\' C)
'/\' D)
'/\' E)
'/\' F)));
set H = (
EqClass (z,(
CompF (B,G))));
A6: (A
'/\' (((C
'/\' D)
'/\' E)
'/\' F))
= ((A
'/\' ((C
'/\' D)
'/\' E))
'/\' F) by
PARTIT1: 14
.= (((A
'/\' (C
'/\' D))
'/\' E)
'/\' F) by
PARTIT1: 14
.= ((((A
'/\' C)
'/\' D)
'/\' E)
'/\' F) by
PARTIT1: 14;
GG
= ((
EqClass (u,(((B
'/\' C)
'/\' D)
'/\' E)))
/\ (
EqClass (u,F))) by
Th1;
then GG
= (((
EqClass (u,((B
'/\' C)
'/\' D)))
/\ (
EqClass (u,E)))
/\ (
EqClass (u,F))) by
Th1;
then GG
= ((((
EqClass (u,(B
'/\' C)))
/\ (
EqClass (u,D)))
/\ (
EqClass (u,E)))
/\ (
EqClass (u,F))) by
Th1;
then
A7: (GG
/\ I)
= ((((((
EqClass (u,B))
/\ (
EqClass (u,C)))
/\ (
EqClass (u,D)))
/\ (
EqClass (u,E)))
/\ (
EqClass (u,F)))
/\ (
EqClass (z,A))) by
Th1;
A8: (h
. B)
= (
EqClass (u,B)) by
A3,
Th37;
A9: (h
. F)
= (
EqClass (u,F)) by
A3,
Th37;
A10: (h
. E)
= (
EqClass (u,E)) by
A3,
Th37;
A11: (h
. D)
= (
EqClass (u,D)) by
A3,
Th37;
A12: (h
. C)
= (
EqClass (u,C)) by
A3,
Th37;
A13: (
rng h)
=
{(h
. A), (h
. B), (h
. C), (h
. D), (h
. E), (h
. F)} by
Th39;
(
rng h)
c= (
bool Y)
proof
let t be
object;
assume
A14: t
in (
rng h);
now
per cases by
A13,
A14,
ENUMSET1:def 4;
case t
= (h
. A);
hence thesis by
A5;
end;
case t
= (h
. B);
hence thesis by
A8;
end;
case t
= (h
. C);
hence thesis by
A12;
end;
case t
= (h
. D);
hence thesis by
A11;
end;
case t
= (h
. E);
hence thesis by
A10;
end;
case t
= (h
. F);
hence thesis by
A9;
end;
end;
hence thesis;
end;
then
reconsider FF = (
rng h) as
Subset-Family of Y;
A15: (
dom h)
= G by
A2,
Th38;
then A
in (
dom h) by
A2,
ENUMSET1:def 4;
then
A16: (h
. A)
in (
rng h) by
FUNCT_1:def 3;
then
A17: (
Intersect FF)
= (
meet (
rng h)) by
SETFAM_1:def 9;
for d be
set st d
in G holds (h
. d)
in d
proof
let d be
set;
assume
A18: d
in G;
now
per cases by
A2,
A18,
ENUMSET1:def 4;
case d
= A;
hence thesis by
A5;
end;
case d
= B;
hence thesis by
A8;
end;
case d
= C;
hence thesis by
A12;
end;
case d
= D;
hence thesis by
A11;
end;
case d
= E;
hence thesis by
A10;
end;
case d
= F;
hence thesis by
A9;
end;
end;
hence thesis;
end;
then (
Intersect FF)
<>
{} by
A1,
A15,
BVFUNC_2:def 5;
then
consider m be
object such that
A19: m
in (
Intersect FF) by
XBOOLE_0:def 1;
D
in (
dom h) by
A2,
A15,
ENUMSET1:def 4;
then (h
. D)
in (
rng h) by
FUNCT_1:def 3;
then m
in (h
. D) by
A17,
A19,
SETFAM_1:def 1;
then
A20: m
in (
EqClass (u,D)) by
A3,
Th37;
C
in (
dom h) by
A2,
A15,
ENUMSET1:def 4;
then (h
. C)
in (
rng h) by
FUNCT_1:def 3;
then m
in (h
. C) by
A17,
A19,
SETFAM_1:def 1;
then
A21: m
in (
EqClass (u,C)) by
A3,
Th37;
B
in (
dom h) by
A2,
A15,
ENUMSET1:def 4;
then (h
. B)
in (
rng h) by
FUNCT_1:def 3;
then m
in (h
. B) by
A17,
A19,
SETFAM_1:def 1;
then m
in (
EqClass (u,B)) by
A3,
Th37;
then m
in ((
EqClass (u,B))
/\ (
EqClass (u,C))) by
A21,
XBOOLE_0:def 4;
then
A22: m
in (((
EqClass (u,B))
/\ (
EqClass (u,C)))
/\ (
EqClass (u,D))) by
A20,
XBOOLE_0:def 4;
F
in (
dom h) by
A2,
A15,
ENUMSET1:def 4;
then (h
. F)
in (
rng h) by
FUNCT_1:def 3;
then m
in (h
. F) by
A17,
A19,
SETFAM_1:def 1;
then
A23: m
in (
EqClass (u,F)) by
A3,
Th37;
E
in (
dom h) by
A2,
A15,
ENUMSET1:def 4;
then (h
. E)
in (
rng h) by
FUNCT_1:def 3;
then m
in (h
. E) by
A17,
A19,
SETFAM_1:def 1;
then m
in (
EqClass (u,E)) by
A3,
Th37;
then m
in ((((
EqClass (u,B))
/\ (
EqClass (u,C)))
/\ (
EqClass (u,D)))
/\ (
EqClass (u,E))) by
A22,
XBOOLE_0:def 4;
then
A24: m
in (((((
EqClass (u,B))
/\ (
EqClass (u,C)))
/\ (
EqClass (u,D)))
/\ (
EqClass (u,E)))
/\ (
EqClass (u,F))) by
A23,
XBOOLE_0:def 4;
m
in (h
. A) by
A16,
A17,
A19,
SETFAM_1:def 1;
then m
in (
EqClass (z,A)) by
A3,
Th37;
then (GG
/\ I)
<>
{} by
A7,
A24,
XBOOLE_0:def 4;
then
consider p be
object such that
A25: p
in (GG
/\ I) by
XBOOLE_0:def 1;
reconsider p as
Element of Y by
A25;
A26: p
in GG by
A25,
XBOOLE_0:def 4;
set L = (
EqClass (z,(((C
'/\' D)
'/\' E)
'/\' F)));
GG
= (
EqClass (u,(((B
'/\' (C
'/\' D))
'/\' E)
'/\' F))) by
PARTIT1: 14;
then GG
= (
EqClass (u,((B
'/\' ((C
'/\' D)
'/\' E))
'/\' F))) by
PARTIT1: 14;
then GG
= (
EqClass (u,(B
'/\' (((C
'/\' D)
'/\' E)
'/\' F)))) by
PARTIT1: 14;
then
A27: GG
c= L by
A4,
BVFUNC11: 3;
A28: z
in H by
EQREL_1:def 6;
set K = (
EqClass (p,(((C
'/\' D)
'/\' E)
'/\' F)));
p
in K & p
in I by
A25,
EQREL_1:def 6,
XBOOLE_0:def 4;
then
A29: p
in (I
/\ K) by
XBOOLE_0:def 4;
then (I
/\ K)
in (
INTERSECTION (A,(((C
'/\' D)
'/\' E)
'/\' F))) & not (I
/\ K)
in
{
{} } by
SETFAM_1:def 5,
TARSKI:def 1;
then (I
/\ K)
in ((
INTERSECTION (A,(((C
'/\' D)
'/\' E)
'/\' F)))
\
{
{} }) by
XBOOLE_0:def 5;
then
A30: (I
/\ K)
in (A
'/\' (((C
'/\' D)
'/\' E)
'/\' F)) by
PARTIT1:def 4;
p
in (
EqClass (p,(((C
'/\' D)
'/\' E)
'/\' F))) by
EQREL_1:def 6;
then K
meets L by
A27,
A26,
XBOOLE_0: 3;
then K
= L by
EQREL_1: 41;
then
A31: z
in K by
EQREL_1:def 6;
z
in I by
EQREL_1:def 6;
then
A32: z
in (I
/\ K) by
A31,
XBOOLE_0:def 4;
(
CompF (B,G))
= ((((A
'/\' C)
'/\' D)
'/\' E)
'/\' F) by
A2,
A3,
Th32;
then
A33: (I
/\ K)
= H or (I
/\ K)
misses H by
A30,
A6,
EQREL_1:def 4;
GG
= (
EqClass (u,(
CompF (A,G)))) by
A2,
A3,
Th31;
hence thesis by
A29,
A26,
A32,
A28,
A33,
XBOOLE_0: 3;
end;
begin
reserve Y for non
empty
set,
G for
Subset of (
PARTITIONS Y),
A,B,C,D,E,F,J,M for
a_partition of Y,
x,x1,x2,x3,x4,x5,x6,x7,x8,x9 for
set;
theorem ::
BVFUNC14:42
Th42: G
=
{A, B, C, D, E, F, J} & A
<> B & A
<> C & A
<> D & A
<> E & A
<> F & A
<> J & B
<> C & B
<> D & B
<> E & B
<> F & B
<> J & C
<> D & C
<> E & C
<> F & C
<> J & D
<> E & D
<> F & D
<> J & E
<> F & E
<> J & F
<> J implies (
CompF (A,G))
= (((((B
'/\' C)
'/\' D)
'/\' E)
'/\' F)
'/\' J)
proof
assume that
A1: G
=
{A, B, C, D, E, F, J} and
A2: A
<> B and
A3: A
<> C and
A4: A
<> D & A
<> E and
A5: A
<> F & A
<> J and
A6: B
<> C & B
<> D & B
<> E & B
<> F & B
<> J & C
<> D & C
<> E & C
<> F & C
<> J & D
<> E & D
<> F & D
<> J & E
<> F & E
<> J & F
<> J;
A7: (G
\
{A})
= ((
{A}
\/
{B, C, D, E, F, J})
\
{A}) by
A1,
ENUMSET1: 16;
( not D
in
{A}) & not E
in
{A} by
A4,
TARSKI:def 1;
then
A8: (
{D, E}
\
{A})
=
{D, E} by
ZFMISC_1: 63;
A9: ( not F
in
{A}) & not J
in
{A} by
A5,
TARSKI:def 1;
A10: not C
in
{A} by
A3,
TARSKI:def 1;
A11: not B
in
{A} by
A2,
TARSKI:def 1;
(
{B, C, D, E, F, J}
\
{A})
= ((
{B}
\/
{C, D, E, F, J})
\
{A}) by
ENUMSET1: 11
.= ((
{B}
\
{A})
\/ (
{C, D, E, F, J}
\
{A})) by
XBOOLE_1: 42
.= (
{B}
\/ (
{C, D, E, F, J}
\
{A})) by
A11,
ZFMISC_1: 59
.= (
{B}
\/ ((
{C}
\/
{D, E, F, J})
\
{A})) by
ENUMSET1: 7
.= (
{B}
\/ ((
{C}
\
{A})
\/ (
{D, E, F, J}
\
{A}))) by
XBOOLE_1: 42
.= (
{B}
\/ ((
{C}
\
{A})
\/ ((
{D, E}
\/
{F, J})
\
{A}))) by
ENUMSET1: 5
.= (
{B}
\/ ((
{C}
\
{A})
\/ ((
{D, E}
\
{A})
\/ (
{F, J}
\
{A})))) by
XBOOLE_1: 42
.= (
{B}
\/ ((
{C}
\
{A})
\/ (
{D, E}
\/
{F, J}))) by
A9,
A8,
ZFMISC_1: 63
.= (
{B}
\/ (
{C}
\/ (
{D, E}
\/
{F, J}))) by
A10,
ZFMISC_1: 59
.= (
{B}
\/ (
{C}
\/
{D, E, F, J})) by
ENUMSET1: 5
.= (
{B}
\/
{C, D, E, F, J}) by
ENUMSET1: 7
.=
{B, C, D, E, F, J} by
ENUMSET1: 11;
then
A12: (G
\
{A})
= ((
{A}
\
{A})
\/
{B, C, D, E, F, J}) by
A7,
XBOOLE_1: 42
.= (
{}
\/
{B, C, D, E, F, J}) by
XBOOLE_1: 37;
A13: (
'/\' (G
\
{A}))
c= (((((B
'/\' C)
'/\' D)
'/\' E)
'/\' F)
'/\' J)
proof
let x be
object;
reconsider xx = x as
set by
TARSKI: 1;
assume x
in (
'/\' (G
\
{A}));
then
consider h be
Function, FF be
Subset-Family of Y such that
A14: (
dom h)
= (G
\
{A}) and
A15: (
rng h)
= FF and
A16: for d be
set st d
in (G
\
{A}) holds (h
. d)
in d and
A17: x
= (
Intersect FF) and
A18: x
<>
{} by
BVFUNC_2:def 1;
A19: C
in (G
\
{A}) by
A12,
ENUMSET1:def 4;
then
A20: (h
. C)
in C by
A16;
set mbcd = (((h
. B)
/\ (h
. C))
/\ (h
. D));
A21: E
in (G
\
{A}) by
A12,
ENUMSET1:def 4;
then
A22: (h
. E)
in (
rng h) by
A14,
FUNCT_1:def 3;
set mbc = ((h
. B)
/\ (h
. C));
A23: B
in (G
\
{A}) by
A12,
ENUMSET1:def 4;
then (h
. B)
in B by
A16;
then
A24: mbc
in (
INTERSECTION (B,C)) by
A20,
SETFAM_1:def 5;
A25: (h
. B)
in (
rng h) by
A14,
A23,
FUNCT_1:def 3;
then
A26: (
Intersect FF)
= (
meet (
rng h)) by
A15,
SETFAM_1:def 9;
A27: (h
. C)
in (
rng h) by
A14,
A19,
FUNCT_1:def 3;
A28: F
in (G
\
{A}) by
A12,
ENUMSET1:def 4;
then
A29: (h
. F)
in (
rng h) by
A14,
FUNCT_1:def 3;
set mbcdef = (((((h
. B)
/\ (h
. C))
/\ (h
. D))
/\ (h
. E))
/\ (h
. F));
set mbcde = ((((h
. B)
/\ (h
. C))
/\ (h
. D))
/\ (h
. E));
A30: not x
in
{
{} } by
A18,
TARSKI:def 1;
A31: J
in (G
\
{A}) by
A12,
ENUMSET1:def 4;
then
A32: (h
. J)
in (
rng h) by
A14,
FUNCT_1:def 3;
A33: D
in (G
\
{A}) by
A12,
ENUMSET1:def 4;
then
A34: (h
. D)
in (
rng h) by
A14,
FUNCT_1:def 3;
A35: xx
c= ((((((h
. B)
/\ (h
. C))
/\ (h
. D))
/\ (h
. E))
/\ (h
. F))
/\ (h
. J))
proof
let m be
object;
assume
A36: m
in xx;
then m
in (h
. B) & m
in (h
. C) by
A17,
A25,
A27,
A26,
SETFAM_1:def 1;
then
A37: m
in ((h
. B)
/\ (h
. C)) by
XBOOLE_0:def 4;
m
in (h
. D) by
A17,
A34,
A26,
A36,
SETFAM_1:def 1;
then
A38: m
in (((h
. B)
/\ (h
. C))
/\ (h
. D)) by
A37,
XBOOLE_0:def 4;
m
in (h
. E) by
A17,
A22,
A26,
A36,
SETFAM_1:def 1;
then
A39: m
in ((((h
. B)
/\ (h
. C))
/\ (h
. D))
/\ (h
. E)) by
A38,
XBOOLE_0:def 4;
m
in (h
. F) by
A17,
A29,
A26,
A36,
SETFAM_1:def 1;
then
A40: m
in (((((h
. B)
/\ (h
. C))
/\ (h
. D))
/\ (h
. E))
/\ (h
. F)) by
A39,
XBOOLE_0:def 4;
m
in (h
. J) by
A17,
A32,
A26,
A36,
SETFAM_1:def 1;
hence thesis by
A40,
XBOOLE_0:def 4;
end;
then mbcd
<>
{} by
A18;
then
A41: not mbcd
in
{
{} } by
TARSKI:def 1;
((((((h
. B)
/\ (h
. C))
/\ (h
. D))
/\ (h
. E))
/\ (h
. F))
/\ (h
. J))
c= xx
proof
A42: (
rng h)
c=
{(h
. B), (h
. C), (h
. D), (h
. E), (h
. F), (h
. J)}
proof
let u be
object;
assume u
in (
rng h);
then
consider x1 be
object such that
A43: x1
in (
dom h) and
A44: u
= (h
. x1) by
FUNCT_1:def 3;
x1
= B or x1
= C or x1
= D or x1
= E or x1
= F or x1
= J by
A12,
A14,
A43,
ENUMSET1:def 4;
hence thesis by
A44,
ENUMSET1:def 4;
end;
let m be
object;
assume
A45: m
in ((((((h
. B)
/\ (h
. C))
/\ (h
. D))
/\ (h
. E))
/\ (h
. F))
/\ (h
. J));
then
A46: m
in (h
. J) by
XBOOLE_0:def 4;
A47: m
in (((((h
. B)
/\ (h
. C))
/\ (h
. D))
/\ (h
. E))
/\ (h
. F)) by
A45,
XBOOLE_0:def 4;
then
A48: m
in ((((h
. B)
/\ (h
. C))
/\ (h
. D))
/\ (h
. E)) by
XBOOLE_0:def 4;
then
A49: m
in (h
. E) by
XBOOLE_0:def 4;
A50: m
in (((h
. B)
/\ (h
. C))
/\ (h
. D)) by
A48,
XBOOLE_0:def 4;
then
A51: m
in (h
. D) by
XBOOLE_0:def 4;
m
in ((h
. B)
/\ (h
. C)) by
A50,
XBOOLE_0:def 4;
then
A52: m
in (h
. B) & m
in (h
. C) by
XBOOLE_0:def 4;
m
in (h
. F) by
A47,
XBOOLE_0:def 4;
then for y be
set holds y
in (
rng h) implies m
in y by
A52,
A51,
A49,
A46,
A42,
ENUMSET1:def 4;
hence thesis by
A17,
A25,
A26,
SETFAM_1:def 1;
end;
then
A53: ((((((h
. B)
/\ (h
. C))
/\ (h
. D))
/\ (h
. E))
/\ (h
. F))
/\ (h
. J))
= x by
A35,
XBOOLE_0:def 10;
mbc
<>
{} by
A18,
A35;
then not mbc
in
{
{} } by
TARSKI:def 1;
then mbc
in ((
INTERSECTION (B,C))
\
{
{} }) by
A24,
XBOOLE_0:def 5;
then
A54: mbc
in (B
'/\' C) by
PARTIT1:def 4;
(h
. D)
in D by
A16,
A33;
then mbcd
in (
INTERSECTION ((B
'/\' C),D)) by
A54,
SETFAM_1:def 5;
then mbcd
in ((
INTERSECTION ((B
'/\' C),D))
\
{
{} }) by
A41,
XBOOLE_0:def 5;
then
A55: mbcd
in ((B
'/\' C)
'/\' D) by
PARTIT1:def 4;
mbcde
<>
{} by
A18,
A35;
then
A56: not mbcde
in
{
{} } by
TARSKI:def 1;
(h
. E)
in E by
A16,
A21;
then mbcde
in (
INTERSECTION (((B
'/\' C)
'/\' D),E)) by
A55,
SETFAM_1:def 5;
then mbcde
in ((
INTERSECTION (((B
'/\' C)
'/\' D),E))
\
{
{} }) by
A56,
XBOOLE_0:def 5;
then
A57: mbcde
in (((B
'/\' C)
'/\' D)
'/\' E) by
PARTIT1:def 4;
mbcdef
<>
{} by
A18,
A35;
then
A58: not mbcdef
in
{
{} } by
TARSKI:def 1;
(h
. F)
in F by
A16,
A28;
then mbcdef
in (
INTERSECTION ((((B
'/\' C)
'/\' D)
'/\' E),F)) by
A57,
SETFAM_1:def 5;
then mbcdef
in ((
INTERSECTION ((((B
'/\' C)
'/\' D)
'/\' E),F))
\
{
{} }) by
A58,
XBOOLE_0:def 5;
then
A59: mbcdef
in ((((B
'/\' C)
'/\' D)
'/\' E)
'/\' F) by
PARTIT1:def 4;
(h
. J)
in J by
A16,
A31;
then x
in (
INTERSECTION (((((B
'/\' C)
'/\' D)
'/\' E)
'/\' F),J)) by
A53,
A59,
SETFAM_1:def 5;
then x
in ((
INTERSECTION (((((B
'/\' C)
'/\' D)
'/\' E)
'/\' F),J))
\
{
{} }) by
A30,
XBOOLE_0:def 5;
hence thesis by
PARTIT1:def 4;
end;
A60: (((((B
'/\' C)
'/\' D)
'/\' E)
'/\' F)
'/\' J)
c= (
'/\' (G
\
{A}))
proof
let x be
object;
reconsider xx = x as
set by
TARSKI: 1;
assume
A61: x
in (((((B
'/\' C)
'/\' D)
'/\' E)
'/\' F)
'/\' J);
then
A62: x
<>
{} by
EQREL_1:def 4;
x
in ((
INTERSECTION (((((B
'/\' C)
'/\' D)
'/\' E)
'/\' F),J))
\
{
{} }) by
A61,
PARTIT1:def 4;
then
consider bcdef,j be
set such that
A63: bcdef
in ((((B
'/\' C)
'/\' D)
'/\' E)
'/\' F) and
A64: j
in J and
A65: x
= (bcdef
/\ j) by
SETFAM_1:def 5;
bcdef
in ((
INTERSECTION ((((B
'/\' C)
'/\' D)
'/\' E),F))
\
{
{} }) by
A63,
PARTIT1:def 4;
then
consider bcde,f be
set such that
A66: bcde
in (((B
'/\' C)
'/\' D)
'/\' E) and
A67: f
in F and
A68: bcdef
= (bcde
/\ f) by
SETFAM_1:def 5;
bcde
in ((
INTERSECTION (((B
'/\' C)
'/\' D),E))
\
{
{} }) by
A66,
PARTIT1:def 4;
then
consider bcd,e be
set such that
A69: bcd
in ((B
'/\' C)
'/\' D) and
A70: e
in E and
A71: bcde
= (bcd
/\ e) by
SETFAM_1:def 5;
bcd
in ((
INTERSECTION ((B
'/\' C),D))
\
{
{} }) by
A69,
PARTIT1:def 4;
then
consider bc,d be
set such that
A72: bc
in (B
'/\' C) and
A73: d
in D and
A74: bcd
= (bc
/\ d) by
SETFAM_1:def 5;
bc
in ((
INTERSECTION (B,C))
\
{
{} }) by
A72,
PARTIT1:def 4;
then
consider b,c be
set such that
A75: b
in B & c
in C and
A76: bc
= (b
/\ c) by
SETFAM_1:def 5;
set h = ((((((B
.--> b)
+* (C
.--> c))
+* (D
.--> d))
+* (E
.--> e))
+* (F
.--> f))
+* (J
.--> j));
A77: (h
. B)
= b by
A6,
Th37;
A78: (
dom h)
=
{J, B, C, D, E, F} by
Th38
.= (
{J}
\/
{B, C, D, E, F}) by
ENUMSET1: 11
.=
{B, C, D, E, F, J} by
ENUMSET1: 15;
then D
in (
dom h) by
ENUMSET1:def 4;
then
A79: (h
. D)
in (
rng h) by
FUNCT_1:def 3;
A80: for p be
set st p
in (G
\
{A}) holds (h
. p)
in p
proof
let p be
set;
assume p
in (G
\
{A});
then p
= B or p
= C or p
= D or p
= E or p
= F or p
= J by
A12,
ENUMSET1:def 4;
hence thesis by
A6,
A64,
A67,
A70,
A73,
A75,
Th37;
end;
E
in (
dom h) by
A78,
ENUMSET1:def 4;
then
A81: (h
. E)
in (
rng h) by
FUNCT_1:def 3;
C
in (
dom h) by
A78,
ENUMSET1:def 4;
then
A82: (h
. C)
in (
rng h) by
FUNCT_1:def 3;
A83: (h
. C)
= c by
A6,
Th37;
A84: (
rng h)
c=
{(h
. B), (h
. C), (h
. D), (h
. E), (h
. F), (h
. J)}
proof
let t be
object;
assume t
in (
rng h);
then
consider x1 be
object such that
A85: x1
in (
dom h) and
A86: t
= (h
. x1) by
FUNCT_1:def 3;
x1
= B or x1
= C or x1
= D or x1
= E or x1
= F or x1
= J by
A78,
A85,
ENUMSET1:def 4;
hence thesis by
A86,
ENUMSET1:def 4;
end;
J
in (
dom h) by
A78,
ENUMSET1:def 4;
then
A87: (h
. J)
in (
rng h) by
FUNCT_1:def 3;
F
in (
dom h) by
A78,
ENUMSET1:def 4;
then
A88: (h
. F)
in (
rng h) by
FUNCT_1:def 3;
B
in (
dom h) by
A78,
ENUMSET1:def 4;
then
A89: (h
. B)
in (
rng h) by
FUNCT_1:def 3;
{(h
. B), (h
. C), (h
. D), (h
. E), (h
. F), (h
. J)}
c= (
rng h) by
A79,
A89,
A82,
A81,
A88,
A87,
ENUMSET1:def 4;
then
A90: (
rng h)
=
{(h
. B), (h
. C), (h
. D), (h
. E), (h
. F), (h
. J)} by
A84,
XBOOLE_0:def 10;
A91: (h
. J)
= j by
A6,
Th37;
A92: (h
. F)
= f by
A6,
Th37;
A93: (h
. E)
= e by
A6,
Th37;
A94: (h
. D)
= d by
A6,
Th37;
(
rng h)
c= (
bool Y)
proof
let t be
object;
assume t
in (
rng h);
then t
= (h
. D) or t
= (h
. B) or t
= (h
. C) or t
= (h
. E) or t
= (h
. F) or t
= (h
. J) by
A84,
ENUMSET1:def 4;
hence thesis by
A64,
A67,
A70,
A73,
A75,
A94,
A77,
A83,
A93,
A92,
A91;
end;
then
reconsider FF = (
rng h) as
Subset-Family of Y;
A95: (
dom ((((((B
.--> b)
+* (C
.--> c))
+* (D
.--> d))
+* (E
.--> e))
+* (F
.--> f))
+* (J
.--> j)))
=
{J, B, C, D, E, F} by
Th38
.= (
{J}
\/
{B, C, D, E, F}) by
ENUMSET1: 11
.=
{B, C, D, E, F, J} by
ENUMSET1: 15;
reconsider h as
Function;
A96: xx
c= (
Intersect FF)
proof
let u be
object;
assume
A97: u
in xx;
for y be
set holds y
in FF implies u
in y
proof
let y be
set;
assume
A98: y
in FF;
now
per cases by
A84,
A98,
ENUMSET1:def 4;
case
A99: y
= (h
. D);
u
in (((d
/\ ((b
/\ c)
/\ e))
/\ f)
/\ j) by
A65,
A68,
A71,
A74,
A76,
A97,
XBOOLE_1: 16;
then u
in ((d
/\ (((b
/\ c)
/\ e)
/\ f))
/\ j) by
XBOOLE_1: 16;
then u
in (d
/\ ((((b
/\ c)
/\ e)
/\ f)
/\ j)) by
XBOOLE_1: 16;
hence thesis by
A94,
A99,
XBOOLE_0:def 4;
end;
case
A100: y
= (h
. B);
u
in ((((c
/\ (d
/\ b))
/\ e)
/\ f)
/\ j) by
A65,
A68,
A71,
A74,
A76,
A97,
XBOOLE_1: 16;
then u
in (((c
/\ ((d
/\ b)
/\ e))
/\ f)
/\ j) by
XBOOLE_1: 16;
then u
in (((c
/\ ((d
/\ e)
/\ b))
/\ f)
/\ j) by
XBOOLE_1: 16;
then u
in ((c
/\ (((d
/\ e)
/\ b)
/\ f))
/\ j) by
XBOOLE_1: 16;
then u
in (c
/\ ((((d
/\ e)
/\ b)
/\ f)
/\ j)) by
XBOOLE_1: 16;
then u
in (c
/\ (((d
/\ e)
/\ (f
/\ b))
/\ j)) by
XBOOLE_1: 16;
then u
in (c
/\ ((d
/\ e)
/\ ((f
/\ b)
/\ j))) by
XBOOLE_1: 16;
then u
in (c
/\ ((d
/\ e)
/\ (f
/\ (j
/\ b)))) by
XBOOLE_1: 16;
then u
in ((c
/\ (d
/\ e))
/\ (f
/\ (j
/\ b))) by
XBOOLE_1: 16;
then u
in (((c
/\ (d
/\ e))
/\ f)
/\ (j
/\ b)) by
XBOOLE_1: 16;
then u
in ((((c
/\ (d
/\ e))
/\ f)
/\ j)
/\ b) by
XBOOLE_1: 16;
hence thesis by
A77,
A100,
XBOOLE_0:def 4;
end;
case
A101: y
= (h
. C);
u
in ((((c
/\ (d
/\ b))
/\ e)
/\ f)
/\ j) by
A65,
A68,
A71,
A74,
A76,
A97,
XBOOLE_1: 16;
then u
in (((c
/\ ((d
/\ b)
/\ e))
/\ f)
/\ j) by
XBOOLE_1: 16;
then u
in (((c
/\ ((d
/\ e)
/\ b))
/\ f)
/\ j) by
XBOOLE_1: 16;
then u
in ((c
/\ (((d
/\ e)
/\ b)
/\ f))
/\ j) by
XBOOLE_1: 16;
then u
in (c
/\ ((((d
/\ e)
/\ b)
/\ f)
/\ j)) by
XBOOLE_1: 16;
hence thesis by
A83,
A101,
XBOOLE_0:def 4;
end;
case
A102: y
= (h
. E);
u
in ((((b
/\ c)
/\ d)
/\ (f
/\ e))
/\ j) by
A65,
A68,
A71,
A74,
A76,
A97,
XBOOLE_1: 16;
then u
in (((b
/\ c)
/\ d)
/\ ((f
/\ e)
/\ j)) by
XBOOLE_1: 16;
then u
in (((b
/\ c)
/\ d)
/\ ((f
/\ j)
/\ e)) by
XBOOLE_1: 16;
then u
in ((((b
/\ c)
/\ d)
/\ (f
/\ j))
/\ e) by
XBOOLE_1: 16;
hence thesis by
A93,
A102,
XBOOLE_0:def 4;
end;
case
A103: y
= (h
. F);
u
in (((((b
/\ c)
/\ d)
/\ e)
/\ j)
/\ f) by
A65,
A68,
A71,
A74,
A76,
A97,
XBOOLE_1: 16;
hence thesis by
A92,
A103,
XBOOLE_0:def 4;
end;
case y
= (h
. J);
hence thesis by
A65,
A91,
A97,
XBOOLE_0:def 4;
end;
end;
hence thesis;
end;
then u
in (
meet FF) by
A90,
SETFAM_1:def 1;
hence thesis by
A90,
SETFAM_1:def 9;
end;
A104: (
Intersect FF)
= (
meet (
rng h)) by
A79,
SETFAM_1:def 9;
(
Intersect FF)
c= xx
proof
let t be
object;
assume
A105: t
in (
Intersect FF);
(h
. C)
in (
rng h) by
A90,
ENUMSET1:def 4;
then
A106: t
in c by
A83,
A104,
A105,
SETFAM_1:def 1;
(h
. B)
in (
rng h) by
A90,
ENUMSET1:def 4;
then t
in b by
A77,
A104,
A105,
SETFAM_1:def 1;
then
A107: t
in (b
/\ c) by
A106,
XBOOLE_0:def 4;
(h
. D)
in (
rng h) by
A90,
ENUMSET1:def 4;
then t
in d by
A94,
A104,
A105,
SETFAM_1:def 1;
then
A108: t
in ((b
/\ c)
/\ d) by
A107,
XBOOLE_0:def 4;
(h
. E)
in (
rng h) by
A90,
ENUMSET1:def 4;
then t
in e by
A93,
A104,
A105,
SETFAM_1:def 1;
then
A109: t
in (((b
/\ c)
/\ d)
/\ e) by
A108,
XBOOLE_0:def 4;
(h
. F)
in (
rng h) by
A90,
ENUMSET1:def 4;
then t
in f by
A92,
A104,
A105,
SETFAM_1:def 1;
then
A110: t
in ((((b
/\ c)
/\ d)
/\ e)
/\ f) by
A109,
XBOOLE_0:def 4;
(h
. J)
in (
rng h) by
A90,
ENUMSET1:def 4;
then t
in j by
A91,
A104,
A105,
SETFAM_1:def 1;
hence thesis by
A65,
A68,
A71,
A74,
A76,
A110,
XBOOLE_0:def 4;
end;
then x
= (
Intersect FF) by
A96,
XBOOLE_0:def 10;
hence thesis by
A12,
A95,
A80,
A62,
BVFUNC_2:def 1;
end;
(
CompF (A,G))
= (
'/\' (G
\
{A})) by
BVFUNC_2:def 7;
hence thesis by
A60,
A13,
XBOOLE_0:def 10;
end;
theorem ::
BVFUNC14:43
Th43: G
=
{A, B, C, D, E, F, J} & A
<> B & A
<> C & A
<> D & A
<> E & A
<> F & A
<> J & B
<> C & B
<> D & B
<> E & B
<> F & B
<> J & C
<> D & C
<> E & C
<> F & C
<> J & D
<> E & D
<> F & D
<> J & E
<> F & E
<> J & F
<> J implies (
CompF (B,G))
= (((((A
'/\' C)
'/\' D)
'/\' E)
'/\' F)
'/\' J)
proof
{A, B, C, D, E, F, J}
= (
{A, B}
\/
{C, D, E, F, J}) by
ENUMSET1: 17
.=
{B, A, C, D, E, F, J} by
ENUMSET1: 17;
hence thesis by
Th42;
end;
theorem ::
BVFUNC14:44
Th44: G
=
{A, B, C, D, E, F, J} & A
<> B & A
<> C & A
<> D & A
<> E & A
<> F & A
<> J & B
<> C & B
<> D & B
<> E & B
<> F & B
<> J & C
<> D & C
<> E & C
<> F & C
<> J & D
<> E & D
<> F & D
<> J & E
<> F & E
<> J & F
<> J implies (
CompF (C,G))
= (((((A
'/\' B)
'/\' D)
'/\' E)
'/\' F)
'/\' J)
proof
{A, B, C, D, E, F, J}
= (
{A, B, C}
\/
{D, E, F, J}) by
ENUMSET1: 18
.= ((
{A}
\/
{B, C})
\/
{D, E, F, J}) by
ENUMSET1: 2
.= (
{A, C, B}
\/
{D, E, F, J}) by
ENUMSET1: 2
.= ((
{A, C}
\/
{B})
\/
{D, E, F, J}) by
ENUMSET1: 3
.= (
{C, A, B}
\/
{D, E, F, J}) by
ENUMSET1: 3
.=
{C, A, B, D, E, F, J} by
ENUMSET1: 18;
hence thesis by
Th42;
end;
theorem ::
BVFUNC14:45
Th45: G
=
{A, B, C, D, E, F, J} & A
<> B & A
<> C & A
<> D & A
<> E & A
<> F & A
<> J & B
<> C & B
<> D & B
<> E & B
<> F & B
<> J & C
<> D & C
<> E & C
<> F & C
<> J & D
<> E & D
<> F & D
<> J & E
<> F & E
<> J & F
<> J implies (
CompF (D,G))
= (((((A
'/\' B)
'/\' C)
'/\' E)
'/\' F)
'/\' J)
proof
{A, B, C, D, E, F, J}
= (
{A, B}
\/
{C, D, E, F, J}) by
ENUMSET1: 17
.= (
{A, B}
\/ (
{C, D}
\/
{E, F, J})) by
ENUMSET1: 8
.= (
{A, B}
\/
{D, C, E, F, J}) by
ENUMSET1: 8
.=
{A, B, D, C, E, F, J} by
ENUMSET1: 17;
hence thesis by
Th44;
end;
theorem ::
BVFUNC14:46
Th46: G
=
{A, B, C, D, E, F, J} & A
<> B & A
<> C & A
<> D & A
<> E & A
<> F & A
<> J & B
<> C & B
<> D & B
<> E & B
<> F & B
<> J & C
<> D & C
<> E & C
<> F & C
<> J & D
<> E & D
<> F & D
<> J & E
<> F & E
<> J & F
<> J implies (
CompF (E,G))
= (((((A
'/\' B)
'/\' C)
'/\' D)
'/\' F)
'/\' J)
proof
{A, B, C, D, E, F, J}
= (
{A, B, C}
\/
{D, E, F, J}) by
ENUMSET1: 18
.= (
{A, B, C}
\/ (
{D, E}
\/
{F, J})) by
ENUMSET1: 5
.= (
{A, B, C}
\/
{E, D, F, J}) by
ENUMSET1: 5
.=
{A, B, C, E, D, F, J} by
ENUMSET1: 18;
hence thesis by
Th45;
end;
theorem ::
BVFUNC14:47
Th47: G
=
{A, B, C, D, E, F, J} & A
<> B & A
<> C & A
<> D & A
<> E & A
<> F & A
<> J & B
<> C & B
<> D & B
<> E & B
<> F & B
<> J & C
<> D & C
<> E & C
<> F & C
<> J & D
<> E & D
<> F & D
<> J & E
<> F & E
<> J & F
<> J implies (
CompF (F,G))
= (((((A
'/\' B)
'/\' C)
'/\' D)
'/\' E)
'/\' J)
proof
{A, B, C, D, E, F, J}
= (
{A, B, C, D}
\/
{E, F, J}) by
ENUMSET1: 19
.= (
{A, B, C, D}
\/
{F, E, J}) by
ENUMSET1: 58
.=
{A, B, C, D, F, E, J} by
ENUMSET1: 19;
hence thesis by
Th46;
end;
theorem ::
BVFUNC14:48
G
=
{A, B, C, D, E, F, J} & A
<> B & A
<> C & A
<> D & A
<> E & A
<> F & A
<> J & B
<> C & B
<> D & B
<> E & B
<> F & B
<> J & C
<> D & C
<> E & C
<> F & C
<> J & D
<> E & D
<> F & D
<> J & E
<> F & E
<> J & F
<> J implies (
CompF (J,G))
= (((((A
'/\' B)
'/\' C)
'/\' D)
'/\' E)
'/\' F)
proof
{A, B, C, D, E, F, J}
= (
{A, B, C, D, E}
\/
{F, J}) by
ENUMSET1: 20
.=
{A, B, C, D, E, J, F} by
ENUMSET1: 20;
hence thesis by
Th47;
end;
theorem ::
BVFUNC14:49
Th49: for A,B,C,D,E,F,J be
set, h be
Function, A9,B9,C9,D9,E9,F9,J9 be
set st A
<> B & A
<> C & A
<> D & A
<> E & A
<> F & A
<> J & B
<> C & B
<> D & B
<> E & B
<> F & B
<> J & C
<> D & C
<> E & C
<> F & C
<> J & D
<> E & D
<> F & D
<> J & E
<> F & E
<> J & F
<> J & h
= (((((((B
.--> B9)
+* (C
.--> C9))
+* (D
.--> D9))
+* (E
.--> E9))
+* (F
.--> F9))
+* (J
.--> J9))
+* (A
.--> A9)) holds (h
. A)
= A9 & (h
. B)
= B9 & (h
. C)
= C9 & (h
. D)
= D9 & (h
. E)
= E9 & (h
. F)
= F9 & (h
. J)
= J9
proof
let A,B,C,D,E,F,J be
set;
let h be
Function;
let A9,B9,C9,D9,E9,F9,J9 be
set;
assume that
A1: A
<> B and
A2: A
<> C and
A3: A
<> D and
A4: A
<> E and
A5: A
<> F and
A6: A
<> J and
A7: B
<> C & B
<> D & B
<> E & B
<> F & B
<> J & C
<> D & C
<> E & C
<> F & C
<> J & D
<> E & D
<> F & D
<> J & E
<> F & E
<> J & F
<> J and
A8: h
= (((((((B
.--> B9)
+* (C
.--> C9))
+* (D
.--> D9))
+* (E
.--> E9))
+* (F
.--> F9))
+* (J
.--> J9))
+* (A
.--> A9));
A
in (
dom (A
.--> A9)) by
TARSKI:def 1;
then
A10: (h
. A)
= ((A
.--> A9)
. A) by
A8,
FUNCT_4: 13;
not J
in (
dom (A
.--> A9)) by
A6,
TARSKI:def 1;
then
A11: (h
. J)
= (((((((B
.--> B9)
+* (C
.--> C9))
+* (D
.--> D9))
+* (E
.--> E9))
+* (F
.--> F9))
+* (J
.--> J9))
. J) by
A8,
FUNCT_4: 11
.= J9 by
A7,
Th37;
not F
in (
dom (A
.--> A9)) by
A5,
TARSKI:def 1;
then
A12: (h
. F)
= (((((((B
.--> B9)
+* (C
.--> C9))
+* (D
.--> D9))
+* (E
.--> E9))
+* (F
.--> F9))
+* (J
.--> J9))
. F) by
A8,
FUNCT_4: 11
.= F9 by
A7,
Th37;
not E
in (
dom (A
.--> A9)) by
A4,
TARSKI:def 1;
then
A13: (h
. E)
= (((((((B
.--> B9)
+* (C
.--> C9))
+* (D
.--> D9))
+* (E
.--> E9))
+* (F
.--> F9))
+* (J
.--> J9))
. E) by
A8,
FUNCT_4: 11
.= E9 by
A7,
Th37;
not D
in (
dom (A
.--> A9)) by
A3,
TARSKI:def 1;
then
A14: (h
. D)
= (((((((B
.--> B9)
+* (C
.--> C9))
+* (D
.--> D9))
+* (E
.--> E9))
+* (F
.--> F9))
+* (J
.--> J9))
. D) by
A8,
FUNCT_4: 11
.= D9 by
A7,
Th37;
not C
in (
dom (A
.--> A9)) by
A2,
TARSKI:def 1;
then
A15: (h
. C)
= (((((((B
.--> B9)
+* (C
.--> C9))
+* (D
.--> D9))
+* (E
.--> E9))
+* (F
.--> F9))
+* (J
.--> J9))
. C) by
A8,
FUNCT_4: 11
.= C9 by
A7,
Th37;
not B
in (
dom (A
.--> A9)) by
A1,
TARSKI:def 1;
then (h
. B)
= (((((((B
.--> B9)
+* (C
.--> C9))
+* (D
.--> D9))
+* (E
.--> E9))
+* (F
.--> F9))
+* (J
.--> J9))
. B) by
A8,
FUNCT_4: 11
.= B9 by
A7,
Th37;
hence thesis by
A10,
A15,
A14,
A13,
A12,
A11,
FUNCOP_1: 72;
end;
theorem ::
BVFUNC14:50
Th50: for A,B,C,D,E,F,J be
set, h be
Function, A9,B9,C9,D9,E9,F9,J9 be
set st h
= (((((((B
.--> B9)
+* (C
.--> C9))
+* (D
.--> D9))
+* (E
.--> E9))
+* (F
.--> F9))
+* (J
.--> J9))
+* (A
.--> A9)) holds (
dom h)
=
{A, B, C, D, E, F, J}
proof
let A,B,C,D,E,F,J be
set;
let h be
Function;
let A9,B9,C9,D9,E9,F9,J9 be
set;
assume h
= (((((((B
.--> B9)
+* (C
.--> C9))
+* (D
.--> D9))
+* (E
.--> E9))
+* (F
.--> F9))
+* (J
.--> J9))
+* (A
.--> A9));
then (
dom h)
= ((
dom ((((((B
.--> B9)
+* (C
.--> C9))
+* (D
.--> D9))
+* (E
.--> E9))
+* (F
.--> F9))
+* (J
.--> J9)))
\/ (
dom (A
.--> A9))) by
FUNCT_4:def 1
.= (
{J, B, C, D, E, F}
\/ (
dom (A
.--> A9))) by
Th38
.= ((
{B, C, D, E, F}
\/
{J})
\/
{A}) by
ENUMSET1: 11
.= (
{B, C, D, E, F, J}
\/
{A}) by
ENUMSET1: 15
.=
{A, B, C, D, E, F, J} by
ENUMSET1: 16;
hence thesis;
end;
theorem ::
BVFUNC14:51
Th51: for A,B,C,D,E,F,J be
set, h be
Function, A9,B9,C9,D9,E9,F9,J9 be
set st h
= (((((((B
.--> B9)
+* (C
.--> C9))
+* (D
.--> D9))
+* (E
.--> E9))
+* (F
.--> F9))
+* (J
.--> J9))
+* (A
.--> A9)) holds (
rng h)
=
{(h
. A), (h
. B), (h
. C), (h
. D), (h
. E), (h
. F), (h
. J)}
proof
let A,B,C,D,E,F,J be
set;
let h be
Function;
let A9,B9,C9,D9,E9,F9,J9 be
set;
assume h
= (((((((B
.--> B9)
+* (C
.--> C9))
+* (D
.--> D9))
+* (E
.--> E9))
+* (F
.--> F9))
+* (J
.--> J9))
+* (A
.--> A9));
then
A1: (
dom h)
=
{A, B, C, D, E, F, J} by
Th50;
then B
in (
dom h) by
ENUMSET1:def 5;
then
A2: (h
. B)
in (
rng h) by
FUNCT_1:def 3;
F
in (
dom h) by
A1,
ENUMSET1:def 5;
then
A3: (h
. F)
in (
rng h) by
FUNCT_1:def 3;
E
in (
dom h) by
A1,
ENUMSET1:def 5;
then
A4: (h
. E)
in (
rng h) by
FUNCT_1:def 3;
D
in (
dom h) by
A1,
ENUMSET1:def 5;
then
A5: (h
. D)
in (
rng h) by
FUNCT_1:def 3;
C
in (
dom h) by
A1,
ENUMSET1:def 5;
then
A6: (h
. C)
in (
rng h) by
FUNCT_1:def 3;
J
in (
dom h) by
A1,
ENUMSET1:def 5;
then
A7: (h
. J)
in (
rng h) by
FUNCT_1:def 3;
A8: (
rng h)
c=
{(h
. A), (h
. B), (h
. C), (h
. D), (h
. E), (h
. F), (h
. J)}
proof
let t be
object;
assume t
in (
rng h);
then
consider x1 be
object such that
A9: x1
in (
dom h) and
A10: t
= (h
. x1) by
FUNCT_1:def 3;
x1
= A or x1
= B or x1
= C or x1
= D or x1
= E or x1
= F or x1
= J by
A1,
A9,
ENUMSET1:def 5;
hence thesis by
A10,
ENUMSET1:def 5;
end;
A
in (
dom h) by
A1,
ENUMSET1:def 5;
then
A11: (h
. A)
in (
rng h) by
FUNCT_1:def 3;
{(h
. A), (h
. B), (h
. C), (h
. D), (h
. E), (h
. F), (h
. J)}
c= (
rng h) by
A11,
A2,
A6,
A5,
A4,
A3,
A7,
ENUMSET1:def 5;
hence thesis by
A8,
XBOOLE_0:def 10;
end;
theorem ::
BVFUNC14:52
for G be
Subset of (
PARTITIONS Y), A,B,C,D,E,F,J be
a_partition of Y, z,u be
Element of Y st G is
independent & G
=
{A, B, C, D, E, F, J} & A
<> B & A
<> C & A
<> D & A
<> E & A
<> F & A
<> J & B
<> C & B
<> D & B
<> E & B
<> F & B
<> J & C
<> D & C
<> E & C
<> F & C
<> J & D
<> E & D
<> F & D
<> J & E
<> F & E
<> J & F
<> J holds (
EqClass (u,(((((B
'/\' C)
'/\' D)
'/\' E)
'/\' F)
'/\' J)))
meets (
EqClass (z,A))
proof
let G be
Subset of (
PARTITIONS Y);
let A,B,C,D,E,F,J be
a_partition of Y;
let z,u be
Element of Y;
assume that
A1: G is
independent and
A2: G
=
{A, B, C, D, E, F, J} and
A3: A
<> B & A
<> C & A
<> D & A
<> E & A
<> F & A
<> J & B
<> C & B
<> D & B
<> E & B
<> F & B
<> J & C
<> D & C
<> E & C
<> F & C
<> J & D
<> E & D
<> F & D
<> J & E
<> F & E
<> J & F
<> J;
set h = (((((((B
.--> (
EqClass (u,B)))
+* (C
.--> (
EqClass (u,C))))
+* (D
.--> (
EqClass (u,D))))
+* (E
.--> (
EqClass (u,E))))
+* (F
.--> (
EqClass (u,F))))
+* (J
.--> (
EqClass (u,J))))
+* (A
.--> (
EqClass (z,A))));
A4: (h
. A)
= (
EqClass (z,A)) by
A3,
Th49;
reconsider GG = (
EqClass (u,(((((B
'/\' C)
'/\' D)
'/\' E)
'/\' F)
'/\' J))) as
set;
reconsider I = (
EqClass (z,A)) as
set;
GG
= ((
EqClass (u,((((B
'/\' C)
'/\' D)
'/\' E)
'/\' F)))
/\ (
EqClass (u,J))) by
Th1;
then GG
= (((
EqClass (u,(((B
'/\' C)
'/\' D)
'/\' E)))
/\ (
EqClass (u,F)))
/\ (
EqClass (u,J))) by
Th1;
then GG
= ((((
EqClass (u,((B
'/\' C)
'/\' D)))
/\ (
EqClass (u,E)))
/\ (
EqClass (u,F)))
/\ (
EqClass (u,J))) by
Th1;
then GG
= (((((
EqClass (u,(B
'/\' C)))
/\ (
EqClass (u,D)))
/\ (
EqClass (u,E)))
/\ (
EqClass (u,F)))
/\ (
EqClass (u,J))) by
Th1;
then
A5: (GG
/\ I)
= (((((((
EqClass (u,B))
/\ (
EqClass (u,C)))
/\ (
EqClass (u,D)))
/\ (
EqClass (u,E)))
/\ (
EqClass (u,F)))
/\ (
EqClass (u,J)))
/\ (
EqClass (z,A))) by
Th1;
A6: (h
. B)
= (
EqClass (u,B)) by
A3,
Th49;
A7: (h
. F)
= (
EqClass (u,F)) by
A3,
Th49;
A8: (h
. E)
= (
EqClass (u,E)) by
A3,
Th49;
A9: (h
. J)
= (
EqClass (u,J)) by
A3,
Th49;
A10: (h
. D)
= (
EqClass (u,D)) by
A3,
Th49;
A11: (h
. C)
= (
EqClass (u,C)) by
A3,
Th49;
A12: (
rng h)
=
{(h
. A), (h
. B), (h
. C), (h
. D), (h
. E), (h
. F), (h
. J)} by
Th51;
(
rng h)
c= (
bool Y)
proof
let t be
object;
assume t
in (
rng h);
then t
= (h
. A) or t
= (h
. B) or t
= (h
. C) or t
= (h
. D) or t
= (h
. E) or t
= (h
. F) or t
= (h
. J) by
A12,
ENUMSET1:def 5;
hence thesis by
A4,
A6,
A11,
A10,
A8,
A7,
A9;
end;
then
reconsider FF = (
rng h) as
Subset-Family of Y;
A13: (
dom h)
= G by
A2,
Th50;
then A
in (
dom h) by
A2,
ENUMSET1:def 5;
then
A14: (h
. A)
in (
rng h) by
FUNCT_1:def 3;
then
A15: (
Intersect FF)
= (
meet (
rng h)) by
SETFAM_1:def 9;
for d be
set st d
in G holds (h
. d)
in d
proof
let d be
set;
assume d
in G;
then d
= A or d
= B or d
= C or d
= D or d
= E or d
= F or d
= J by
A2,
ENUMSET1:def 5;
hence thesis by
A4,
A6,
A11,
A10,
A8,
A7,
A9;
end;
then (
Intersect FF)
<>
{} by
A1,
A13,
BVFUNC_2:def 5;
then
consider m be
object such that
A16: m
in (
Intersect FF) by
XBOOLE_0:def 1;
C
in (
dom h) by
A2,
A13,
ENUMSET1:def 5;
then (h
. C)
in (
rng h) by
FUNCT_1:def 3;
then
A17: m
in (
EqClass (u,C)) by
A11,
A15,
A16,
SETFAM_1:def 1;
B
in (
dom h) by
A2,
A13,
ENUMSET1:def 5;
then (h
. B)
in (
rng h) by
FUNCT_1:def 3;
then m
in (
EqClass (u,B)) by
A6,
A15,
A16,
SETFAM_1:def 1;
then
A18: m
in ((
EqClass (u,B))
/\ (
EqClass (u,C))) by
A17,
XBOOLE_0:def 4;
D
in (
dom h) by
A2,
A13,
ENUMSET1:def 5;
then (h
. D)
in (
rng h) by
FUNCT_1:def 3;
then m
in (
EqClass (u,D)) by
A10,
A15,
A16,
SETFAM_1:def 1;
then
A19: m
in (((
EqClass (u,B))
/\ (
EqClass (u,C)))
/\ (
EqClass (u,D))) by
A18,
XBOOLE_0:def 4;
E
in (
dom h) by
A2,
A13,
ENUMSET1:def 5;
then (h
. E)
in (
rng h) by
FUNCT_1:def 3;
then m
in (
EqClass (u,E)) by
A8,
A15,
A16,
SETFAM_1:def 1;
then
A20: m
in ((((
EqClass (u,B))
/\ (
EqClass (u,C)))
/\ (
EqClass (u,D)))
/\ (
EqClass (u,E))) by
A19,
XBOOLE_0:def 4;
F
in (
dom h) by
A2,
A13,
ENUMSET1:def 5;
then (h
. F)
in (
rng h) by
FUNCT_1:def 3;
then m
in (
EqClass (u,F)) by
A7,
A15,
A16,
SETFAM_1:def 1;
then
A21: m
in (((((
EqClass (u,B))
/\ (
EqClass (u,C)))
/\ (
EqClass (u,D)))
/\ (
EqClass (u,E)))
/\ (
EqClass (u,F))) by
A20,
XBOOLE_0:def 4;
J
in (
dom h) by
A2,
A13,
ENUMSET1:def 5;
then (h
. J)
in (
rng h) by
FUNCT_1:def 3;
then m
in (
EqClass (u,J)) by
A9,
A15,
A16,
SETFAM_1:def 1;
then
A22: m
in ((((((
EqClass (u,B))
/\ (
EqClass (u,C)))
/\ (
EqClass (u,D)))
/\ (
EqClass (u,E)))
/\ (
EqClass (u,F)))
/\ (
EqClass (u,J))) by
A21,
XBOOLE_0:def 4;
m
in (
EqClass (z,A)) by
A4,
A14,
A15,
A16,
SETFAM_1:def 1;
then m
in ((
EqClass (u,(((((B
'/\' C)
'/\' D)
'/\' E)
'/\' F)
'/\' J)))
/\ (
EqClass (z,A))) by
A5,
A22,
XBOOLE_0:def 4;
hence thesis by
XBOOLE_0:def 7;
end;
theorem ::
BVFUNC14:53
for G be
Subset of (
PARTITIONS Y), A,B,C,D,E,F,J be
a_partition of Y, z,u be
Element of Y st G is
independent & G
=
{A, B, C, D, E, F, J} & A
<> B & A
<> C & A
<> D & A
<> E & A
<> F & A
<> J & B
<> C & B
<> D & B
<> E & B
<> F & B
<> J & C
<> D & C
<> E & C
<> F & C
<> J & D
<> E & D
<> F & D
<> J & E
<> F & E
<> J & F
<> J & (
EqClass (z,((((C
'/\' D)
'/\' E)
'/\' F)
'/\' J)))
= (
EqClass (u,((((C
'/\' D)
'/\' E)
'/\' F)
'/\' J))) holds (
EqClass (u,(
CompF (A,G))))
meets (
EqClass (z,(
CompF (B,G))))
proof
let G be
Subset of (
PARTITIONS Y);
let A,B,C,D,E,F,J be
a_partition of Y;
let z,u be
Element of Y;
assume that
A1: G is
independent and
A2: G
=
{A, B, C, D, E, F, J} and
A3: A
<> B & A
<> C & A
<> D & A
<> E & A
<> F & A
<> J & B
<> C & B
<> D & B
<> E & B
<> F & B
<> J & C
<> D & C
<> E & C
<> F & C
<> J & D
<> E & D
<> F & D
<> J & E
<> F & E
<> J & F
<> J and
A4: (
EqClass (z,((((C
'/\' D)
'/\' E)
'/\' F)
'/\' J)))
= (
EqClass (u,((((C
'/\' D)
'/\' E)
'/\' F)
'/\' J)));
set h = (((((((B
.--> (
EqClass (u,B)))
+* (C
.--> (
EqClass (u,C))))
+* (D
.--> (
EqClass (u,D))))
+* (E
.--> (
EqClass (u,E))))
+* (F
.--> (
EqClass (u,F))))
+* (J
.--> (
EqClass (u,J))))
+* (A
.--> (
EqClass (z,A))));
A5: (h
. A)
= (
EqClass (z,A)) by
A3,
Th49;
reconsider L = (
EqClass (z,((((C
'/\' D)
'/\' E)
'/\' F)
'/\' J))) as
set;
reconsider GG = (
EqClass (u,(((((B
'/\' C)
'/\' D)
'/\' E)
'/\' F)
'/\' J))) as
set;
reconsider I = (
EqClass (z,A)) as
set;
GG
= ((
EqClass (u,((((B
'/\' C)
'/\' D)
'/\' E)
'/\' F)))
/\ (
EqClass (u,J))) by
Th1;
then GG
= (((
EqClass (u,(((B
'/\' C)
'/\' D)
'/\' E)))
/\ (
EqClass (u,F)))
/\ (
EqClass (u,J))) by
Th1;
then GG
= ((((
EqClass (u,((B
'/\' C)
'/\' D)))
/\ (
EqClass (u,E)))
/\ (
EqClass (u,F)))
/\ (
EqClass (u,J))) by
Th1;
then GG
= (((((
EqClass (u,(B
'/\' C)))
/\ (
EqClass (u,D)))
/\ (
EqClass (u,E)))
/\ (
EqClass (u,F)))
/\ (
EqClass (u,J))) by
Th1;
then
A6: (GG
/\ I)
= (((((((
EqClass (u,B))
/\ (
EqClass (u,C)))
/\ (
EqClass (u,D)))
/\ (
EqClass (u,E)))
/\ (
EqClass (u,F)))
/\ (
EqClass (u,J)))
/\ (
EqClass (z,A))) by
Th1;
A7: (
CompF (A,G))
= (((((B
'/\' C)
'/\' D)
'/\' E)
'/\' F)
'/\' J) by
A2,
A3,
Th42;
reconsider HH = (
EqClass (z,(
CompF (B,G)))) as
set;
A8: z
in HH by
EQREL_1:def 6;
A9: (A
'/\' ((((C
'/\' D)
'/\' E)
'/\' F)
'/\' J))
= ((A
'/\' (((C
'/\' D)
'/\' E)
'/\' F))
'/\' J) by
PARTIT1: 14
.= (((A
'/\' ((C
'/\' D)
'/\' E))
'/\' F)
'/\' J) by
PARTIT1: 14
.= ((((A
'/\' (C
'/\' D))
'/\' E)
'/\' F)
'/\' J) by
PARTIT1: 14
.= (((((A
'/\' C)
'/\' D)
'/\' E)
'/\' F)
'/\' J) by
PARTIT1: 14;
A10: (h
. B)
= (
EqClass (u,B)) by
A3,
Th49;
A11: (h
. F)
= (
EqClass (u,F)) by
A3,
Th49;
A12: (h
. E)
= (
EqClass (u,E)) by
A3,
Th49;
A13: (h
. J)
= (
EqClass (u,J)) by
A3,
Th49;
A14: (h
. D)
= (
EqClass (u,D)) by
A3,
Th49;
A15: (h
. C)
= (
EqClass (u,C)) by
A3,
Th49;
A16: (
rng h)
=
{(h
. A), (h
. B), (h
. C), (h
. D), (h
. E), (h
. F), (h
. J)} by
Th51;
(
rng h)
c= (
bool Y)
proof
let t be
object;
assume t
in (
rng h);
then t
= (h
. A) or t
= (h
. B) or t
= (h
. C) or t
= (h
. D) or t
= (h
. E) or t
= (h
. F) or t
= (h
. J) by
A16,
ENUMSET1:def 5;
hence thesis by
A5,
A10,
A15,
A14,
A12,
A11,
A13;
end;
then
reconsider FF = (
rng h) as
Subset-Family of Y;
A17: (
dom h)
= G by
A2,
Th50;
then A
in (
dom h) by
A2,
ENUMSET1:def 5;
then
A18: (h
. A)
in (
rng h) by
FUNCT_1:def 3;
then
A19: (
Intersect FF)
= (
meet (
rng h)) by
SETFAM_1:def 9;
for d be
set st d
in G holds (h
. d)
in d
proof
let d be
set;
assume d
in G;
then d
= A or d
= B or d
= C or d
= D or d
= E or d
= F or d
= J by
A2,
ENUMSET1:def 5;
hence thesis by
A5,
A10,
A15,
A14,
A12,
A11,
A13;
end;
then (
Intersect FF)
<>
{} by
A1,
A17,
BVFUNC_2:def 5;
then
consider m be
object such that
A20: m
in (
Intersect FF) by
XBOOLE_0:def 1;
C
in (
dom h) by
A2,
A17,
ENUMSET1:def 5;
then (h
. C)
in (
rng h) by
FUNCT_1:def 3;
then
A21: m
in (
EqClass (u,C)) by
A15,
A19,
A20,
SETFAM_1:def 1;
B
in (
dom h) by
A2,
A17,
ENUMSET1:def 5;
then (h
. B)
in (
rng h) by
FUNCT_1:def 3;
then m
in (
EqClass (u,B)) by
A10,
A19,
A20,
SETFAM_1:def 1;
then
A22: m
in ((
EqClass (u,B))
/\ (
EqClass (u,C))) by
A21,
XBOOLE_0:def 4;
D
in (
dom h) by
A2,
A17,
ENUMSET1:def 5;
then (h
. D)
in (
rng h) by
FUNCT_1:def 3;
then m
in (
EqClass (u,D)) by
A14,
A19,
A20,
SETFAM_1:def 1;
then
A23: m
in (((
EqClass (u,B))
/\ (
EqClass (u,C)))
/\ (
EqClass (u,D))) by
A22,
XBOOLE_0:def 4;
E
in (
dom h) by
A2,
A17,
ENUMSET1:def 5;
then (h
. E)
in (
rng h) by
FUNCT_1:def 3;
then m
in (
EqClass (u,E)) by
A12,
A19,
A20,
SETFAM_1:def 1;
then
A24: m
in ((((
EqClass (u,B))
/\ (
EqClass (u,C)))
/\ (
EqClass (u,D)))
/\ (
EqClass (u,E))) by
A23,
XBOOLE_0:def 4;
F
in (
dom h) by
A2,
A17,
ENUMSET1:def 5;
then (h
. F)
in (
rng h) by
FUNCT_1:def 3;
then m
in (
EqClass (u,F)) by
A11,
A19,
A20,
SETFAM_1:def 1;
then
A25: m
in (((((
EqClass (u,B))
/\ (
EqClass (u,C)))
/\ (
EqClass (u,D)))
/\ (
EqClass (u,E)))
/\ (
EqClass (u,F))) by
A24,
XBOOLE_0:def 4;
J
in (
dom h) by
A2,
A17,
ENUMSET1:def 5;
then (h
. J)
in (
rng h) by
FUNCT_1:def 3;
then m
in (
EqClass (u,J)) by
A13,
A19,
A20,
SETFAM_1:def 1;
then
A26: m
in ((((((
EqClass (u,B))
/\ (
EqClass (u,C)))
/\ (
EqClass (u,D)))
/\ (
EqClass (u,E)))
/\ (
EqClass (u,F)))
/\ (
EqClass (u,J))) by
A25,
XBOOLE_0:def 4;
m
in (
EqClass (z,A)) by
A5,
A18,
A19,
A20,
SETFAM_1:def 1;
then
A27: (GG
/\ I)
<>
{} by
A6,
A26,
XBOOLE_0:def 4;
then
consider p be
object such that
A28: p
in (GG
/\ I) by
XBOOLE_0:def 1;
(GG
/\ I)
in (
INTERSECTION (A,(((((B
'/\' C)
'/\' D)
'/\' E)
'/\' F)
'/\' J))) & not (GG
/\ I)
in
{
{} } by
A27,
SETFAM_1:def 5,
TARSKI:def 1;
then (GG
/\ I)
in ((
INTERSECTION (A,(((((B
'/\' C)
'/\' D)
'/\' E)
'/\' F)
'/\' J)))
\
{
{} }) by
XBOOLE_0:def 5;
then (GG
/\ I)
in (A
'/\' (((((B
'/\' C)
'/\' D)
'/\' E)
'/\' F)
'/\' J)) by
PARTIT1:def 4;
then
reconsider p as
Element of Y by
A28;
A29: p
in GG by
A28,
XBOOLE_0:def 4;
reconsider K = (
EqClass (p,((((C
'/\' D)
'/\' E)
'/\' F)
'/\' J))) as
set;
A30: p
in (
EqClass (p,((((C
'/\' D)
'/\' E)
'/\' F)
'/\' J))) by
EQREL_1:def 6;
GG
= (
EqClass (u,((((B
'/\' (C
'/\' D))
'/\' E)
'/\' F)
'/\' J))) by
PARTIT1: 14;
then GG
= (
EqClass (u,(((B
'/\' ((C
'/\' D)
'/\' E))
'/\' F)
'/\' J))) by
PARTIT1: 14;
then GG
= (
EqClass (u,((B
'/\' (((C
'/\' D)
'/\' E)
'/\' F))
'/\' J))) by
PARTIT1: 14;
then GG
= (
EqClass (u,(B
'/\' ((((C
'/\' D)
'/\' E)
'/\' F)
'/\' J)))) by
PARTIT1: 14;
then GG
c= L by
A4,
BVFUNC11: 3;
then K
meets L by
A29,
A30,
XBOOLE_0: 3;
then K
= L by
EQREL_1: 41;
then
A31: z
in K by
EQREL_1:def 6;
p
in K & p
in I by
A28,
EQREL_1:def 6,
XBOOLE_0:def 4;
then
A32: p
in (I
/\ K) by
XBOOLE_0:def 4;
then (I
/\ K)
in (
INTERSECTION (A,((((C
'/\' D)
'/\' E)
'/\' F)
'/\' J))) & not (I
/\ K)
in
{
{} } by
SETFAM_1:def 5,
TARSKI:def 1;
then (I
/\ K)
in ((
INTERSECTION (A,((((C
'/\' D)
'/\' E)
'/\' F)
'/\' J)))
\
{
{} }) by
XBOOLE_0:def 5;
then
A33: (I
/\ K)
in (A
'/\' ((((C
'/\' D)
'/\' E)
'/\' F)
'/\' J)) by
PARTIT1:def 4;
z
in I by
EQREL_1:def 6;
then z
in (I
/\ K) by
A31,
XBOOLE_0:def 4;
then
A34: (I
/\ K)
meets HH by
A8,
XBOOLE_0: 3;
(
CompF (B,G))
= (((((A
'/\' C)
'/\' D)
'/\' E)
'/\' F)
'/\' J) by
A2,
A3,
Th43;
then p
in HH by
A32,
A33,
A34,
A9,
EQREL_1:def 4;
hence thesis by
A7,
A29,
XBOOLE_0: 3;
end;
theorem ::
BVFUNC14:54
Th54: G
=
{A, B, C, D, E, F, J, M} & A
<> B & A
<> C & A
<> D & A
<> E & A
<> F & A
<> J & A
<> M & B
<> C & B
<> D & B
<> E & B
<> F & B
<> J & B
<> M & C
<> D & C
<> E & C
<> F & C
<> J & C
<> M & D
<> E & D
<> F & D
<> J & D
<> M & E
<> F & E
<> J & E
<> M & F
<> J & F
<> M & J
<> M implies (
CompF (A,G))
= ((((((B
'/\' C)
'/\' D)
'/\' E)
'/\' F)
'/\' J)
'/\' M)
proof
assume that
A1: G
=
{A, B, C, D, E, F, J, M} and
A2: A
<> B and
A3: A
<> C and
A4: A
<> D & A
<> E and
A5: A
<> F & A
<> J and
A6: A
<> M and
A7: B
<> C & B
<> D & B
<> E & B
<> F & B
<> J & B
<> M & C
<> D & C
<> E & C
<> F & C
<> J & C
<> M & D
<> E & D
<> F & D
<> J & D
<> M & E
<> F & E
<> J & E
<> M & F
<> J & F
<> M & J
<> M;
A8: not B
in
{A} by
A2,
TARSKI:def 1;
(G
\
{A})
= ((
{A}
\/
{B, C, D, E, F, J, M})
\
{A}) by
A1,
ENUMSET1: 22;
then
A9: (G
\
{A})
= ((
{A}
\
{A})
\/ (
{B, C, D, E, F, J, M}
\
{A})) by
XBOOLE_1: 42;
A10: ( not D
in
{A}) & not E
in
{A} by
A4,
TARSKI:def 1;
A11: not C
in
{A} by
A3,
TARSKI:def 1;
A12: not M
in
{A} by
A6,
TARSKI:def 1;
A13: ( not F
in
{A}) & not J
in
{A} by
A5,
TARSKI:def 1;
(
{B, C, D, E, F, J, M}
\
{A})
= ((
{B}
\/
{C, D, E, F, J, M})
\
{A}) by
ENUMSET1: 16
.= ((
{B}
\
{A})
\/ (
{C, D, E, F, J, M}
\
{A})) by
XBOOLE_1: 42
.= (
{B}
\/ (
{C, D, E, F, J, M}
\
{A})) by
A8,
ZFMISC_1: 59
.= (
{B}
\/ ((
{C}
\/
{D, E, F, J, M})
\
{A})) by
ENUMSET1: 11
.= (
{B}
\/ ((
{C}
\
{A})
\/ (
{D, E, F, J, M}
\
{A}))) by
XBOOLE_1: 42
.= (
{B}
\/ ((
{C}
\
{A})
\/ ((
{D, E}
\/
{F, J, M})
\
{A}))) by
ENUMSET1: 8
.= (
{B}
\/ ((
{C}
\
{A})
\/ ((
{D, E}
\
{A})
\/ (
{F, J, M}
\
{A})))) by
XBOOLE_1: 42
.= (
{B}
\/ ((
{C}
\
{A})
\/ (
{D, E}
\/ (
{F, J, M}
\
{A})))) by
A10,
ZFMISC_1: 63
.= (
{B}
\/ ((
{C}
\
{A})
\/ (
{D, E}
\/ ((
{F, J}
\/
{M})
\
{A})))) by
ENUMSET1: 3
.= (
{B}
\/ ((
{C}
\
{A})
\/ (
{D, E}
\/ ((
{F, J}
\
{A})
\/ (
{M}
\
{A}))))) by
XBOOLE_1: 42
.= (
{B}
\/ ((
{C}
\
{A})
\/ (
{D, E}
\/ (
{F, J}
\/ (
{M}
\
{A}))))) by
A13,
ZFMISC_1: 63
.= (
{B}
\/ (
{C}
\/ (
{D, E}
\/ (
{F, J}
\/ (
{M}
\
{A}))))) by
A11,
ZFMISC_1: 59
.= (
{B}
\/ (
{C}
\/ (
{D, E}
\/ (
{F, J}
\/
{M})))) by
A12,
ZFMISC_1: 59
.= (
{B}
\/ (
{C}
\/ (
{D, E}
\/
{F, J, M}))) by
ENUMSET1: 3
.= (
{B}
\/ (
{C}
\/
{D, E, F, J, M})) by
ENUMSET1: 8
.= (
{B}
\/
{C, D, E, F, J, M}) by
ENUMSET1: 11
.=
{B, C, D, E, F, J, M} by
ENUMSET1: 16;
then
A14: (G
\
{A})
= (
{}
\/
{B, C, D, E, F, J, M}) by
A9,
XBOOLE_1: 37;
A15: (
'/\' (G
\
{A}))
c= ((((((B
'/\' C)
'/\' D)
'/\' E)
'/\' F)
'/\' J)
'/\' M)
proof
let x be
object;
reconsider xx = x as
set by
TARSKI: 1;
assume x
in (
'/\' (G
\
{A}));
then
consider h be
Function, FF be
Subset-Family of Y such that
A16: (
dom h)
= (G
\
{A}) and
A17: (
rng h)
= FF and
A18: for d be
set st d
in (G
\
{A}) holds (h
. d)
in d and
A19: x
= (
Intersect FF) and
A20: x
<>
{} by
BVFUNC_2:def 1;
A21: C
in (G
\
{A}) by
A14,
ENUMSET1:def 5;
then
A22: (h
. C)
in C by
A18;
set mbcdef = (((((h
. B)
/\ (h
. C))
/\ (h
. D))
/\ (h
. E))
/\ (h
. F));
set mbcde = ((((h
. B)
/\ (h
. C))
/\ (h
. D))
/\ (h
. E));
set mbcdefj = ((((((h
. B)
/\ (h
. C))
/\ (h
. D))
/\ (h
. E))
/\ (h
. F))
/\ (h
. J));
A23: not x
in
{
{} } by
A20,
TARSKI:def 1;
A24: J
in (G
\
{A}) by
A14,
ENUMSET1:def 5;
then
A25: (h
. J)
in (
rng h) by
A16,
FUNCT_1:def 3;
set mbc = ((h
. B)
/\ (h
. C));
A26: B
in (G
\
{A}) by
A14,
ENUMSET1:def 5;
then (h
. B)
in B by
A18;
then
A27: mbc
in (
INTERSECTION (B,C)) by
A22,
SETFAM_1:def 5;
A28: (h
. B)
in (
rng h) by
A16,
A26,
FUNCT_1:def 3;
then
A29: (
Intersect FF)
= (
meet (
rng h)) by
A17,
SETFAM_1:def 9;
A30: (h
. C)
in (
rng h) by
A16,
A21,
FUNCT_1:def 3;
A31: F
in (G
\
{A}) by
A14,
ENUMSET1:def 5;
then
A32: (h
. F)
in (
rng h) by
A16,
FUNCT_1:def 3;
set mbcd = (((h
. B)
/\ (h
. C))
/\ (h
. D));
A33: E
in (G
\
{A}) by
A14,
ENUMSET1:def 5;
then
A34: (h
. E)
in (
rng h) by
A16,
FUNCT_1:def 3;
A35: M
in (G
\
{A}) by
A14,
ENUMSET1:def 5;
then
A36: (h
. M)
in (
rng h) by
A16,
FUNCT_1:def 3;
A37: D
in (G
\
{A}) by
A14,
ENUMSET1:def 5;
then
A38: (h
. D)
in (
rng h) by
A16,
FUNCT_1:def 3;
A39: xx
c= (((((((h
. B)
/\ (h
. C))
/\ (h
. D))
/\ (h
. E))
/\ (h
. F))
/\ (h
. J))
/\ (h
. M))
proof
let p be
object;
assume
A40: p
in xx;
then p
in (h
. B) & p
in (h
. C) by
A19,
A28,
A30,
A29,
SETFAM_1:def 1;
then
A41: p
in ((h
. B)
/\ (h
. C)) by
XBOOLE_0:def 4;
p
in (h
. D) by
A19,
A38,
A29,
A40,
SETFAM_1:def 1;
then
A42: p
in (((h
. B)
/\ (h
. C))
/\ (h
. D)) by
A41,
XBOOLE_0:def 4;
p
in (h
. E) by
A19,
A34,
A29,
A40,
SETFAM_1:def 1;
then
A43: p
in ((((h
. B)
/\ (h
. C))
/\ (h
. D))
/\ (h
. E)) by
A42,
XBOOLE_0:def 4;
p
in (h
. F) by
A19,
A32,
A29,
A40,
SETFAM_1:def 1;
then
A44: p
in (((((h
. B)
/\ (h
. C))
/\ (h
. D))
/\ (h
. E))
/\ (h
. F)) by
A43,
XBOOLE_0:def 4;
p
in (h
. J) by
A19,
A25,
A29,
A40,
SETFAM_1:def 1;
then
A45: p
in ((((((h
. B)
/\ (h
. C))
/\ (h
. D))
/\ (h
. E))
/\ (h
. F))
/\ (h
. J)) by
A44,
XBOOLE_0:def 4;
p
in (h
. M) by
A19,
A36,
A29,
A40,
SETFAM_1:def 1;
hence thesis by
A45,
XBOOLE_0:def 4;
end;
then mbcd
<>
{} by
A20;
then
A46: not mbcd
in
{
{} } by
TARSKI:def 1;
(((((((h
. B)
/\ (h
. C))
/\ (h
. D))
/\ (h
. E))
/\ (h
. F))
/\ (h
. J))
/\ (h
. M))
c= xx
proof
A47: (
rng h)
c=
{(h
. B), (h
. C), (h
. D), (h
. E), (h
. F), (h
. J), (h
. M)}
proof
let u be
object;
assume u
in (
rng h);
then
consider x1 be
object such that
A48: x1
in (
dom h) and
A49: u
= (h
. x1) by
FUNCT_1:def 3;
x1
= B or x1
= C or x1
= D or x1
= E or x1
= F or x1
= J or x1
= M by
A14,
A16,
A48,
ENUMSET1:def 5;
hence thesis by
A49,
ENUMSET1:def 5;
end;
let p be
object;
assume
A50: p
in (((((((h
. B)
/\ (h
. C))
/\ (h
. D))
/\ (h
. E))
/\ (h
. F))
/\ (h
. J))
/\ (h
. M));
then
A51: p
in (h
. M) by
XBOOLE_0:def 4;
A52: p
in ((((((h
. B)
/\ (h
. C))
/\ (h
. D))
/\ (h
. E))
/\ (h
. F))
/\ (h
. J)) by
A50,
XBOOLE_0:def 4;
then
A53: p
in (h
. J) by
XBOOLE_0:def 4;
A54: p
in (((((h
. B)
/\ (h
. C))
/\ (h
. D))
/\ (h
. E))
/\ (h
. F)) by
A52,
XBOOLE_0:def 4;
then
A55: p
in ((((h
. B)
/\ (h
. C))
/\ (h
. D))
/\ (h
. E)) by
XBOOLE_0:def 4;
then
A56: p
in (h
. E) by
XBOOLE_0:def 4;
A57: p
in (((h
. B)
/\ (h
. C))
/\ (h
. D)) by
A55,
XBOOLE_0:def 4;
then
A58: p
in (h
. D) by
XBOOLE_0:def 4;
p
in ((h
. B)
/\ (h
. C)) by
A57,
XBOOLE_0:def 4;
then
A59: p
in (h
. B) & p
in (h
. C) by
XBOOLE_0:def 4;
p
in (h
. F) by
A54,
XBOOLE_0:def 4;
then for y be
set holds y
in (
rng h) implies p
in y by
A59,
A58,
A56,
A53,
A51,
A47,
ENUMSET1:def 5;
hence thesis by
A19,
A28,
A29,
SETFAM_1:def 1;
end;
then
A60: (((((((h
. B)
/\ (h
. C))
/\ (h
. D))
/\ (h
. E))
/\ (h
. F))
/\ (h
. J))
/\ (h
. M))
= x by
A39,
XBOOLE_0:def 10;
mbc
<>
{} by
A20,
A39;
then not mbc
in
{
{} } by
TARSKI:def 1;
then mbc
in ((
INTERSECTION (B,C))
\
{
{} }) by
A27,
XBOOLE_0:def 5;
then
A61: mbc
in (B
'/\' C) by
PARTIT1:def 4;
(h
. D)
in D by
A18,
A37;
then mbcd
in (
INTERSECTION ((B
'/\' C),D)) by
A61,
SETFAM_1:def 5;
then mbcd
in ((
INTERSECTION ((B
'/\' C),D))
\
{
{} }) by
A46,
XBOOLE_0:def 5;
then
A62: mbcd
in ((B
'/\' C)
'/\' D) by
PARTIT1:def 4;
mbcde
<>
{} by
A20,
A39;
then
A63: not mbcde
in
{
{} } by
TARSKI:def 1;
(h
. E)
in E by
A18,
A33;
then mbcde
in (
INTERSECTION (((B
'/\' C)
'/\' D),E)) by
A62,
SETFAM_1:def 5;
then mbcde
in ((
INTERSECTION (((B
'/\' C)
'/\' D),E))
\
{
{} }) by
A63,
XBOOLE_0:def 5;
then
A64: mbcde
in (((B
'/\' C)
'/\' D)
'/\' E) by
PARTIT1:def 4;
mbcdef
<>
{} by
A20,
A39;
then
A65: not mbcdef
in
{
{} } by
TARSKI:def 1;
(h
. F)
in F by
A18,
A31;
then mbcdef
in (
INTERSECTION ((((B
'/\' C)
'/\' D)
'/\' E),F)) by
A64,
SETFAM_1:def 5;
then mbcdef
in ((
INTERSECTION ((((B
'/\' C)
'/\' D)
'/\' E),F))
\
{
{} }) by
A65,
XBOOLE_0:def 5;
then
A66: mbcdef
in ((((B
'/\' C)
'/\' D)
'/\' E)
'/\' F) by
PARTIT1:def 4;
mbcdefj
<>
{} by
A20,
A39;
then
A67: not mbcdefj
in
{
{} } by
TARSKI:def 1;
(h
. J)
in J by
A18,
A24;
then mbcdefj
in (
INTERSECTION (((((B
'/\' C)
'/\' D)
'/\' E)
'/\' F),J)) by
A66,
SETFAM_1:def 5;
then mbcdefj
in ((
INTERSECTION (((((B
'/\' C)
'/\' D)
'/\' E)
'/\' F),J))
\
{
{} }) by
A67,
XBOOLE_0:def 5;
then
A68: mbcdefj
in (((((B
'/\' C)
'/\' D)
'/\' E)
'/\' F)
'/\' J) by
PARTIT1:def 4;
(h
. M)
in M by
A18,
A35;
then x
in (
INTERSECTION ((((((B
'/\' C)
'/\' D)
'/\' E)
'/\' F)
'/\' J),M)) by
A60,
A68,
SETFAM_1:def 5;
then x
in ((
INTERSECTION ((((((B
'/\' C)
'/\' D)
'/\' E)
'/\' F)
'/\' J),M))
\
{
{} }) by
A23,
XBOOLE_0:def 5;
hence thesis by
PARTIT1:def 4;
end;
A69: ((((((B
'/\' C)
'/\' D)
'/\' E)
'/\' F)
'/\' J)
'/\' M)
c= (
'/\' (G
\
{A}))
proof
let x be
object;
reconsider xx = x as
set by
TARSKI: 1;
assume
A70: x
in ((((((B
'/\' C)
'/\' D)
'/\' E)
'/\' F)
'/\' J)
'/\' M);
then
A71: x
<>
{} by
EQREL_1:def 4;
x
in ((
INTERSECTION ((((((B
'/\' C)
'/\' D)
'/\' E)
'/\' F)
'/\' J),M))
\
{
{} }) by
A70,
PARTIT1:def 4;
then
consider bcdefj,m be
set such that
A72: bcdefj
in (((((B
'/\' C)
'/\' D)
'/\' E)
'/\' F)
'/\' J) and
A73: m
in M and
A74: x
= (bcdefj
/\ m) by
SETFAM_1:def 5;
bcdefj
in ((
INTERSECTION (((((B
'/\' C)
'/\' D)
'/\' E)
'/\' F),J))
\
{
{} }) by
A72,
PARTIT1:def 4;
then
consider bcdef,j be
set such that
A75: bcdef
in ((((B
'/\' C)
'/\' D)
'/\' E)
'/\' F) and
A76: j
in J and
A77: bcdefj
= (bcdef
/\ j) by
SETFAM_1:def 5;
bcdef
in ((
INTERSECTION ((((B
'/\' C)
'/\' D)
'/\' E),F))
\
{
{} }) by
A75,
PARTIT1:def 4;
then
consider bcde,f be
set such that
A78: bcde
in (((B
'/\' C)
'/\' D)
'/\' E) and
A79: f
in F and
A80: bcdef
= (bcde
/\ f) by
SETFAM_1:def 5;
bcde
in ((
INTERSECTION (((B
'/\' C)
'/\' D),E))
\
{
{} }) by
A78,
PARTIT1:def 4;
then
consider bcd,e be
set such that
A81: bcd
in ((B
'/\' C)
'/\' D) and
A82: e
in E and
A83: bcde
= (bcd
/\ e) by
SETFAM_1:def 5;
bcd
in ((
INTERSECTION ((B
'/\' C),D))
\
{
{} }) by
A81,
PARTIT1:def 4;
then
consider bc,d be
set such that
A84: bc
in (B
'/\' C) and
A85: d
in D and
A86: bcd
= (bc
/\ d) by
SETFAM_1:def 5;
bc
in ((
INTERSECTION (B,C))
\
{
{} }) by
A84,
PARTIT1:def 4;
then
consider b,c be
set such that
A87: b
in B & c
in C and
A88: bc
= (b
/\ c) by
SETFAM_1:def 5;
set h = (((((((B
.--> b)
+* (C
.--> c))
+* (D
.--> d))
+* (E
.--> e))
+* (F
.--> f))
+* (J
.--> j))
+* (M
.--> m));
A89: (h
. B)
= b by
A7,
Th49;
A90: (
dom (((((((B
.--> b)
+* (C
.--> c))
+* (D
.--> d))
+* (E
.--> e))
+* (F
.--> f))
+* (J
.--> j))
+* (M
.--> m)))
=
{M, B, C, D, E, F, J} by
Th50
.= (
{M}
\/
{B, C, D, E, F, J}) by
ENUMSET1: 16
.=
{B, C, D, E, F, J, M} by
ENUMSET1: 21;
then
A91: E
in (
dom h) & F
in (
dom h) by
ENUMSET1:def 5;
A92: D
in (
dom h) by
A90,
ENUMSET1:def 5;
then
A93: (h
. D)
in (
rng h) by
FUNCT_1:def 3;
A94: J
in (
dom h) & M
in (
dom h) by
A90,
ENUMSET1:def 5;
A95: B
in (
dom h) & C
in (
dom h) by
A90,
ENUMSET1:def 5;
A96:
{(h
. B), (h
. C), (h
. D), (h
. E), (h
. F), (h
. J), (h
. M)}
c= (
rng h)
proof
let t be
object;
assume t
in
{(h
. B), (h
. C), (h
. D), (h
. E), (h
. F), (h
. J), (h
. M)};
then t
= (h
. D) or t
= (h
. B) or t
= (h
. C) or t
= (h
. E) or t
= (h
. F) or t
= (h
. J) or t
= (h
. M) by
ENUMSET1:def 5;
hence thesis by
A92,
A95,
A91,
A94,
FUNCT_1:def 3;
end;
A97: for p be
set st p
in (G
\
{A}) holds (h
. p)
in p
proof
let p be
set;
assume p
in (G
\
{A});
then p
= D or p
= B or p
= C or p
= E or p
= F or p
= J or p
= M by
A14,
ENUMSET1:def 5;
hence thesis by
A7,
A73,
A76,
A79,
A82,
A85,
A87,
Th49;
end;
A98: (h
. C)
= c by
A7,
Th49;
A99: (h
. M)
= m by
A7,
Th49;
A100: (h
. J)
= j by
A7,
Th49;
A101: (h
. F)
= f by
A7,
Th49;
A102: (
rng h)
c=
{(h
. B), (h
. C), (h
. D), (h
. E), (h
. F), (h
. J), (h
. M)}
proof
let t be
object;
assume t
in (
rng h);
then
consider x1 be
object such that
A103: x1
in (
dom h) and
A104: t
= (h
. x1) by
FUNCT_1:def 3;
x1
= D or x1
= B or x1
= C or x1
= E or x1
= F or x1
= J or x1
= M by
A90,
A103,
ENUMSET1:def 5;
hence thesis by
A104,
ENUMSET1:def 5;
end;
then
A105: (
rng h)
=
{(h
. B), (h
. C), (h
. D), (h
. E), (h
. F), (h
. J), (h
. M)} by
A96,
XBOOLE_0:def 10;
A106: (h
. E)
= e by
A7,
Th49;
A107: (h
. D)
= d by
A7,
Th49;
(
rng h)
c= (
bool Y)
proof
let t be
object;
assume t
in (
rng h);
then t
= (h
. D) or t
= (h
. B) or t
= (h
. C) or t
= (h
. E) or t
= (h
. F) or t
= (h
. J) or t
= (h
. M) by
A102,
ENUMSET1:def 5;
hence thesis by
A73,
A76,
A79,
A82,
A85,
A87,
A107,
A89,
A98,
A106,
A101,
A100,
A99;
end;
then
reconsider FF = (
rng h) as
Subset-Family of Y;
reconsider h as
Function;
A108: xx
c= (
Intersect FF)
proof
let u be
object;
assume
A109: u
in xx;
for y be
set holds y
in FF implies u
in y
proof
let y be
set;
assume
A110: y
in FF;
now
per cases by
A102,
A110,
ENUMSET1:def 5;
case
A111: y
= (h
. D);
u
in ((((d
/\ ((b
/\ c)
/\ e))
/\ f)
/\ j)
/\ m) by
A74,
A77,
A80,
A83,
A86,
A88,
A109,
XBOOLE_1: 16;
then u
in (((d
/\ (((b
/\ c)
/\ e)
/\ f))
/\ j)
/\ m) by
XBOOLE_1: 16;
then u
in ((d
/\ ((((b
/\ c)
/\ e)
/\ f)
/\ j))
/\ m) by
XBOOLE_1: 16;
then u
in (d
/\ (((((b
/\ c)
/\ e)
/\ f)
/\ j)
/\ m)) by
XBOOLE_1: 16;
hence thesis by
A107,
A111,
XBOOLE_0:def 4;
end;
case
A112: y
= (h
. B);
u
in (((((c
/\ (d
/\ b))
/\ e)
/\ f)
/\ j)
/\ m) by
A74,
A77,
A80,
A83,
A86,
A88,
A109,
XBOOLE_1: 16;
then u
in ((((c
/\ ((d
/\ b)
/\ e))
/\ f)
/\ j)
/\ m) by
XBOOLE_1: 16;
then u
in ((((c
/\ ((d
/\ e)
/\ b))
/\ f)
/\ j)
/\ m) by
XBOOLE_1: 16;
then u
in (((c
/\ (((d
/\ e)
/\ b)
/\ f))
/\ j)
/\ m) by
XBOOLE_1: 16;
then u
in ((c
/\ ((((d
/\ e)
/\ b)
/\ f)
/\ j))
/\ m) by
XBOOLE_1: 16;
then u
in ((c
/\ (((d
/\ e)
/\ (f
/\ b))
/\ j))
/\ m) by
XBOOLE_1: 16;
then u
in ((c
/\ ((d
/\ e)
/\ ((f
/\ b)
/\ j)))
/\ m) by
XBOOLE_1: 16;
then u
in ((c
/\ ((d
/\ e)
/\ (f
/\ (j
/\ b))))
/\ m) by
XBOOLE_1: 16;
then u
in (((c
/\ (d
/\ e))
/\ (f
/\ (j
/\ b)))
/\ m) by
XBOOLE_1: 16;
then u
in ((((c
/\ (d
/\ e))
/\ f)
/\ (j
/\ b))
/\ m) by
XBOOLE_1: 16;
then u
in (((((c
/\ (d
/\ e))
/\ f)
/\ j)
/\ b)
/\ m) by
XBOOLE_1: 16;
then u
in (((((c
/\ (d
/\ e))
/\ f)
/\ j)
/\ m)
/\ b) by
XBOOLE_1: 16;
hence thesis by
A89,
A112,
XBOOLE_0:def 4;
end;
case
A113: y
= (h
. C);
u
in (((((c
/\ (d
/\ b))
/\ e)
/\ f)
/\ j)
/\ m) by
A74,
A77,
A80,
A83,
A86,
A88,
A109,
XBOOLE_1: 16;
then u
in ((((c
/\ ((d
/\ b)
/\ e))
/\ f)
/\ j)
/\ m) by
XBOOLE_1: 16;
then u
in ((((c
/\ ((d
/\ e)
/\ b))
/\ f)
/\ j)
/\ m) by
XBOOLE_1: 16;
then u
in (((c
/\ (((d
/\ e)
/\ b)
/\ f))
/\ j)
/\ m) by
XBOOLE_1: 16;
then u
in ((c
/\ ((((d
/\ e)
/\ b)
/\ f)
/\ j))
/\ m) by
XBOOLE_1: 16;
then u
in (c
/\ (((((d
/\ e)
/\ b)
/\ f)
/\ j)
/\ m)) by
XBOOLE_1: 16;
hence thesis by
A98,
A113,
XBOOLE_0:def 4;
end;
case
A114: y
= (h
. E);
u
in (((((b
/\ c)
/\ d)
/\ (f
/\ e))
/\ j)
/\ m) by
A74,
A77,
A80,
A83,
A86,
A88,
A109,
XBOOLE_1: 16;
then u
in ((((b
/\ c)
/\ d)
/\ ((f
/\ e)
/\ j))
/\ m) by
XBOOLE_1: 16;
then u
in ((((b
/\ c)
/\ d)
/\ ((f
/\ j)
/\ e))
/\ m) by
XBOOLE_1: 16;
then u
in (((((b
/\ c)
/\ d)
/\ (f
/\ j))
/\ e)
/\ m) by
XBOOLE_1: 16;
then u
in (((((b
/\ c)
/\ d)
/\ (f
/\ j))
/\ m)
/\ e) by
XBOOLE_1: 16;
hence thesis by
A106,
A114,
XBOOLE_0:def 4;
end;
case
A115: y
= (h
. F);
u
in ((((((b
/\ c)
/\ d)
/\ e)
/\ j)
/\ f)
/\ m) by
A74,
A77,
A80,
A83,
A86,
A88,
A109,
XBOOLE_1: 16;
then u
in ((((((b
/\ c)
/\ d)
/\ e)
/\ j)
/\ m)
/\ f) by
XBOOLE_1: 16;
hence thesis by
A101,
A115,
XBOOLE_0:def 4;
end;
case
A116: y
= (h
. J);
u
in ((((((b
/\ c)
/\ d)
/\ e)
/\ f)
/\ m)
/\ j) by
A74,
A77,
A80,
A83,
A86,
A88,
A109,
XBOOLE_1: 16;
hence thesis by
A100,
A116,
XBOOLE_0:def 4;
end;
case y
= (h
. M);
hence thesis by
A74,
A99,
A109,
XBOOLE_0:def 4;
end;
end;
hence thesis;
end;
then u
in (
meet FF) by
A105,
SETFAM_1:def 1;
hence thesis by
A105,
SETFAM_1:def 9;
end;
A117: (
Intersect FF)
= (
meet (
rng h)) by
A93,
SETFAM_1:def 9;
(
Intersect FF)
c= xx
proof
let t be
object;
assume
A118: t
in (
Intersect FF);
(h
. C)
in
{(h
. B), (h
. C), (h
. D), (h
. E), (h
. F), (h
. J), (h
. M)} by
ENUMSET1:def 5;
then
A119: t
in c by
A98,
A96,
A117,
A118,
SETFAM_1:def 1;
(h
. B)
in
{(h
. B), (h
. C), (h
. D), (h
. E), (h
. F), (h
. J), (h
. M)} by
ENUMSET1:def 5;
then t
in b by
A89,
A96,
A117,
A118,
SETFAM_1:def 1;
then
A120: t
in (b
/\ c) by
A119,
XBOOLE_0:def 4;
(h
. D)
in
{(h
. B), (h
. C), (h
. D), (h
. E), (h
. F), (h
. J), (h
. M)} by
ENUMSET1:def 5;
then t
in d by
A107,
A96,
A117,
A118,
SETFAM_1:def 1;
then
A121: t
in ((b
/\ c)
/\ d) by
A120,
XBOOLE_0:def 4;
(h
. E)
in
{(h
. B), (h
. C), (h
. D), (h
. E), (h
. F), (h
. J), (h
. M)} by
ENUMSET1:def 5;
then t
in e by
A106,
A96,
A117,
A118,
SETFAM_1:def 1;
then
A122: t
in (((b
/\ c)
/\ d)
/\ e) by
A121,
XBOOLE_0:def 4;
(h
. F)
in
{(h
. B), (h
. C), (h
. D), (h
. E), (h
. F), (h
. J), (h
. M)} by
ENUMSET1:def 5;
then t
in f by
A101,
A96,
A117,
A118,
SETFAM_1:def 1;
then
A123: t
in ((((b
/\ c)
/\ d)
/\ e)
/\ f) by
A122,
XBOOLE_0:def 4;
(h
. J)
in
{(h
. B), (h
. C), (h
. D), (h
. E), (h
. F), (h
. J), (h
. M)} by
ENUMSET1:def 5;
then t
in j by
A100,
A96,
A117,
A118,
SETFAM_1:def 1;
then
A124: t
in (((((b
/\ c)
/\ d)
/\ e)
/\ f)
/\ j) by
A123,
XBOOLE_0:def 4;
(h
. M)
in
{(h
. B), (h
. C), (h
. D), (h
. E), (h
. F), (h
. J), (h
. M)} by
ENUMSET1:def 5;
then t
in m by
A99,
A96,
A117,
A118,
SETFAM_1:def 1;
hence thesis by
A74,
A77,
A80,
A83,
A86,
A88,
A124,
XBOOLE_0:def 4;
end;
then x
= (
Intersect FF) by
A108,
XBOOLE_0:def 10;
hence thesis by
A14,
A90,
A97,
A71,
BVFUNC_2:def 1;
end;
(
CompF (A,G))
= (
'/\' (G
\
{A})) by
BVFUNC_2:def 7;
hence thesis by
A69,
A15,
XBOOLE_0:def 10;
end;
theorem ::
BVFUNC14:55
Th55: G
=
{A, B, C, D, E, F, J, M} & A
<> B & A
<> C & A
<> D & A
<> E & A
<> F & A
<> J & A
<> M & B
<> C & B
<> D & B
<> E & B
<> F & B
<> J & B
<> M & C
<> D & C
<> E & C
<> F & C
<> J & C
<> M & D
<> E & D
<> F & D
<> J & D
<> M & E
<> F & E
<> J & E
<> M & F
<> J & F
<> M & J
<> M implies (
CompF (B,G))
= ((((((A
'/\' C)
'/\' D)
'/\' E)
'/\' F)
'/\' J)
'/\' M)
proof
{A, B, C, D, E, F, J, M}
= (
{A, B}
\/
{C, D, E, F, J, M}) by
ENUMSET1: 23
.=
{B, A, C, D, E, F, J, M} by
ENUMSET1: 23;
hence thesis by
Th54;
end;
theorem ::
BVFUNC14:56
Th56: G
=
{A, B, C, D, E, F, J, M} & A
<> B & A
<> C & A
<> D & A
<> E & A
<> F & A
<> J & A
<> M & B
<> C & B
<> D & B
<> E & B
<> F & B
<> J & B
<> M & C
<> D & C
<> E & C
<> F & C
<> J & C
<> M & D
<> E & D
<> F & D
<> J & D
<> M & E
<> F & E
<> J & E
<> M & F
<> J & F
<> M & J
<> M implies (
CompF (C,G))
= ((((((A
'/\' B)
'/\' D)
'/\' E)
'/\' F)
'/\' J)
'/\' M)
proof
{A, B, C, D, E, F, J, M}
= (
{A, B, C}
\/
{D, E, F, J, M}) by
ENUMSET1: 24
.= ((
{A}
\/
{B, C})
\/
{D, E, F, J, M}) by
ENUMSET1: 2
.= (
{A, C, B}
\/
{D, E, F, J, M}) by
ENUMSET1: 2
.=
{A, C, B, D, E, F, J, M} by
ENUMSET1: 24;
hence thesis by
Th55;
end;
theorem ::
BVFUNC14:57
Th57: G
=
{A, B, C, D, E, F, J, M} & A
<> B & A
<> C & A
<> D & A
<> E & A
<> F & A
<> J & A
<> M & B
<> C & B
<> D & B
<> E & B
<> F & B
<> J & B
<> M & C
<> D & C
<> E & C
<> F & C
<> J & C
<> M & D
<> E & D
<> F & D
<> J & D
<> M & E
<> F & E
<> J & E
<> M & F
<> J & F
<> M & J
<> M implies (
CompF (D,G))
= ((((((A
'/\' B)
'/\' C)
'/\' E)
'/\' F)
'/\' J)
'/\' M)
proof
{A, B, C, D, E, F, J, M}
= (
{A, B}
\/
{C, D, E, F, J, M}) by
ENUMSET1: 23
.= (
{A, B}
\/ (
{C, D}
\/
{E, F, J, M})) by
ENUMSET1: 12
.= (
{A, B}
\/
{D, C, E, F, J, M}) by
ENUMSET1: 12
.=
{A, B, D, C, E, F, J, M} by
ENUMSET1: 23;
hence thesis by
Th56;
end;
theorem ::
BVFUNC14:58
Th58: G
=
{A, B, C, D, E, F, J, M} & A
<> B & A
<> C & A
<> D & A
<> E & A
<> F & A
<> J & A
<> M & B
<> C & B
<> D & B
<> E & B
<> F & B
<> J & B
<> M & C
<> D & C
<> E & C
<> F & C
<> J & C
<> M & D
<> E & D
<> F & D
<> J & D
<> M & E
<> F & E
<> J & E
<> M & F
<> J & F
<> M & J
<> M implies (
CompF (E,G))
= ((((((A
'/\' B)
'/\' C)
'/\' D)
'/\' F)
'/\' J)
'/\' M)
proof
{A, B, C, D, E, F, J, M}
= (
{A, B, C}
\/
{D, E, F, J, M}) by
ENUMSET1: 24
.= (
{A, B, C}
\/ (
{D, E}
\/
{F, J, M})) by
ENUMSET1: 8
.= (
{A, B, C}
\/
{E, D, F, J, M}) by
ENUMSET1: 8
.=
{A, B, C, E, D, F, J, M} by
ENUMSET1: 24;
hence thesis by
Th57;
end;
theorem ::
BVFUNC14:59
Th59: G
=
{A, B, C, D, E, F, J, M} & A
<> B & A
<> C & A
<> D & A
<> E & A
<> F & A
<> J & A
<> M & B
<> C & B
<> D & B
<> E & B
<> F & B
<> J & B
<> M & C
<> D & C
<> E & C
<> F & C
<> J & C
<> M & D
<> E & D
<> F & D
<> J & D
<> M & E
<> F & E
<> J & E
<> M & F
<> J & F
<> M & J
<> M implies (
CompF (F,G))
= ((((((A
'/\' B)
'/\' C)
'/\' D)
'/\' E)
'/\' J)
'/\' M)
proof
{A, B, C, D, E, F, J, M}
= (
{A, B, C, D}
\/
{E, F, J, M}) by
ENUMSET1: 25
.= (
{A, B, C, D}
\/ (
{E, F}
\/
{J, M})) by
ENUMSET1: 5
.= (
{A, B, C, D}
\/
{F, E, J, M}) by
ENUMSET1: 5
.=
{A, B, C, D, F, E, J, M} by
ENUMSET1: 25;
hence thesis by
Th58;
end;
theorem ::
BVFUNC14:60
Th60: G
=
{A, B, C, D, E, F, J, M} & A
<> B & A
<> C & A
<> D & A
<> E & A
<> F & A
<> J & A
<> M & B
<> C & B
<> D & B
<> E & B
<> F & B
<> J & B
<> M & C
<> D & C
<> E & C
<> F & C
<> J & C
<> M & D
<> E & D
<> F & D
<> J & D
<> M & E
<> F & E
<> J & E
<> M & F
<> J & F
<> M & J
<> M implies (
CompF (J,G))
= ((((((A
'/\' B)
'/\' C)
'/\' D)
'/\' E)
'/\' F)
'/\' M)
proof
{A, B, C, D, E, F, J, M}
= (
{A, B, C, D, E}
\/
{F, J, M}) by
ENUMSET1: 26
.= (
{A, B, C, D, E}
\/ (
{J, F}
\/
{M})) by
ENUMSET1: 3
.= (
{A, B, C, D, E}
\/
{J, F, M}) by
ENUMSET1: 3
.=
{A, B, C, D, E, J, F, M} by
ENUMSET1: 26;
hence thesis by
Th59;
end;
theorem ::
BVFUNC14:61
G
=
{A, B, C, D, E, F, J, M} & A
<> B & A
<> C & A
<> D & A
<> E & A
<> F & A
<> J & A
<> M & B
<> C & B
<> D & B
<> E & B
<> F & B
<> J & B
<> M & C
<> D & C
<> E & C
<> F & C
<> J & C
<> M & D
<> E & D
<> F & D
<> J & D
<> M & E
<> F & E
<> J & E
<> M & F
<> J & F
<> M & J
<> M implies (
CompF (M,G))
= ((((((A
'/\' B)
'/\' C)
'/\' D)
'/\' E)
'/\' F)
'/\' J)
proof
{A, B, C, D, E, F, J, M}
= (
{A, B, C, D, E, F}
\/
{J, M}) by
ENUMSET1: 27
.=
{A, B, C, D, E, F, M, J} by
ENUMSET1: 27;
hence thesis by
Th60;
end;
theorem ::
BVFUNC14:62
Th62: for A,B,C,D,E,F,J,M be
set, h be
Function, A9,B9,C9,D9,E9,F9,J9,M9 be
set st A
<> B & A
<> C & A
<> D & A
<> E & A
<> F & A
<> J & B
<> C & B
<> D & B
<> E & B
<> F & B
<> J & B
<> M & C
<> D & C
<> E & C
<> F & C
<> J & C
<> M & D
<> E & D
<> F & D
<> J & D
<> M & E
<> F & E
<> J & E
<> M & F
<> J & F
<> M & J
<> M & h
= ((((((((B
.--> B9)
+* (C
.--> C9))
+* (D
.--> D9))
+* (E
.--> E9))
+* (F
.--> F9))
+* (J
.--> J9))
+* (M
.--> M9))
+* (A
.--> A9)) holds (h
. B)
= B9 & (h
. C)
= C9 & (h
. D)
= D9 & (h
. E)
= E9 & (h
. F)
= F9 & (h
. J)
= J9
proof
let A,B,C,D,E,F,J,M be
set;
let h be
Function;
let A9,B9,C9,D9,E9,F9,J9,M9 be
set;
assume that
A1: A
<> B and
A2: A
<> C and
A3: A
<> D and
A4: A
<> E and
A5: A
<> F and
A6: A
<> J and
A7: B
<> C & B
<> D & B
<> E & B
<> F & B
<> J & B
<> M & C
<> D & C
<> E & C
<> F & C
<> J & C
<> M & D
<> E & D
<> F & D
<> J & D
<> M & E
<> F & E
<> J & E
<> M & F
<> J & F
<> M & J
<> M and
A8: h
= ((((((((B
.--> B9)
+* (C
.--> C9))
+* (D
.--> D9))
+* (E
.--> E9))
+* (F
.--> F9))
+* (J
.--> J9))
+* (M
.--> M9))
+* (A
.--> A9));
not C
in (
dom (A
.--> A9)) by
A2,
TARSKI:def 1;
then
A10: (h
. C)
= ((((((((B
.--> B9)
+* (C
.--> C9))
+* (D
.--> D9))
+* (E
.--> E9))
+* (F
.--> F9))
+* (J
.--> J9))
+* (M
.--> M9))
. C) by
A8,
FUNCT_4: 11;
not J
in (
dom (A
.--> A9)) by
A6,
TARSKI:def 1;
then
A11: (h
. J)
= ((((((((B
.--> B9)
+* (C
.--> C9))
+* (D
.--> D9))
+* (E
.--> E9))
+* (F
.--> F9))
+* (J
.--> J9))
+* (M
.--> M9))
. J) by
A8,
FUNCT_4: 11
.= J9 by
A7,
Th49;
not F
in (
dom (A
.--> A9)) by
A5,
TARSKI:def 1;
then
A12: (h
. F)
= ((((((((B
.--> B9)
+* (C
.--> C9))
+* (D
.--> D9))
+* (E
.--> E9))
+* (F
.--> F9))
+* (J
.--> J9))
+* (M
.--> M9))
. F) by
A8,
FUNCT_4: 11
.= F9 by
A7,
Th49;
not E
in (
dom (A
.--> A9)) by
A4,
TARSKI:def 1;
then
A13: (h
. E)
= ((((((((B
.--> B9)
+* (C
.--> C9))
+* (D
.--> D9))
+* (E
.--> E9))
+* (F
.--> F9))
+* (J
.--> J9))
+* (M
.--> M9))
. E) by
A8,
FUNCT_4: 11
.= E9 by
A7,
Th49;
not D
in (
dom (A
.--> A9)) by
A3,
TARSKI:def 1;
then
A14: (h
. D)
= ((((((((B
.--> B9)
+* (C
.--> C9))
+* (D
.--> D9))
+* (E
.--> E9))
+* (F
.--> F9))
+* (J
.--> J9))
+* (M
.--> M9))
. D) by
A8,
FUNCT_4: 11
.= D9 by
A7,
Th49;
not B
in (
dom (A
.--> A9)) by
A1,
TARSKI:def 1;
then (h
. B)
= ((((((((B
.--> B9)
+* (C
.--> C9))
+* (D
.--> D9))
+* (E
.--> E9))
+* (F
.--> F9))
+* (J
.--> J9))
+* (M
.--> M9))
. B) by
A8,
FUNCT_4: 11
.= B9 by
A7,
Th49;
hence thesis by
A7,
A10,
A14,
A13,
A12,
A11,
Th49;
end;
theorem ::
BVFUNC14:63
Th63: for A,B,C,D,E,F,J,M be
set, h be
Function, A9,B9,C9,D9,E9,F9,J9,M9 be
set st h
= ((((((((B
.--> B9)
+* (C
.--> C9))
+* (D
.--> D9))
+* (E
.--> E9))
+* (F
.--> F9))
+* (J
.--> J9))
+* (M
.--> M9))
+* (A
.--> A9)) holds (
dom h)
=
{A, B, C, D, E, F, J, M}
proof
let A,B,C,D,E,F,J,M be
set;
let h be
Function;
let A9,B9,C9,D9,E9,F9,J9,M9 be
set;
assume
A1: h
= ((((((((B
.--> B9)
+* (C
.--> C9))
+* (D
.--> D9))
+* (E
.--> E9))
+* (F
.--> F9))
+* (J
.--> J9))
+* (M
.--> M9))
+* (A
.--> A9));
A2: (
dom (A
.--> A9))
=
{A};
(
dom (((((((B
.--> B9)
+* (C
.--> C9))
+* (D
.--> D9))
+* (E
.--> E9))
+* (F
.--> F9))
+* (J
.--> J9))
+* (M
.--> M9)))
=
{M, B, C, D, E, F, J} by
Th50
.= (
{M}
\/
{B, C, D, E, F, J}) by
ENUMSET1: 16
.=
{B, C, D, E, F, J, M} by
ENUMSET1: 21;
then (
dom ((((((((B
.--> B9)
+* (C
.--> C9))
+* (D
.--> D9))
+* (E
.--> E9))
+* (F
.--> F9))
+* (J
.--> J9))
+* (M
.--> M9))
+* (A
.--> A9)))
= (
{B, C, D, E, F, J, M}
\/
{A}) by
A2,
FUNCT_4:def 1
.=
{A, B, C, D, E, F, J, M} by
ENUMSET1: 22;
hence thesis by
A1;
end;
theorem ::
BVFUNC14:64
Th64: for A,B,C,D,E,F,J,M be
set, h be
Function, A9,B9,C9,D9,E9,F9,J9,M9 be
set st h
= ((((((((B
.--> B9)
+* (C
.--> C9))
+* (D
.--> D9))
+* (E
.--> E9))
+* (F
.--> F9))
+* (J
.--> J9))
+* (M
.--> M9))
+* (A
.--> A9)) holds (
rng h)
=
{(h
. A), (h
. B), (h
. C), (h
. D), (h
. E), (h
. F), (h
. J), (h
. M)}
proof
let A,B,C,D,E,F,J,M be
set;
let h be
Function;
let A9,B9,C9,D9,E9,F9,J9,M9 be
set;
assume h
= ((((((((B
.--> B9)
+* (C
.--> C9))
+* (D
.--> D9))
+* (E
.--> E9))
+* (F
.--> F9))
+* (J
.--> J9))
+* (M
.--> M9))
+* (A
.--> A9));
then
A1: (
dom h)
=
{A, B, C, D, E, F, J, M} by
Th63;
then B
in (
dom h) by
ENUMSET1:def 6;
then
A2: (h
. B)
in (
rng h) by
FUNCT_1:def 3;
M
in (
dom h) by
A1,
ENUMSET1:def 6;
then
A3: (h
. M)
in (
rng h) by
FUNCT_1:def 3;
J
in (
dom h) by
A1,
ENUMSET1:def 6;
then
A4: (h
. J)
in (
rng h) by
FUNCT_1:def 3;
F
in (
dom h) by
A1,
ENUMSET1:def 6;
then
A5: (h
. F)
in (
rng h) by
FUNCT_1:def 3;
E
in (
dom h) by
A1,
ENUMSET1:def 6;
then
A6: (h
. E)
in (
rng h) by
FUNCT_1:def 3;
A7: (
rng h)
c=
{(h
. A), (h
. B), (h
. C), (h
. D), (h
. E), (h
. F), (h
. J), (h
. M)}
proof
let t be
object;
assume t
in (
rng h);
then
consider x1 be
object such that
A8: x1
in (
dom h) and
A9: t
= (h
. x1) by
FUNCT_1:def 3;
x1
= A or x1
= B or x1
= C or x1
= D or x1
= E or x1
= F or x1
= J or x1
= M by
A1,
A8,
ENUMSET1:def 6;
hence thesis by
A9,
ENUMSET1:def 6;
end;
D
in (
dom h) by
A1,
ENUMSET1:def 6;
then
A10: (h
. D)
in (
rng h) by
FUNCT_1:def 3;
C
in (
dom h) by
A1,
ENUMSET1:def 6;
then
A11: (h
. C)
in (
rng h) by
FUNCT_1:def 3;
A
in (
dom h) by
A1,
ENUMSET1:def 6;
then
A12: (h
. A)
in (
rng h) by
FUNCT_1:def 3;
{(h
. A), (h
. B), (h
. C), (h
. D), (h
. E), (h
. F), (h
. J), (h
. M)}
c= (
rng h) by
A12,
A2,
A11,
A10,
A6,
A5,
A4,
A3,
ENUMSET1:def 6;
hence thesis by
A7,
XBOOLE_0:def 10;
end;
theorem ::
BVFUNC14:65
for G be
Subset of (
PARTITIONS Y), A,B,C,D,E,F,J,M be
a_partition of Y, z,u be
Element of Y st G is
independent & G
=
{A, B, C, D, E, F, J, M} & A
<> B & A
<> C & A
<> D & A
<> E & A
<> F & A
<> J & A
<> M & B
<> C & B
<> D & B
<> E & B
<> F & B
<> J & B
<> M & C
<> D & C
<> E & C
<> F & C
<> J & C
<> M & D
<> E & D
<> F & D
<> J & D
<> M & E
<> F & E
<> J & E
<> M & F
<> J & F
<> M & J
<> M holds ((
EqClass (u,((((((B
'/\' C)
'/\' D)
'/\' E)
'/\' F)
'/\' J)
'/\' M)))
/\ (
EqClass (z,A)))
<>
{}
proof
let G be
Subset of (
PARTITIONS Y);
let A,B,C,D,E,F,J,M be
a_partition of Y;
let z,u be
Element of Y;
assume that
A1: G is
independent and
A2: G
=
{A, B, C, D, E, F, J, M} and
A3: A
<> B & A
<> C & A
<> D & A
<> E & A
<> F & A
<> J and
A4: A
<> M and
A5: B
<> C & B
<> D & B
<> E & B
<> F & B
<> J & B
<> M & C
<> D & C
<> E & C
<> F & C
<> J & C
<> M & D
<> E & D
<> F & D
<> J & D
<> M & E
<> F & E
<> J & E
<> M & F
<> J & F
<> M & J
<> M;
set h = ((((((((B
.--> (
EqClass (u,B)))
+* (C
.--> (
EqClass (u,C))))
+* (D
.--> (
EqClass (u,D))))
+* (E
.--> (
EqClass (u,E))))
+* (F
.--> (
EqClass (u,F))))
+* (J
.--> (
EqClass (u,J))))
+* (M
.--> (
EqClass (u,M))))
+* (A
.--> (
EqClass (z,A))));
A6: (h
. B)
= (
EqClass (u,B)) by
A3,
A5,
Th62;
reconsider GG = (
EqClass (u,((((((B
'/\' C)
'/\' D)
'/\' E)
'/\' F)
'/\' J)
'/\' M))) as
set;
reconsider I = (
EqClass (z,A)) as
set;
GG
= ((
EqClass (u,(((((B
'/\' C)
'/\' D)
'/\' E)
'/\' F)
'/\' J)))
/\ (
EqClass (u,M))) by
Th1;
then GG
= (((
EqClass (u,((((B
'/\' C)
'/\' D)
'/\' E)
'/\' F)))
/\ (
EqClass (u,J)))
/\ (
EqClass (u,M))) by
Th1;
then GG
= ((((
EqClass (u,(((B
'/\' C)
'/\' D)
'/\' E)))
/\ (
EqClass (u,F)))
/\ (
EqClass (u,J)))
/\ (
EqClass (u,M))) by
Th1;
then GG
= (((((
EqClass (u,((B
'/\' C)
'/\' D)))
/\ (
EqClass (u,E)))
/\ (
EqClass (u,F)))
/\ (
EqClass (u,J)))
/\ (
EqClass (u,M))) by
Th1;
then GG
= ((((((
EqClass (u,(B
'/\' C)))
/\ (
EqClass (u,D)))
/\ (
EqClass (u,E)))
/\ (
EqClass (u,F)))
/\ (
EqClass (u,J)))
/\ (
EqClass (u,M))) by
Th1;
then
A7: (GG
/\ I)
= ((((((((
EqClass (u,B))
/\ (
EqClass (u,C)))
/\ (
EqClass (u,D)))
/\ (
EqClass (u,E)))
/\ (
EqClass (u,F)))
/\ (
EqClass (u,J)))
/\ (
EqClass (u,M)))
/\ (
EqClass (z,A))) by
Th1;
A8: (h
. A)
= (
EqClass (z,A)) by
FUNCT_7: 94;
A9: (h
. C)
= (
EqClass (u,C)) by
A3,
A5,
Th62;
A10: (h
. M)
= (
EqClass (u,M)) by
A4,
Lm1;
A11: (h
. J)
= (
EqClass (u,J)) by
A3,
A5,
Th62;
A12: (h
. F)
= (
EqClass (u,F)) by
A3,
A5,
Th62;
A13: (h
. E)
= (
EqClass (u,E)) by
A3,
A5,
Th62;
A14: (h
. D)
= (
EqClass (u,D)) by
A3,
A5,
Th62;
A15: (
rng h)
=
{(h
. A), (h
. B), (h
. C), (h
. D), (h
. E), (h
. F), (h
. J), (h
. M)} by
Th64;
(
rng h)
c= (
bool Y)
proof
let t be
object;
assume t
in (
rng h);
then t
= (h
. A) or t
= (h
. B) or t
= (h
. C) or t
= (h
. D) or t
= (h
. E) or t
= (h
. F) or t
= (h
. J) or t
= (h
. M) by
A15,
ENUMSET1:def 6;
hence thesis by
A8,
A6,
A9,
A14,
A13,
A12,
A11,
A10;
end;
then
reconsider FF = (
rng h) as
Subset-Family of Y;
A16: (
dom h)
= G by
A2,
Th63;
then A
in (
dom h) by
A2,
ENUMSET1:def 6;
then
A17: (h
. A)
in (
rng h) by
FUNCT_1:def 3;
then
A18: (
Intersect FF)
= (
meet (
rng h)) by
SETFAM_1:def 9;
for d be
set st d
in G holds (h
. d)
in d
proof
let d be
set;
assume d
in G;
then d
= A or d
= B or d
= C or d
= D or d
= E or d
= F or d
= J or d
= M by
A2,
ENUMSET1:def 6;
hence thesis by
A8,
A6,
A9,
A14,
A13,
A12,
A11,
A10;
end;
then (
Intersect FF)
<>
{} by
A1,
A16,
BVFUNC_2:def 5;
then
consider m be
object such that
A19: m
in (
Intersect FF) by
XBOOLE_0:def 1;
C
in (
dom h) by
A2,
A16,
ENUMSET1:def 6;
then (h
. C)
in (
rng h) by
FUNCT_1:def 3;
then
A20: m
in (
EqClass (u,C)) by
A9,
A18,
A19,
SETFAM_1:def 1;
B
in (
dom h) by
A2,
A16,
ENUMSET1:def 6;
then (h
. B)
in (
rng h) by
FUNCT_1:def 3;
then m
in (
EqClass (u,B)) by
A6,
A18,
A19,
SETFAM_1:def 1;
then
A21: m
in ((
EqClass (u,B))
/\ (
EqClass (u,C))) by
A20,
XBOOLE_0:def 4;
D
in (
dom h) by
A2,
A16,
ENUMSET1:def 6;
then (h
. D)
in (
rng h) by
FUNCT_1:def 3;
then m
in (
EqClass (u,D)) by
A14,
A18,
A19,
SETFAM_1:def 1;
then
A22: m
in (((
EqClass (u,B))
/\ (
EqClass (u,C)))
/\ (
EqClass (u,D))) by
A21,
XBOOLE_0:def 4;
E
in (
dom h) by
A2,
A16,
ENUMSET1:def 6;
then (h
. E)
in (
rng h) by
FUNCT_1:def 3;
then m
in (
EqClass (u,E)) by
A13,
A18,
A19,
SETFAM_1:def 1;
then
A23: m
in ((((
EqClass (u,B))
/\ (
EqClass (u,C)))
/\ (
EqClass (u,D)))
/\ (
EqClass (u,E))) by
A22,
XBOOLE_0:def 4;
F
in (
dom h) by
A2,
A16,
ENUMSET1:def 6;
then (h
. F)
in (
rng h) by
FUNCT_1:def 3;
then m
in (
EqClass (u,F)) by
A12,
A18,
A19,
SETFAM_1:def 1;
then
A24: m
in (((((
EqClass (u,B))
/\ (
EqClass (u,C)))
/\ (
EqClass (u,D)))
/\ (
EqClass (u,E)))
/\ (
EqClass (u,F))) by
A23,
XBOOLE_0:def 4;
J
in (
dom h) by
A2,
A16,
ENUMSET1:def 6;
then (h
. J)
in (
rng h) by
FUNCT_1:def 3;
then m
in (
EqClass (u,J)) by
A11,
A18,
A19,
SETFAM_1:def 1;
then
A25: m
in ((((((
EqClass (u,B))
/\ (
EqClass (u,C)))
/\ (
EqClass (u,D)))
/\ (
EqClass (u,E)))
/\ (
EqClass (u,F)))
/\ (
EqClass (u,J))) by
A24,
XBOOLE_0:def 4;
M
in (
dom h) by
A2,
A16,
ENUMSET1:def 6;
then (h
. M)
in (
rng h) by
FUNCT_1:def 3;
then m
in (
EqClass (u,M)) by
A10,
A18,
A19,
SETFAM_1:def 1;
then
A26: m
in (((((((
EqClass (u,B))
/\ (
EqClass (u,C)))
/\ (
EqClass (u,D)))
/\ (
EqClass (u,E)))
/\ (
EqClass (u,F)))
/\ (
EqClass (u,J)))
/\ (
EqClass (u,M))) by
A25,
XBOOLE_0:def 4;
m
in (
EqClass (z,A)) by
A8,
A17,
A18,
A19,
SETFAM_1:def 1;
hence thesis by
A7,
A26,
XBOOLE_0:def 4;
end;
theorem ::
BVFUNC14:66
for G be
Subset of (
PARTITIONS Y), A,B,C,D,E,F,J,M be
a_partition of Y, z,u be
Element of Y st G is
independent & G
=
{A, B, C, D, E, F, J, M} & A
<> B & A
<> C & A
<> D & A
<> E & A
<> F & A
<> J & A
<> M & B
<> C & B
<> D & B
<> E & B
<> F & B
<> J & B
<> M & C
<> D & C
<> E & C
<> F & C
<> J & C
<> M & D
<> E & D
<> F & D
<> J & D
<> M & E
<> F & E
<> J & E
<> M & F
<> J & F
<> M & J
<> M & (
EqClass (z,(((((C
'/\' D)
'/\' E)
'/\' F)
'/\' J)
'/\' M)))
= (
EqClass (u,(((((C
'/\' D)
'/\' E)
'/\' F)
'/\' J)
'/\' M))) holds (
EqClass (u,(
CompF (A,G))))
meets (
EqClass (z,(
CompF (B,G))))
proof
let G be
Subset of (
PARTITIONS Y);
let A,B,C,D,E,F,J,M be
a_partition of Y;
let z,u be
Element of Y;
assume that
A1: G is
independent and
A2: G
=
{A, B, C, D, E, F, J, M} and
A3: A
<> B & A
<> C & A
<> D & A
<> E & A
<> F & A
<> J and
A4: A
<> M and
A5: B
<> C & B
<> D & B
<> E & B
<> F & B
<> J & B
<> M & C
<> D & C
<> E & C
<> F & C
<> J & C
<> M & D
<> E & D
<> F & D
<> J & D
<> M & E
<> F & E
<> J & E
<> M & F
<> J & F
<> M & J
<> M and
A6: (
EqClass (z,(((((C
'/\' D)
'/\' E)
'/\' F)
'/\' J)
'/\' M)))
= (
EqClass (u,(((((C
'/\' D)
'/\' E)
'/\' F)
'/\' J)
'/\' M)));
set h = ((((((((B
.--> (
EqClass (u,B)))
+* (C
.--> (
EqClass (u,C))))
+* (D
.--> (
EqClass (u,D))))
+* (E
.--> (
EqClass (u,E))))
+* (F
.--> (
EqClass (u,F))))
+* (J
.--> (
EqClass (u,J))))
+* (M
.--> (
EqClass (u,M))))
+* (A
.--> (
EqClass (z,A))));
A7: (h
. B)
= (
EqClass (u,B)) by
A3,
A5,
Th62;
set HH = (
EqClass (z,(
CompF (B,G)))), I = (
EqClass (z,A)), GG = (
EqClass (u,((((((B
'/\' C)
'/\' D)
'/\' E)
'/\' F)
'/\' J)
'/\' M)));
A8: GG
= (
EqClass (u,(
CompF (A,G)))) by
A2,
A3,
A4,
A5,
Th54;
GG
= ((
EqClass (u,(((((B
'/\' C)
'/\' D)
'/\' E)
'/\' F)
'/\' J)))
/\ (
EqClass (u,M))) by
Th1;
then GG
= (((
EqClass (u,((((B
'/\' C)
'/\' D)
'/\' E)
'/\' F)))
/\ (
EqClass (u,J)))
/\ (
EqClass (u,M))) by
Th1;
then GG
= ((((
EqClass (u,(((B
'/\' C)
'/\' D)
'/\' E)))
/\ (
EqClass (u,F)))
/\ (
EqClass (u,J)))
/\ (
EqClass (u,M))) by
Th1;
then GG
= (((((
EqClass (u,((B
'/\' C)
'/\' D)))
/\ (
EqClass (u,E)))
/\ (
EqClass (u,F)))
/\ (
EqClass (u,J)))
/\ (
EqClass (u,M))) by
Th1;
then GG
= ((((((
EqClass (u,(B
'/\' C)))
/\ (
EqClass (u,D)))
/\ (
EqClass (u,E)))
/\ (
EqClass (u,F)))
/\ (
EqClass (u,J)))
/\ (
EqClass (u,M))) by
Th1;
then
A9: (GG
/\ I)
= ((((((((
EqClass (u,B))
/\ (
EqClass (u,C)))
/\ (
EqClass (u,D)))
/\ (
EqClass (u,E)))
/\ (
EqClass (u,F)))
/\ (
EqClass (u,J)))
/\ (
EqClass (u,M)))
/\ (
EqClass (z,A))) by
Th1;
A10: (h
. A)
= (
EqClass (z,A)) by
FUNCT_7: 94;
A11: (h
. C)
= (
EqClass (u,C)) by
A3,
A5,
Th62;
A12: (h
. M)
= (
EqClass (u,M)) by
A4,
Lm1;
A13: (h
. J)
= (
EqClass (u,J)) by
A3,
A5,
Th62;
A14: (h
. F)
= (
EqClass (u,F)) by
A3,
A5,
Th62;
A15: (h
. E)
= (
EqClass (u,E)) by
A3,
A5,
Th62;
A16: (h
. D)
= (
EqClass (u,D)) by
A3,
A5,
Th62;
A17: (
rng h)
=
{(h
. A), (h
. B), (h
. C), (h
. D), (h
. E), (h
. F), (h
. J), (h
. M)} by
Th64;
(
rng h)
c= (
bool Y)
proof
let t be
object;
assume t
in (
rng h);
then t
= (h
. A) or t
= (h
. B) or t
= (h
. C) or t
= (h
. D) or t
= (h
. E) or t
= (h
. F) or t
= (h
. J) or t
= (h
. M) by
A17,
ENUMSET1:def 6;
hence thesis by
A10,
A7,
A11,
A16,
A15,
A14,
A13,
A12;
end;
then
reconsider FF = (
rng h) as
Subset-Family of Y;
A18: (
dom h)
= G by
A2,
Th63;
then A
in (
dom h) by
A2,
ENUMSET1:def 6;
then
A19: (h
. A)
in (
rng h) by
FUNCT_1:def 3;
then
A20: (
Intersect FF)
= (
meet (
rng h)) by
SETFAM_1:def 9;
for d be
set st d
in G holds (h
. d)
in d
proof
let d be
set;
assume d
in G;
then d
= A or d
= B or d
= C or d
= D or d
= E or d
= F or d
= J or d
= M by
A2,
ENUMSET1:def 6;
hence thesis by
A10,
A7,
A11,
A16,
A15,
A14,
A13,
A12;
end;
then (
Intersect FF)
<>
{} by
A1,
A18,
BVFUNC_2:def 5;
then
consider m be
object such that
A21: m
in (
Intersect FF) by
XBOOLE_0:def 1;
C
in (
dom h) by
A2,
A18,
ENUMSET1:def 6;
then (h
. C)
in (
rng h) by
FUNCT_1:def 3;
then
A22: m
in (
EqClass (u,C)) by
A11,
A20,
A21,
SETFAM_1:def 1;
B
in (
dom h) by
A2,
A18,
ENUMSET1:def 6;
then (h
. B)
in (
rng h) by
FUNCT_1:def 3;
then m
in (
EqClass (u,B)) by
A7,
A20,
A21,
SETFAM_1:def 1;
then
A23: m
in ((
EqClass (u,B))
/\ (
EqClass (u,C))) by
A22,
XBOOLE_0:def 4;
D
in (
dom h) by
A2,
A18,
ENUMSET1:def 6;
then (h
. D)
in (
rng h) by
FUNCT_1:def 3;
then m
in (
EqClass (u,D)) by
A16,
A20,
A21,
SETFAM_1:def 1;
then
A24: m
in (((
EqClass (u,B))
/\ (
EqClass (u,C)))
/\ (
EqClass (u,D))) by
A23,
XBOOLE_0:def 4;
E
in (
dom h) by
A2,
A18,
ENUMSET1:def 6;
then (h
. E)
in (
rng h) by
FUNCT_1:def 3;
then m
in (
EqClass (u,E)) by
A15,
A20,
A21,
SETFAM_1:def 1;
then
A25: m
in ((((
EqClass (u,B))
/\ (
EqClass (u,C)))
/\ (
EqClass (u,D)))
/\ (
EqClass (u,E))) by
A24,
XBOOLE_0:def 4;
F
in (
dom h) by
A2,
A18,
ENUMSET1:def 6;
then (h
. F)
in (
rng h) by
FUNCT_1:def 3;
then m
in (
EqClass (u,F)) by
A14,
A20,
A21,
SETFAM_1:def 1;
then
A26: m
in (((((
EqClass (u,B))
/\ (
EqClass (u,C)))
/\ (
EqClass (u,D)))
/\ (
EqClass (u,E)))
/\ (
EqClass (u,F))) by
A25,
XBOOLE_0:def 4;
J
in (
dom h) by
A2,
A18,
ENUMSET1:def 6;
then (h
. J)
in (
rng h) by
FUNCT_1:def 3;
then m
in (
EqClass (u,J)) by
A13,
A20,
A21,
SETFAM_1:def 1;
then
A27: m
in ((((((
EqClass (u,B))
/\ (
EqClass (u,C)))
/\ (
EqClass (u,D)))
/\ (
EqClass (u,E)))
/\ (
EqClass (u,F)))
/\ (
EqClass (u,J))) by
A26,
XBOOLE_0:def 4;
M
in (
dom h) by
A2,
A18,
ENUMSET1:def 6;
then (h
. M)
in (
rng h) by
FUNCT_1:def 3;
then m
in (
EqClass (u,M)) by
A12,
A20,
A21,
SETFAM_1:def 1;
then
A28: m
in (((((((
EqClass (u,B))
/\ (
EqClass (u,C)))
/\ (
EqClass (u,D)))
/\ (
EqClass (u,E)))
/\ (
EqClass (u,F)))
/\ (
EqClass (u,J)))
/\ (
EqClass (u,M))) by
A27,
XBOOLE_0:def 4;
m
in (
EqClass (z,A)) by
A10,
A19,
A20,
A21,
SETFAM_1:def 1;
then (GG
/\ I)
<>
{} by
A9,
A28,
XBOOLE_0:def 4;
then
consider p be
object such that
A29: p
in (GG
/\ I) by
XBOOLE_0:def 1;
reconsider p as
Element of Y by
A29;
reconsider K = (
EqClass (p,(((((C
'/\' D)
'/\' E)
'/\' F)
'/\' J)
'/\' M))) as
set;
A30: p
in GG by
A29,
XBOOLE_0:def 4;
reconsider L = (
EqClass (z,(((((C
'/\' D)
'/\' E)
'/\' F)
'/\' J)
'/\' M))) as
set;
A31: p
in (
EqClass (p,(((((C
'/\' D)
'/\' E)
'/\' F)
'/\' J)
'/\' M))) by
EQREL_1:def 6;
GG
= (
EqClass (u,(((((B
'/\' (C
'/\' D))
'/\' E)
'/\' F)
'/\' J)
'/\' M))) by
PARTIT1: 14;
then GG
= (
EqClass (u,((((B
'/\' ((C
'/\' D)
'/\' E))
'/\' F)
'/\' J)
'/\' M))) by
PARTIT1: 14;
then GG
= (
EqClass (u,(((B
'/\' (((C
'/\' D)
'/\' E)
'/\' F))
'/\' J)
'/\' M))) by
PARTIT1: 14;
then GG
= (
EqClass (u,((B
'/\' ((((C
'/\' D)
'/\' E)
'/\' F)
'/\' J))
'/\' M))) by
PARTIT1: 14;
then GG
= (
EqClass (u,(B
'/\' (((((C
'/\' D)
'/\' E)
'/\' F)
'/\' J)
'/\' M)))) by
PARTIT1: 14;
then GG
c= L by
A6,
BVFUNC11: 3;
then K
meets L by
A30,
A31,
XBOOLE_0: 3;
then K
= L by
EQREL_1: 41;
then
A32: z
in K by
EQREL_1:def 6;
A33: z
in HH by
EQREL_1:def 6;
z
in I by
EQREL_1:def 6;
then z
in (I
/\ K) by
A32,
XBOOLE_0:def 4;
then
A34: (I
/\ K)
meets HH by
A33,
XBOOLE_0: 3;
A35: (A
'/\' (((((C
'/\' D)
'/\' E)
'/\' F)
'/\' J)
'/\' M))
= ((A
'/\' ((((C
'/\' D)
'/\' E)
'/\' F)
'/\' J))
'/\' M) by
PARTIT1: 14
.= (((A
'/\' (((C
'/\' D)
'/\' E)
'/\' F))
'/\' J)
'/\' M) by
PARTIT1: 14
.= ((((A
'/\' ((C
'/\' D)
'/\' E))
'/\' F)
'/\' J)
'/\' M) by
PARTIT1: 14
.= (((((A
'/\' (C
'/\' D))
'/\' E)
'/\' F)
'/\' J)
'/\' M) by
PARTIT1: 14
.= ((((((A
'/\' C)
'/\' D)
'/\' E)
'/\' F)
'/\' J)
'/\' M) by
PARTIT1: 14;
p
in K & p
in I by
A29,
EQREL_1:def 6,
XBOOLE_0:def 4;
then
A36: p
in (I
/\ K) by
XBOOLE_0:def 4;
then (I
/\ K)
in (
INTERSECTION (A,(((((C
'/\' D)
'/\' E)
'/\' F)
'/\' J)
'/\' M))) & not (I
/\ K)
in
{
{} } by
SETFAM_1:def 5,
TARSKI:def 1;
then (I
/\ K)
in ((
INTERSECTION (A,(((((C
'/\' D)
'/\' E)
'/\' F)
'/\' J)
'/\' M)))
\
{
{} }) by
XBOOLE_0:def 5;
then
A37: (I
/\ K)
in (A
'/\' (((((C
'/\' D)
'/\' E)
'/\' F)
'/\' J)
'/\' M)) by
PARTIT1:def 4;
(
CompF (B,G))
= ((((((A
'/\' C)
'/\' D)
'/\' E)
'/\' F)
'/\' J)
'/\' M) by
A2,
A3,
A4,
A5,
Th55;
then p
in HH by
A36,
A37,
A34,
A35,
EQREL_1:def 4;
hence thesis by
A8,
A30,
XBOOLE_0: 3;
end;
Lm3:
{x1, x2, x3, x4, x5, x6, x7, x8, x9}
= (
{x1, x2, x3, x4}
\/
{x5, x6, x7, x8, x9})
proof
now
let x be
object;
A1: x
in
{x5, x6, x7, x8, x9} iff x
= x5 or x
= x6 or x
= x7 or x
= x8 or x
= x9 by
ENUMSET1:def 3;
x
in
{x1, x2, x3, x4} iff x
= x1 or x
= x2 or x
= x3 or x
= x4 by
ENUMSET1:def 2;
hence x
in
{x1, x2, x3, x4, x5, x6, x7, x8, x9} iff x
in (
{x1, x2, x3, x4}
\/
{x5, x6, x7, x8, x9}) by
A1,
ENUMSET1:def 7,
XBOOLE_0:def 3;
end;
hence thesis by
TARSKI: 2;
end;
theorem ::
BVFUNC14:67
Th67: for G be
Subset of (
PARTITIONS Y), A,B,C,D,E,F,J,M,N be
a_partition of Y st G
=
{A, B, C, D, E, F, J, M, N} & A
<> B & A
<> C & A
<> D & A
<> E & A
<> F & A
<> J & A
<> M & A
<> N & B
<> C & B
<> D & B
<> E & B
<> F & B
<> J & B
<> M & B
<> N & C
<> D & C
<> E & C
<> F & C
<> J & C
<> M & C
<> N & D
<> E & D
<> F & D
<> J & D
<> M & D
<> N & E
<> F & E
<> J & E
<> M & E
<> N & F
<> J & F
<> M & F
<> N & J
<> M & J
<> N & M
<> N holds (
CompF (A,G))
= (((((((B
'/\' C)
'/\' D)
'/\' E)
'/\' F)
'/\' J)
'/\' M)
'/\' N)
proof
let G be
Subset of (
PARTITIONS Y);
let A,B,C,D,E,F,J,M,N be
a_partition of Y;
assume that
A1: G
=
{A, B, C, D, E, F, J, M, N} and
A2: A
<> B and
A3: A
<> C and
A4: A
<> D & A
<> E and
A5: A
<> F & A
<> J and
A6: A
<> M & A
<> N and
A7: B
<> C & B
<> D & B
<> E & B
<> F & B
<> J & B
<> M & B
<> N & C
<> D & C
<> E & C
<> F & C
<> J & C
<> M & C
<> N & D
<> E & D
<> F & D
<> J & D
<> M & D
<> N & E
<> F & E
<> J & E
<> M & E
<> N & F
<> J & F
<> M & F
<> N & J
<> M & J
<> N and
A8: M
<> N;
A9: not B
in
{A} by
A2,
TARSKI:def 1;
( not D
in
{A}) & not E
in
{A} by
A4,
TARSKI:def 1;
then
A10: (
{D, E}
\
{A})
=
{D, E} by
ZFMISC_1: 63;
A11: ( not F
in
{A}) & not J
in
{A} by
A5,
TARSKI:def 1;
A12: not C
in
{A} by
A3,
TARSKI:def 1;
(G
\
{A})
= ((
{A}
\/
{B, C, D, E, F, J, M, N})
\
{A}) by
A1,
ENUMSET1: 77;
then
A13: (G
\
{A})
= ((
{A}
\
{A})
\/ (
{B, C, D, E, F, J, M, N}
\
{A})) by
XBOOLE_1: 42;
A14: ( not M
in
{A}) & not N
in
{A} by
A6,
TARSKI:def 1;
(
{B, C, D, E, F, J, M, N}
\
{A})
= ((
{B}
\/
{C, D, E, F, J, M, N})
\
{A}) by
ENUMSET1: 22
.= ((
{B}
\
{A})
\/ (
{C, D, E, F, J, M, N}
\
{A})) by
XBOOLE_1: 42
.= (
{B}
\/ (
{C, D, E, F, J, M, N}
\
{A})) by
A9,
ZFMISC_1: 59
.= (
{B}
\/ ((
{C}
\/
{D, E, F, J, M, N})
\
{A})) by
ENUMSET1: 16
.= (
{B}
\/ ((
{C}
\
{A})
\/ (
{D, E, F, J, M, N}
\
{A}))) by
XBOOLE_1: 42
.= (
{B}
\/ ((
{C}
\
{A})
\/ ((
{D, E}
\/
{F, J, M, N})
\
{A}))) by
ENUMSET1: 12
.= (
{B}
\/ ((
{C}
\
{A})
\/ ((
{D, E}
\
{A})
\/ (
{F, J, M, N}
\
{A})))) by
XBOOLE_1: 42
.= (
{B}
\/ ((
{C}
\
{A})
\/ (
{D, E}
\/ ((
{F, J}
\/
{M, N})
\
{A})))) by
A10,
ENUMSET1: 5
.= (
{B}
\/ ((
{C}
\
{A})
\/ (
{D, E}
\/ ((
{F, J}
\
{A})
\/ (
{M, N}
\
{A}))))) by
XBOOLE_1: 42
.= (
{B}
\/ ((
{C}
\
{A})
\/ (
{D, E}
\/ (
{F, J}
\/ (
{M, N}
\
{A}))))) by
A11,
ZFMISC_1: 63
.= (
{B}
\/ (
{C}
\/ (
{D, E}
\/ (
{F, J}
\/ (
{M, N}
\
{A}))))) by
A12,
ZFMISC_1: 59
.= (
{B}
\/ (
{C}
\/ (
{D, E}
\/ (
{F, J}
\/
{M, N})))) by
A14,
ZFMISC_1: 63
.= (
{B}
\/ (
{C}
\/ (
{D, E}
\/
{F, J, M, N}))) by
ENUMSET1: 5
.= (
{B}
\/ (
{C}
\/
{D, E, F, J, M, N})) by
ENUMSET1: 12
.= (
{B}
\/
{C, D, E, F, J, M, N}) by
ENUMSET1: 16
.=
{B, C, D, E, F, J, M, N} by
ENUMSET1: 22;
then
A15: (G
\
{A})
= (
{}
\/
{B, C, D, E, F, J, M, N}) by
A13,
XBOOLE_1: 37;
A16: (
'/\' (G
\
{A}))
c= (((((((B
'/\' C)
'/\' D)
'/\' E)
'/\' F)
'/\' J)
'/\' M)
'/\' N)
proof
let x be
object;
reconsider xx = x as
set by
TARSKI: 1;
assume x
in (
'/\' (G
\
{A}));
then
consider h be
Function, FF be
Subset-Family of Y such that
A17: (
dom h)
= (G
\
{A}) and
A18: (
rng h)
= FF and
A19: for d be
set st d
in (G
\
{A}) holds (h
. d)
in d and
A20: x
= (
Intersect FF) and
A21: x
<>
{} by
BVFUNC_2:def 1;
A22: C
in (G
\
{A}) by
A15,
ENUMSET1:def 6;
then
A23: (h
. C)
in C by
A19;
set mbcd = (((h
. B)
/\ (h
. C))
/\ (h
. D));
A24: E
in (G
\
{A}) by
A15,
ENUMSET1:def 6;
then
A25: (h
. E)
in (
rng h) by
A17,
FUNCT_1:def 3;
A26: N
in (G
\
{A}) by
A15,
ENUMSET1:def 6;
then
A27: (h
. N)
in (
rng h) by
A17,
FUNCT_1:def 3;
set mbc = ((h
. B)
/\ (h
. C));
A28: B
in (G
\
{A}) by
A15,
ENUMSET1:def 6;
then (h
. B)
in B by
A19;
then
A29: mbc
in (
INTERSECTION (B,C)) by
A23,
SETFAM_1:def 5;
A30: (h
. B)
in (
rng h) by
A17,
A28,
FUNCT_1:def 3;
then
A31: (
Intersect FF)
= (
meet (
rng h)) by
A18,
SETFAM_1:def 9;
A32: (h
. C)
in (
rng h) by
A17,
A22,
FUNCT_1:def 3;
A33: F
in (G
\
{A}) by
A15,
ENUMSET1:def 6;
then
A34: (h
. F)
in (
rng h) by
A17,
FUNCT_1:def 3;
A35: M
in (G
\
{A}) by
A15,
ENUMSET1:def 6;
then
A36: (h
. M)
in (
rng h) by
A17,
FUNCT_1:def 3;
A37: J
in (G
\
{A}) by
A15,
ENUMSET1:def 6;
then
A38: (h
. J)
in (
rng h) by
A17,
FUNCT_1:def 3;
A39: D
in (G
\
{A}) by
A15,
ENUMSET1:def 6;
then
A40: (h
. D)
in (
rng h) by
A17,
FUNCT_1:def 3;
A41: xx
c= ((((((((h
. B)
/\ (h
. C))
/\ (h
. D))
/\ (h
. E))
/\ (h
. F))
/\ (h
. J))
/\ (h
. M))
/\ (h
. N))
proof
let p be
object;
assume
A42: p
in xx;
then p
in (h
. B) & p
in (h
. C) by
A20,
A30,
A32,
A31,
SETFAM_1:def 1;
then
A43: p
in ((h
. B)
/\ (h
. C)) by
XBOOLE_0:def 4;
p
in (h
. D) by
A20,
A40,
A31,
A42,
SETFAM_1:def 1;
then
A44: p
in (((h
. B)
/\ (h
. C))
/\ (h
. D)) by
A43,
XBOOLE_0:def 4;
p
in (h
. E) by
A20,
A25,
A31,
A42,
SETFAM_1:def 1;
then
A45: p
in ((((h
. B)
/\ (h
. C))
/\ (h
. D))
/\ (h
. E)) by
A44,
XBOOLE_0:def 4;
p
in (h
. F) by
A20,
A34,
A31,
A42,
SETFAM_1:def 1;
then
A46: p
in (((((h
. B)
/\ (h
. C))
/\ (h
. D))
/\ (h
. E))
/\ (h
. F)) by
A45,
XBOOLE_0:def 4;
p
in (h
. J) by
A20,
A38,
A31,
A42,
SETFAM_1:def 1;
then
A47: p
in ((((((h
. B)
/\ (h
. C))
/\ (h
. D))
/\ (h
. E))
/\ (h
. F))
/\ (h
. J)) by
A46,
XBOOLE_0:def 4;
p
in (h
. M) by
A20,
A36,
A31,
A42,
SETFAM_1:def 1;
then
A48: p
in (((((((h
. B)
/\ (h
. C))
/\ (h
. D))
/\ (h
. E))
/\ (h
. F))
/\ (h
. J))
/\ (h
. M)) by
A47,
XBOOLE_0:def 4;
p
in (h
. N) by
A20,
A27,
A31,
A42,
SETFAM_1:def 1;
hence thesis by
A48,
XBOOLE_0:def 4;
end;
then mbcd
<>
{} by
A21;
then
A49: not mbcd
in
{
{} } by
TARSKI:def 1;
mbc
<>
{} by
A21,
A41;
then not mbc
in
{
{} } by
TARSKI:def 1;
then mbc
in ((
INTERSECTION (B,C))
\
{
{} }) by
A29,
XBOOLE_0:def 5;
then
A50: mbc
in (B
'/\' C) by
PARTIT1:def 4;
(h
. D)
in D by
A19,
A39;
then mbcd
in (
INTERSECTION ((B
'/\' C),D)) by
A50,
SETFAM_1:def 5;
then mbcd
in ((
INTERSECTION ((B
'/\' C),D))
\
{
{} }) by
A49,
XBOOLE_0:def 5;
then
A51: mbcd
in ((B
'/\' C)
'/\' D) by
PARTIT1:def 4;
set mbcdefjm = (((((((h
. B)
/\ (h
. C))
/\ (h
. D))
/\ (h
. E))
/\ (h
. F))
/\ (h
. J))
/\ (h
. M));
set mbcdefj = ((((((h
. B)
/\ (h
. C))
/\ (h
. D))
/\ (h
. E))
/\ (h
. F))
/\ (h
. J));
A52: not x
in
{
{} } by
A21,
TARSKI:def 1;
set mbcdef = (((((h
. B)
/\ (h
. C))
/\ (h
. D))
/\ (h
. E))
/\ (h
. F));
set mbcde = ((((h
. B)
/\ (h
. C))
/\ (h
. D))
/\ (h
. E));
mbcdef
<>
{} by
A21,
A41;
then
A53: not mbcdef
in
{
{} } by
TARSKI:def 1;
mbcde
<>
{} by
A21,
A41;
then
A54: not mbcde
in
{
{} } by
TARSKI:def 1;
((((((((h
. B)
/\ (h
. C))
/\ (h
. D))
/\ (h
. E))
/\ (h
. F))
/\ (h
. J))
/\ (h
. M))
/\ (h
. N))
c= xx
proof
A55: (
rng h)
c=
{(h
. B), (h
. C), (h
. D), (h
. E), (h
. F), (h
. J), (h
. M), (h
. N)}
proof
let u be
object;
assume u
in (
rng h);
then
consider x1 be
object such that
A56: x1
in (
dom h) and
A57: u
= (h
. x1) by
FUNCT_1:def 3;
x1
= B or x1
= C or x1
= D or x1
= E or x1
= F or x1
= J or x1
= M or x1
= N by
A15,
A17,
A56,
ENUMSET1:def 6;
hence thesis by
A57,
ENUMSET1:def 6;
end;
let p be
object;
assume
A58: p
in ((((((((h
. B)
/\ (h
. C))
/\ (h
. D))
/\ (h
. E))
/\ (h
. F))
/\ (h
. J))
/\ (h
. M))
/\ (h
. N));
then
A59: p
in (h
. N) by
XBOOLE_0:def 4;
A60: p
in (((((((h
. B)
/\ (h
. C))
/\ (h
. D))
/\ (h
. E))
/\ (h
. F))
/\ (h
. J))
/\ (h
. M)) by
A58,
XBOOLE_0:def 4;
then
A61: p
in ((((((h
. B)
/\ (h
. C))
/\ (h
. D))
/\ (h
. E))
/\ (h
. F))
/\ (h
. J)) by
XBOOLE_0:def 4;
then
A62: p
in (h
. J) by
XBOOLE_0:def 4;
A63: p
in (h
. M) by
A60,
XBOOLE_0:def 4;
A64: p
in (((((h
. B)
/\ (h
. C))
/\ (h
. D))
/\ (h
. E))
/\ (h
. F)) by
A61,
XBOOLE_0:def 4;
then
A65: p
in ((((h
. B)
/\ (h
. C))
/\ (h
. D))
/\ (h
. E)) by
XBOOLE_0:def 4;
then
A66: p
in (h
. E) by
XBOOLE_0:def 4;
A67: p
in (((h
. B)
/\ (h
. C))
/\ (h
. D)) by
A65,
XBOOLE_0:def 4;
then
A68: p
in (h
. D) by
XBOOLE_0:def 4;
p
in ((h
. B)
/\ (h
. C)) by
A67,
XBOOLE_0:def 4;
then
A69: p
in (h
. B) & p
in (h
. C) by
XBOOLE_0:def 4;
p
in (h
. F) by
A64,
XBOOLE_0:def 4;
then for y be
set holds y
in (
rng h) implies p
in y by
A69,
A68,
A66,
A62,
A63,
A59,
A55,
ENUMSET1:def 6;
hence thesis by
A20,
A30,
A31,
SETFAM_1:def 1;
end;
then
A70: ((((((((h
. B)
/\ (h
. C))
/\ (h
. D))
/\ (h
. E))
/\ (h
. F))
/\ (h
. J))
/\ (h
. M))
/\ (h
. N))
= x by
A41,
XBOOLE_0:def 10;
(h
. E)
in E by
A19,
A24;
then mbcde
in (
INTERSECTION (((B
'/\' C)
'/\' D),E)) by
A51,
SETFAM_1:def 5;
then mbcde
in ((
INTERSECTION (((B
'/\' C)
'/\' D),E))
\
{
{} }) by
A54,
XBOOLE_0:def 5;
then
A71: mbcde
in (((B
'/\' C)
'/\' D)
'/\' E) by
PARTIT1:def 4;
(h
. F)
in F by
A19,
A33;
then mbcdef
in (
INTERSECTION ((((B
'/\' C)
'/\' D)
'/\' E),F)) by
A71,
SETFAM_1:def 5;
then mbcdef
in ((
INTERSECTION ((((B
'/\' C)
'/\' D)
'/\' E),F))
\
{
{} }) by
A53,
XBOOLE_0:def 5;
then
A72: mbcdef
in ((((B
'/\' C)
'/\' D)
'/\' E)
'/\' F) by
PARTIT1:def 4;
mbcdefj
<>
{} by
A21,
A41;
then
A73: not mbcdefj
in
{
{} } by
TARSKI:def 1;
(h
. J)
in J by
A19,
A37;
then mbcdefj
in (
INTERSECTION (((((B
'/\' C)
'/\' D)
'/\' E)
'/\' F),J)) by
A72,
SETFAM_1:def 5;
then mbcdefj
in ((
INTERSECTION (((((B
'/\' C)
'/\' D)
'/\' E)
'/\' F),J))
\
{
{} }) by
A73,
XBOOLE_0:def 5;
then
A74: mbcdefj
in (((((B
'/\' C)
'/\' D)
'/\' E)
'/\' F)
'/\' J) by
PARTIT1:def 4;
mbcdefjm
<>
{} by
A21,
A41;
then
A75: not mbcdefjm
in
{
{} } by
TARSKI:def 1;
(h
. M)
in M by
A19,
A35;
then mbcdefjm
in (
INTERSECTION ((((((B
'/\' C)
'/\' D)
'/\' E)
'/\' F)
'/\' J),M)) by
A74,
SETFAM_1:def 5;
then mbcdefjm
in ((
INTERSECTION ((((((B
'/\' C)
'/\' D)
'/\' E)
'/\' F)
'/\' J),M))
\
{
{} }) by
A75,
XBOOLE_0:def 5;
then
A76: mbcdefjm
in ((((((B
'/\' C)
'/\' D)
'/\' E)
'/\' F)
'/\' J)
'/\' M) by
PARTIT1:def 4;
(h
. N)
in N by
A19,
A26;
then x
in (
INTERSECTION (((((((B
'/\' C)
'/\' D)
'/\' E)
'/\' F)
'/\' J)
'/\' M),N)) by
A70,
A76,
SETFAM_1:def 5;
then x
in ((
INTERSECTION (((((((B
'/\' C)
'/\' D)
'/\' E)
'/\' F)
'/\' J)
'/\' M),N))
\
{
{} }) by
A52,
XBOOLE_0:def 5;
hence thesis by
PARTIT1:def 4;
end;
A77: (((((((B
'/\' C)
'/\' D)
'/\' E)
'/\' F)
'/\' J)
'/\' M)
'/\' N)
c= (
'/\' (G
\
{A}))
proof
let x be
object;
reconsider xx = x as
set by
TARSKI: 1;
assume
A78: x
in (((((((B
'/\' C)
'/\' D)
'/\' E)
'/\' F)
'/\' J)
'/\' M)
'/\' N);
then
A79: x
<>
{} by
EQREL_1:def 4;
x
in ((
INTERSECTION (((((((B
'/\' C)
'/\' D)
'/\' E)
'/\' F)
'/\' J)
'/\' M),N))
\
{
{} }) by
A78,
PARTIT1:def 4;
then
consider bcdefjm,n be
set such that
A80: bcdefjm
in ((((((B
'/\' C)
'/\' D)
'/\' E)
'/\' F)
'/\' J)
'/\' M) and
A81: n
in N and
A82: x
= (bcdefjm
/\ n) by
SETFAM_1:def 5;
bcdefjm
in ((
INTERSECTION ((((((B
'/\' C)
'/\' D)
'/\' E)
'/\' F)
'/\' J),M))
\
{
{} }) by
A80,
PARTIT1:def 4;
then
consider bcdefj,m be
set such that
A83: bcdefj
in (((((B
'/\' C)
'/\' D)
'/\' E)
'/\' F)
'/\' J) and
A84: m
in M and
A85: bcdefjm
= (bcdefj
/\ m) by
SETFAM_1:def 5;
bcdefj
in ((
INTERSECTION (((((B
'/\' C)
'/\' D)
'/\' E)
'/\' F),J))
\
{
{} }) by
A83,
PARTIT1:def 4;
then
consider bcdef,j be
set such that
A86: bcdef
in ((((B
'/\' C)
'/\' D)
'/\' E)
'/\' F) and
A87: j
in J and
A88: bcdefj
= (bcdef
/\ j) by
SETFAM_1:def 5;
bcdef
in ((
INTERSECTION ((((B
'/\' C)
'/\' D)
'/\' E),F))
\
{
{} }) by
A86,
PARTIT1:def 4;
then
consider bcde,f be
set such that
A89: bcde
in (((B
'/\' C)
'/\' D)
'/\' E) and
A90: f
in F and
A91: bcdef
= (bcde
/\ f) by
SETFAM_1:def 5;
bcde
in ((
INTERSECTION (((B
'/\' C)
'/\' D),E))
\
{
{} }) by
A89,
PARTIT1:def 4;
then
consider bcd,e be
set such that
A92: bcd
in ((B
'/\' C)
'/\' D) and
A93: e
in E and
A94: bcde
= (bcd
/\ e) by
SETFAM_1:def 5;
bcd
in ((
INTERSECTION ((B
'/\' C),D))
\
{
{} }) by
A92,
PARTIT1:def 4;
then
consider bc,d be
set such that
A95: bc
in (B
'/\' C) and
A96: d
in D and
A97: bcd
= (bc
/\ d) by
SETFAM_1:def 5;
bc
in ((
INTERSECTION (B,C))
\
{
{} }) by
A95,
PARTIT1:def 4;
then
consider b,c be
set such that
A98: b
in B and
A99: c
in C and
A100: bc
= (b
/\ c) by
SETFAM_1:def 5;
set h = ((((((((B
.--> b)
+* (C
.--> c))
+* (D
.--> d))
+* (E
.--> e))
+* (F
.--> f))
+* (J
.--> j))
+* (M
.--> m))
+* (N
.--> n));
A101: (h
. N)
= n by
FUNCT_7: 94;
A102: (
dom ((((((((B
.--> b)
+* (C
.--> c))
+* (D
.--> d))
+* (E
.--> e))
+* (F
.--> f))
+* (J
.--> j))
+* (M
.--> m))
+* (N
.--> n)))
=
{N, B, C, D, E, F, J, M} by
Th63
.= (
{N}
\/
{B, C, D, E, F, J, M}) by
ENUMSET1: 22
.=
{B, C, D, E, F, J, M, N} by
ENUMSET1: 28;
then
A103: C
in (
dom h) by
ENUMSET1:def 6;
A104: for p be
set st p
in (G
\
{A}) holds (h
. p)
in p
proof
let p be
set;
assume p
in (G
\
{A});
then p
= B or p
= C or p
= D or p
= E or p
= F or p
= J or p
= M or p
= N by
A15,
ENUMSET1:def 6;
hence thesis by
A7,
A8,
A81,
A84,
A87,
A90,
A93,
A96,
A98,
A99,
Lm1,
Th62,
FUNCT_7: 94;
end;
A105: D
in (
dom h) by
A102,
ENUMSET1:def 6;
then
A106: (h
. D)
in (
rng h) by
FUNCT_1:def 3;
A107: N
in (
dom h) by
A102,
ENUMSET1:def 6;
A108: M
in (
dom h) by
A102,
ENUMSET1:def 6;
A109: J
in (
dom h) by
A102,
ENUMSET1:def 6;
A110: F
in (
dom h) by
A102,
ENUMSET1:def 6;
A111: (h
. B)
= b by
A7,
Th62;
A112: (
rng h)
c=
{(h
. B), (h
. C), (h
. D), (h
. E), (h
. F), (h
. J), (h
. M), (h
. N)}
proof
let t be
object;
assume t
in (
rng h);
then
consider x1 be
object such that
A113: x1
in (
dom h) and
A114: t
= (h
. x1) by
FUNCT_1:def 3;
now
per cases by
A102,
A113,
ENUMSET1:def 6;
case x1
= D;
hence thesis by
A114,
ENUMSET1:def 6;
end;
case x1
= B;
hence thesis by
A114,
ENUMSET1:def 6;
end;
case x1
= C;
hence thesis by
A114,
ENUMSET1:def 6;
end;
case x1
= E;
hence thesis by
A114,
ENUMSET1:def 6;
end;
case x1
= F;
hence thesis by
A114,
ENUMSET1:def 6;
end;
case x1
= J;
hence thesis by
A114,
ENUMSET1:def 6;
end;
case x1
= M;
hence thesis by
A114,
ENUMSET1:def 6;
end;
case x1
= N;
hence thesis by
A114,
ENUMSET1:def 6;
end;
end;
hence thesis;
end;
A115: (h
. J)
= j by
A7,
Th62;
A116: (h
. F)
= f by
A7,
Th62;
A117: (h
. M)
= m by
A8,
Lm1;
A118: (h
. E)
= e by
A7,
Th62;
A119: (h
. C)
= c by
A7,
Th62;
A120: (h
. D)
= d by
A7,
Th62;
(
rng h)
c= (
bool Y)
proof
let t be
object;
assume
A121: t
in (
rng h);
now
per cases by
A112,
A121,
ENUMSET1:def 6;
case t
= (h
. D);
hence thesis by
A96,
A120;
end;
case t
= (h
. B);
hence thesis by
A98,
A111;
end;
case t
= (h
. C);
hence thesis by
A99,
A119;
end;
case t
= (h
. E);
hence thesis by
A93,
A118;
end;
case t
= (h
. F);
hence thesis by
A90,
A116;
end;
case t
= (h
. J);
hence thesis by
A87,
A115;
end;
case t
= (h
. M);
hence thesis by
A84,
A117;
end;
case t
= (h
. N);
hence thesis by
A81,
A101;
end;
end;
hence thesis;
end;
then
reconsider FF = (
rng h) as
Subset-Family of Y;
A122: E
in (
dom h) by
A102,
ENUMSET1:def 6;
A123: B
in (
dom h) by
A102,
ENUMSET1:def 6;
{(h
. B), (h
. C), (h
. D), (h
. E), (h
. F), (h
. J), (h
. M), (h
. N)}
c= (
rng h)
proof
let t be
object;
assume
A124: t
in
{(h
. B), (h
. C), (h
. D), (h
. E), (h
. F), (h
. J), (h
. M), (h
. N)};
now
per cases by
A124,
ENUMSET1:def 6;
case t
= (h
. D);
hence thesis by
A105,
FUNCT_1:def 3;
end;
case t
= (h
. B);
hence thesis by
A123,
FUNCT_1:def 3;
end;
case t
= (h
. C);
hence thesis by
A103,
FUNCT_1:def 3;
end;
case t
= (h
. E);
hence thesis by
A122,
FUNCT_1:def 3;
end;
case t
= (h
. F);
hence thesis by
A110,
FUNCT_1:def 3;
end;
case t
= (h
. J);
hence thesis by
A109,
FUNCT_1:def 3;
end;
case t
= (h
. M);
hence thesis by
A108,
FUNCT_1:def 3;
end;
case t
= (h
. N);
hence thesis by
A107,
FUNCT_1:def 3;
end;
end;
hence thesis;
end;
then
A125: (
rng h)
=
{(h
. B), (h
. C), (h
. D), (h
. E), (h
. F), (h
. J), (h
. M), (h
. N)} by
A112,
XBOOLE_0:def 10;
reconsider h as
Function;
A126: xx
c= (
Intersect FF)
proof
let u be
object;
assume
A127: u
in xx;
for y be
set holds y
in FF implies u
in y
proof
let y be
set;
assume
A128: y
in FF;
now
per cases by
A112,
A128,
ENUMSET1:def 6;
case
A129: y
= (h
. D);
u
in (((((d
/\ ((b
/\ c)
/\ e))
/\ f)
/\ j)
/\ m)
/\ n) by
A82,
A85,
A88,
A91,
A94,
A97,
A100,
A127,
XBOOLE_1: 16;
then u
in ((((d
/\ (((b
/\ c)
/\ e)
/\ f))
/\ j)
/\ m)
/\ n) by
XBOOLE_1: 16;
then u
in (((d
/\ ((((b
/\ c)
/\ e)
/\ f)
/\ j))
/\ m)
/\ n) by
XBOOLE_1: 16;
then u
in ((d
/\ (((((b
/\ c)
/\ e)
/\ f)
/\ j)
/\ m))
/\ n) by
XBOOLE_1: 16;
then u
in (d
/\ ((((((b
/\ c)
/\ e)
/\ f)
/\ j)
/\ m)
/\ n)) by
XBOOLE_1: 16;
hence thesis by
A120,
A129,
XBOOLE_0:def 4;
end;
case
A130: y
= (h
. B);
u
in ((((((c
/\ (d
/\ b))
/\ e)
/\ f)
/\ j)
/\ m)
/\ n) by
A82,
A85,
A88,
A91,
A94,
A97,
A100,
A127,
XBOOLE_1: 16;
then u
in (((((c
/\ ((d
/\ b)
/\ e))
/\ f)
/\ j)
/\ m)
/\ n) by
XBOOLE_1: 16;
then u
in (((((c
/\ ((d
/\ e)
/\ b))
/\ f)
/\ j)
/\ m)
/\ n) by
XBOOLE_1: 16;
then u
in ((((c
/\ (((d
/\ e)
/\ b)
/\ f))
/\ j)
/\ m)
/\ n) by
XBOOLE_1: 16;
then u
in (((c
/\ ((((d
/\ e)
/\ b)
/\ f)
/\ j))
/\ m)
/\ n) by
XBOOLE_1: 16;
then u
in (((c
/\ (((d
/\ e)
/\ (f
/\ b))
/\ j))
/\ m)
/\ n) by
XBOOLE_1: 16;
then u
in (((c
/\ ((d
/\ e)
/\ ((f
/\ b)
/\ j)))
/\ m)
/\ n) by
XBOOLE_1: 16;
then u
in (((c
/\ ((d
/\ e)
/\ (f
/\ (j
/\ b))))
/\ m)
/\ n) by
XBOOLE_1: 16;
then u
in ((((c
/\ (d
/\ e))
/\ (f
/\ (j
/\ b)))
/\ m)
/\ n) by
XBOOLE_1: 16;
then u
in (((((c
/\ (d
/\ e))
/\ f)
/\ (j
/\ b))
/\ m)
/\ n) by
XBOOLE_1: 16;
then u
in ((((((c
/\ (d
/\ e))
/\ f)
/\ j)
/\ b)
/\ m)
/\ n) by
XBOOLE_1: 16;
then u
in (((((c
/\ (d
/\ e))
/\ f)
/\ j)
/\ (m
/\ b))
/\ n) by
XBOOLE_1: 16;
then u
in ((((c
/\ (d
/\ e))
/\ f)
/\ j)
/\ ((m
/\ b)
/\ n)) by
XBOOLE_1: 16;
then u
in ((((c
/\ (d
/\ e))
/\ f)
/\ j)
/\ (m
/\ (b
/\ n))) by
XBOOLE_1: 16;
then u
in (((((c
/\ (d
/\ e))
/\ f)
/\ j)
/\ m)
/\ (n
/\ b)) by
XBOOLE_1: 16;
then u
in ((((((c
/\ (d
/\ e))
/\ f)
/\ j)
/\ m)
/\ n)
/\ b) by
XBOOLE_1: 16;
hence thesis by
A111,
A130,
XBOOLE_0:def 4;
end;
case
A131: y
= (h
. C);
u
in ((((((c
/\ (d
/\ b))
/\ e)
/\ f)
/\ j)
/\ m)
/\ n) by
A82,
A85,
A88,
A91,
A94,
A97,
A100,
A127,
XBOOLE_1: 16;
then u
in (((((c
/\ ((d
/\ b)
/\ e))
/\ f)
/\ j)
/\ m)
/\ n) by
XBOOLE_1: 16;
then u
in (((((c
/\ ((d
/\ e)
/\ b))
/\ f)
/\ j)
/\ m)
/\ n) by
XBOOLE_1: 16;
then u
in ((((c
/\ (((d
/\ e)
/\ b)
/\ f))
/\ j)
/\ m)
/\ n) by
XBOOLE_1: 16;
then u
in (((c
/\ ((((d
/\ e)
/\ b)
/\ f)
/\ j))
/\ m)
/\ n) by
XBOOLE_1: 16;
then u
in ((c
/\ (((((d
/\ e)
/\ b)
/\ f)
/\ j)
/\ m))
/\ n) by
XBOOLE_1: 16;
then u
in (c
/\ ((((((d
/\ e)
/\ b)
/\ f)
/\ j)
/\ m)
/\ n)) by
XBOOLE_1: 16;
hence thesis by
A119,
A131,
XBOOLE_0:def 4;
end;
case
A132: y
= (h
. E);
u
in ((((((b
/\ c)
/\ d)
/\ (f
/\ e))
/\ j)
/\ m)
/\ n) by
A82,
A85,
A88,
A91,
A94,
A97,
A100,
A127,
XBOOLE_1: 16;
then u
in (((((b
/\ c)
/\ d)
/\ ((f
/\ e)
/\ j))
/\ m)
/\ n) by
XBOOLE_1: 16;
then u
in (((((b
/\ c)
/\ d)
/\ ((f
/\ j)
/\ e))
/\ m)
/\ n) by
XBOOLE_1: 16;
then u
in ((((((b
/\ c)
/\ d)
/\ (f
/\ j))
/\ e)
/\ m)
/\ n) by
XBOOLE_1: 16;
then u
in (((((b
/\ c)
/\ d)
/\ (f
/\ j))
/\ (e
/\ m))
/\ n) by
XBOOLE_1: 16;
then u
in ((((b
/\ c)
/\ d)
/\ (f
/\ j))
/\ ((m
/\ e)
/\ n)) by
XBOOLE_1: 16;
then u
in ((((b
/\ c)
/\ d)
/\ (f
/\ j))
/\ (m
/\ (n
/\ e))) by
XBOOLE_1: 16;
then u
in (((((b
/\ c)
/\ d)
/\ (f
/\ j))
/\ m)
/\ (n
/\ e)) by
XBOOLE_1: 16;
then u
in ((((((b
/\ c)
/\ d)
/\ (f
/\ j))
/\ m)
/\ n)
/\ e) by
XBOOLE_1: 16;
hence thesis by
A118,
A132,
XBOOLE_0:def 4;
end;
case
A133: y
= (h
. F);
u
in (((((((b
/\ c)
/\ d)
/\ e)
/\ j)
/\ f)
/\ m)
/\ n) by
A82,
A85,
A88,
A91,
A94,
A97,
A100,
A127,
XBOOLE_1: 16;
then u
in (((((((b
/\ c)
/\ d)
/\ e)
/\ j)
/\ m)
/\ f)
/\ n) by
XBOOLE_1: 16;
then u
in (((((((b
/\ c)
/\ d)
/\ e)
/\ j)
/\ m)
/\ n)
/\ f) by
XBOOLE_1: 16;
hence thesis by
A116,
A133,
XBOOLE_0:def 4;
end;
case
A134: y
= (h
. J);
u
in (((((((b
/\ c)
/\ d)
/\ e)
/\ f)
/\ m)
/\ j)
/\ n) by
A82,
A85,
A88,
A91,
A94,
A97,
A100,
A127,
XBOOLE_1: 16;
then u
in (((((((b
/\ c)
/\ d)
/\ e)
/\ f)
/\ m)
/\ n)
/\ j) by
XBOOLE_1: 16;
hence thesis by
A115,
A134,
XBOOLE_0:def 4;
end;
case
A135: y
= (h
. M);
u
in (((((((b
/\ c)
/\ d)
/\ e)
/\ f)
/\ j)
/\ n)
/\ m) by
A82,
A85,
A88,
A91,
A94,
A97,
A100,
A127,
XBOOLE_1: 16;
hence thesis by
A117,
A135,
XBOOLE_0:def 4;
end;
case y
= (h
. N);
hence thesis by
A82,
A101,
A127,
XBOOLE_0:def 4;
end;
end;
hence thesis;
end;
then u
in (
meet FF) by
A125,
SETFAM_1:def 1;
hence thesis by
A125,
SETFAM_1:def 9;
end;
A136: (
Intersect FF)
= (
meet (
rng h)) by
A106,
SETFAM_1:def 9;
(
Intersect FF)
c= xx
proof
let t be
object;
assume
A137: t
in (
Intersect FF);
(h
. C)
in (
rng h) by
A125,
ENUMSET1:def 6;
then
A138: t
in c by
A119,
A136,
A137,
SETFAM_1:def 1;
(h
. B)
in (
rng h) by
A125,
ENUMSET1:def 6;
then t
in b by
A111,
A136,
A137,
SETFAM_1:def 1;
then
A139: t
in (b
/\ c) by
A138,
XBOOLE_0:def 4;
(h
. D)
in (
rng h) by
A125,
ENUMSET1:def 6;
then t
in d by
A120,
A136,
A137,
SETFAM_1:def 1;
then
A140: t
in ((b
/\ c)
/\ d) by
A139,
XBOOLE_0:def 4;
(h
. E)
in (
rng h) by
A125,
ENUMSET1:def 6;
then t
in e by
A118,
A136,
A137,
SETFAM_1:def 1;
then
A141: t
in (((b
/\ c)
/\ d)
/\ e) by
A140,
XBOOLE_0:def 4;
(h
. F)
in (
rng h) by
A125,
ENUMSET1:def 6;
then t
in f by
A116,
A136,
A137,
SETFAM_1:def 1;
then
A142: t
in ((((b
/\ c)
/\ d)
/\ e)
/\ f) by
A141,
XBOOLE_0:def 4;
(h
. J)
in (
rng h) by
A125,
ENUMSET1:def 6;
then t
in j by
A115,
A136,
A137,
SETFAM_1:def 1;
then
A143: t
in (((((b
/\ c)
/\ d)
/\ e)
/\ f)
/\ j) by
A142,
XBOOLE_0:def 4;
(h
. M)
in (
rng h) by
A125,
ENUMSET1:def 6;
then t
in m by
A117,
A136,
A137,
SETFAM_1:def 1;
then
A144: t
in ((((((b
/\ c)
/\ d)
/\ e)
/\ f)
/\ j)
/\ m) by
A143,
XBOOLE_0:def 4;
(h
. N)
in (
rng h) by
A125,
ENUMSET1:def 6;
then t
in n by
A101,
A136,
A137,
SETFAM_1:def 1;
hence thesis by
A82,
A85,
A88,
A91,
A94,
A97,
A100,
A144,
XBOOLE_0:def 4;
end;
then x
= (
Intersect FF) by
A126,
XBOOLE_0:def 10;
hence thesis by
A15,
A102,
A104,
A79,
BVFUNC_2:def 1;
end;
(
CompF (A,G))
= (
'/\' (G
\
{A})) by
BVFUNC_2:def 7;
hence thesis by
A77,
A16,
XBOOLE_0:def 10;
end;
theorem ::
BVFUNC14:68
Th68: for G be
Subset of (
PARTITIONS Y), A,B,C,D,E,F,J,M,N be
a_partition of Y st G
=
{A, B, C, D, E, F, J, M, N} & A
<> B & A
<> C & A
<> D & A
<> E & A
<> F & A
<> J & A
<> M & A
<> N & B
<> C & B
<> D & B
<> E & B
<> F & B
<> J & B
<> M & B
<> N & C
<> D & C
<> E & C
<> F & C
<> J & C
<> M & C
<> N & D
<> E & D
<> F & D
<> J & D
<> M & D
<> N & E
<> F & E
<> J & E
<> M & E
<> N & F
<> J & F
<> M & F
<> N & J
<> M & J
<> N & M
<> N holds (
CompF (B,G))
= (((((((A
'/\' C)
'/\' D)
'/\' E)
'/\' F)
'/\' J)
'/\' M)
'/\' N)
proof
let G be
Subset of (
PARTITIONS Y);
let A,B,C,D,E,F,J,M,N be
a_partition of Y;
{A, B, C, D, E, F, J, M, N}
= (
{A, B}
\/
{C, D, E, F, J, M, N}) by
ENUMSET1: 78
.=
{B, A, C, D, E, F, J, M, N} by
ENUMSET1: 78;
hence thesis by
Th67;
end;
theorem ::
BVFUNC14:69
Th69: for G be
Subset of (
PARTITIONS Y), A,B,C,D,E,F,J,M,N be
a_partition of Y st G
=
{A, B, C, D, E, F, J, M, N} & A
<> B & A
<> C & A
<> D & A
<> E & A
<> F & A
<> J & A
<> M & A
<> N & B
<> C & B
<> D & B
<> E & B
<> F & B
<> J & B
<> M & B
<> N & C
<> D & C
<> E & C
<> F & C
<> J & C
<> M & C
<> N & D
<> E & D
<> F & D
<> J & D
<> M & D
<> N & E
<> F & E
<> J & E
<> M & E
<> N & F
<> J & F
<> M & F
<> N & J
<> M & J
<> N & M
<> N holds (
CompF (C,G))
= (((((((A
'/\' B)
'/\' D)
'/\' E)
'/\' F)
'/\' J)
'/\' M)
'/\' N)
proof
let G be
Subset of (
PARTITIONS Y);
let A,B,C,D,E,F,J,M,N be
a_partition of Y;
{A, B, C, D, E, F, J, M, N}
= (
{A, B, C}
\/
{D, E, F, J, M, N}) by
ENUMSET1: 79
.= ((
{A}
\/
{B, C})
\/
{D, E, F, J, M, N}) by
ENUMSET1: 2
.= (
{A, C, B}
\/
{D, E, F, J, M, N}) by
ENUMSET1: 2
.=
{A, C, B, D, E, F, J, M, N} by
ENUMSET1: 79;
hence thesis by
Th68;
end;
theorem ::
BVFUNC14:70
Th70: for G be
Subset of (
PARTITIONS Y), A,B,C,D,E,F,J,M,N be
a_partition of Y st G
=
{A, B, C, D, E, F, J, M, N} & A
<> B & A
<> C & A
<> D & A
<> E & A
<> F & A
<> J & A
<> M & A
<> N & B
<> C & B
<> D & B
<> E & B
<> F & B
<> J & B
<> M & B
<> N & C
<> D & C
<> E & C
<> F & C
<> J & C
<> M & C
<> N & D
<> E & D
<> F & D
<> J & D
<> M & D
<> N & E
<> F & E
<> J & E
<> M & E
<> N & F
<> J & F
<> M & F
<> N & J
<> M & J
<> N & M
<> N holds (
CompF (D,G))
= (((((((A
'/\' B)
'/\' C)
'/\' E)
'/\' F)
'/\' J)
'/\' M)
'/\' N)
proof
let G be
Subset of (
PARTITIONS Y);
let A,B,C,D,E,F,J,M,N be
a_partition of Y;
{A, B, C, D, E, F, J, M, N}
= (
{A, B}
\/
{C, D, E, F, J, M, N}) by
ENUMSET1: 78
.= (
{A, B}
\/ (
{C, D}
\/
{E, F, J, M, N})) by
ENUMSET1: 17
.= (
{A, B}
\/
{D, C, E, F, J, M, N}) by
ENUMSET1: 17
.=
{A, B, D, C, E, F, J, M, N} by
ENUMSET1: 78;
hence thesis by
Th69;
end;
theorem ::
BVFUNC14:71
Th71: for G be
Subset of (
PARTITIONS Y), A,B,C,D,E,F,J,M,N be
a_partition of Y st G
=
{A, B, C, D, E, F, J, M, N} & A
<> B & A
<> C & A
<> D & A
<> E & A
<> F & A
<> J & A
<> M & A
<> N & B
<> C & B
<> D & B
<> E & B
<> F & B
<> J & B
<> M & B
<> N & C
<> D & C
<> E & C
<> F & C
<> J & C
<> M & C
<> N & D
<> E & D
<> F & D
<> J & D
<> M & D
<> N & E
<> F & E
<> J & E
<> M & E
<> N & F
<> J & F
<> M & F
<> N & J
<> M & J
<> N & M
<> N holds (
CompF (E,G))
= (((((((A
'/\' B)
'/\' C)
'/\' D)
'/\' F)
'/\' J)
'/\' M)
'/\' N)
proof
let G be
Subset of (
PARTITIONS Y);
let A,B,C,D,E,F,J,M,N be
a_partition of Y;
{A, B, C, D, E, F, J, M, N}
= (
{A, B, C}
\/
{D, E, F, J, M, N}) by
ENUMSET1: 79
.= (
{A, B, C}
\/ (
{D, E}
\/
{F, J, M, N})) by
ENUMSET1: 12
.= (
{A, B, C}
\/
{E, D, F, J, M, N}) by
ENUMSET1: 12
.=
{A, B, C, E, D, F, J, M, N} by
ENUMSET1: 79;
hence thesis by
Th70;
end;
theorem ::
BVFUNC14:72
Th72: for G be
Subset of (
PARTITIONS Y), A,B,C,D,E,F,J,M,N be
a_partition of Y st G
=
{A, B, C, D, E, F, J, M, N} & A
<> B & A
<> C & A
<> D & A
<> E & A
<> F & A
<> J & A
<> M & A
<> N & B
<> C & B
<> D & B
<> E & B
<> F & B
<> J & B
<> M & B
<> N & C
<> D & C
<> E & C
<> F & C
<> J & C
<> M & C
<> N & D
<> E & D
<> F & D
<> J & D
<> M & D
<> N & E
<> F & E
<> J & E
<> M & E
<> N & F
<> J & F
<> M & F
<> N & J
<> M & J
<> N & M
<> N holds (
CompF (F,G))
= (((((((A
'/\' B)
'/\' C)
'/\' D)
'/\' E)
'/\' J)
'/\' M)
'/\' N)
proof
let G be
Subset of (
PARTITIONS Y);
let A,B,C,D,E,F,J,M,N be
a_partition of Y;
{A, B, C, D, E, F, J, M, N}
= (
{A, B, C, D}
\/
{E, F, J, M, N}) by
Lm3
.= (
{A, B, C, D}
\/ (
{E, F}
\/
{J, M, N})) by
ENUMSET1: 8
.= (
{A, B, C, D}
\/
{F, E, J, M, N}) by
ENUMSET1: 8
.=
{A, B, C, D, F, E, J, M, N} by
Lm3;
hence thesis by
Th71;
end;
theorem ::
BVFUNC14:73
Th73: for G be
Subset of (
PARTITIONS Y), A,B,C,D,E,F,J,M,N be
a_partition of Y st G
=
{A, B, C, D, E, F, J, M, N} & A
<> B & A
<> C & A
<> D & A
<> E & A
<> F & A
<> J & A
<> M & A
<> N & B
<> C & B
<> D & B
<> E & B
<> F & B
<> J & B
<> M & B
<> N & C
<> D & C
<> E & C
<> F & C
<> J & C
<> M & C
<> N & D
<> E & D
<> F & D
<> J & D
<> M & D
<> N & E
<> F & E
<> J & E
<> M & E
<> N & F
<> J & F
<> M & F
<> N & J
<> M & J
<> N & M
<> N holds (
CompF (J,G))
= (((((((A
'/\' B)
'/\' C)
'/\' D)
'/\' E)
'/\' F)
'/\' M)
'/\' N)
proof
let G be
Subset of (
PARTITIONS Y);
let A,B,C,D,E,F,J,M,N be
a_partition of Y;
{A, B, C, D, E, F, J, M, N}
= (
{A, B, C, D, E}
\/
{F, J, M, N}) by
ENUMSET1: 81
.= (
{A, B, C, D, E}
\/ (
{J, F}
\/
{M, N})) by
ENUMSET1: 5
.= (
{A, B, C, D, E}
\/
{J, F, M, N}) by
ENUMSET1: 5
.=
{A, B, C, D, E, J, F, M, N} by
ENUMSET1: 81;
hence thesis by
Th72;
end;
theorem ::
BVFUNC14:74
Th74: for G be
Subset of (
PARTITIONS Y), A,B,C,D,E,F,J,M,N be
a_partition of Y st G
=
{A, B, C, D, E, F, J, M, N} & A
<> B & A
<> C & A
<> D & A
<> E & A
<> F & A
<> J & A
<> M & A
<> N & B
<> C & B
<> D & B
<> E & B
<> F & B
<> J & B
<> M & B
<> N & C
<> D & C
<> E & C
<> F & C
<> J & C
<> M & C
<> N & D
<> E & D
<> F & D
<> J & D
<> M & D
<> N & E
<> F & E
<> J & E
<> M & E
<> N & F
<> J & F
<> M & F
<> N & J
<> M & J
<> N & M
<> N holds (
CompF (M,G))
= (((((((A
'/\' B)
'/\' C)
'/\' D)
'/\' E)
'/\' F)
'/\' J)
'/\' N)
proof
let G be
Subset of (
PARTITIONS Y);
let A,B,C,D,E,F,J,M,N be
a_partition of Y;
{A, B, C, D, E, F, J, M, N}
= (
{A, B, C, D, E, F}
\/
{J, M, N}) by
ENUMSET1: 82
.= (
{A, B, C, D, E, F}
\/ (
{J, M}
\/
{N})) by
ENUMSET1: 3
.= (
{A, B, C, D, E, F}
\/
{M, J, N}) by
ENUMSET1: 3
.=
{A, B, C, D, E, F, M, J, N} by
ENUMSET1: 82;
hence thesis by
Th73;
end;
theorem ::
BVFUNC14:75
for G be
Subset of (
PARTITIONS Y), A,B,C,D,E,F,J,M,N be
a_partition of Y st G
=
{A, B, C, D, E, F, J, M, N} & A
<> B & A
<> C & A
<> D & A
<> E & A
<> F & A
<> J & A
<> M & A
<> N & B
<> C & B
<> D & B
<> E & B
<> F & B
<> J & B
<> M & B
<> N & C
<> D & C
<> E & C
<> F & C
<> J & C
<> M & C
<> N & D
<> E & D
<> F & D
<> J & D
<> M & D
<> N & E
<> F & E
<> J & E
<> M & E
<> N & F
<> J & F
<> M & F
<> N & J
<> M & J
<> N & M
<> N holds (
CompF (N,G))
= (((((((A
'/\' B)
'/\' C)
'/\' D)
'/\' E)
'/\' F)
'/\' J)
'/\' M)
proof
let G be
Subset of (
PARTITIONS Y);
let A,B,C,D,E,F,J,M,N be
a_partition of Y;
{A, B, C, D, E, F, J, M, N}
= (
{A, B, C, D, E, F, J}
\/
{M, N}) by
ENUMSET1: 83
.=
{A, B, C, D, E, F, J, N, M} by
ENUMSET1: 83;
hence thesis by
Th74;
end;
theorem ::
BVFUNC14:76
Th76: for A,B,C,D,E,F,J,M,N be
set, h be
Function, A9,B9,C9,D9,E9,F9,J9,M9,N9 be
set st A
<> B & A
<> C & A
<> D & A
<> E & A
<> F & A
<> J & A
<> M & A
<> N & B
<> C & B
<> D & B
<> E & B
<> F & B
<> J & B
<> M & B
<> N & C
<> D & C
<> E & C
<> F & C
<> J & C
<> M & C
<> N & D
<> E & D
<> F & D
<> J & D
<> M & D
<> N & E
<> F & E
<> J & E
<> M & E
<> N & F
<> J & F
<> M & F
<> N & J
<> M & J
<> N & M
<> N & h
= (((((((((B
.--> B9)
+* (C
.--> C9))
+* (D
.--> D9))
+* (E
.--> E9))
+* (F
.--> F9))
+* (J
.--> J9))
+* (M
.--> M9))
+* (N
.--> N9))
+* (A
.--> A9)) holds (h
. A)
= A9 & (h
. B)
= B9 & (h
. C)
= C9 & (h
. D)
= D9 & (h
. E)
= E9 & (h
. F)
= F9 & (h
. J)
= J9 & (h
. M)
= M9 & (h
. N)
= N9
proof
let A,B,C,D,E,F,J,M,N be
set;
let h be
Function;
let A9,B9,C9,D9,E9,F9,J9,M9,N9 be
set;
assume that
A1: A
<> B and
A2: A
<> C and
A3: A
<> D and
A4: A
<> E and
A5: A
<> F and
A6: A
<> J and
A7: A
<> M and
A8: A
<> N and
A9: B
<> C & B
<> D & B
<> E & B
<> F & B
<> J & B
<> M & B
<> N & C
<> D & C
<> E & C
<> F & C
<> J & C
<> M & C
<> N & D
<> E & D
<> F & D
<> J & D
<> M & D
<> N & E
<> F & E
<> J & E
<> M & E
<> N & F
<> J & F
<> M & F
<> N & J
<> M & J
<> N and
A10: M
<> N and
A11: h
= (((((((((B
.--> B9)
+* (C
.--> C9))
+* (D
.--> D9))
+* (E
.--> E9))
+* (F
.--> F9))
+* (J
.--> J9))
+* (M
.--> M9))
+* (N
.--> N9))
+* (A
.--> A9));
A
in (
dom (A
.--> A9)) by
TARSKI:def 1;
then
A13: (h
. A)
= ((A
.--> A9)
. A) by
A11,
FUNCT_4: 13;
not E
in (
dom (A
.--> A9)) by
A4,
TARSKI:def 1;
then
A14: (h
. E)
= (((((((((B
.--> B9)
+* (C
.--> C9))
+* (D
.--> D9))
+* (E
.--> E9))
+* (F
.--> F9))
+* (J
.--> J9))
+* (M
.--> M9))
+* (N
.--> N9))
. E) by
A11,
FUNCT_4: 11
.= E9 by
A9,
Th62;
not N
in (
dom (A
.--> A9)) by
A8,
TARSKI:def 1;
then
A15: (h
. N)
= (((((((((B
.--> B9)
+* (C
.--> C9))
+* (D
.--> D9))
+* (E
.--> E9))
+* (F
.--> F9))
+* (J
.--> J9))
+* (M
.--> M9))
+* (N
.--> N9))
. N) by
A11,
FUNCT_4: 11
.= N9 by
FUNCT_7: 94;
not D
in (
dom (A
.--> A9)) by
A3,
TARSKI:def 1;
then
A16: (h
. D)
= (((((((((B
.--> B9)
+* (C
.--> C9))
+* (D
.--> D9))
+* (E
.--> E9))
+* (F
.--> F9))
+* (J
.--> J9))
+* (M
.--> M9))
+* (N
.--> N9))
. D) by
A11,
FUNCT_4: 11
.= D9 by
A9,
Th62;
not C
in (
dom (A
.--> A9)) by
A2,
TARSKI:def 1;
then
A17: (h
. C)
= (((((((((B
.--> B9)
+* (C
.--> C9))
+* (D
.--> D9))
+* (E
.--> E9))
+* (F
.--> F9))
+* (J
.--> J9))
+* (M
.--> M9))
+* (N
.--> N9))
. C) by
A11,
FUNCT_4: 11;
not J
in (
dom (A
.--> A9)) by
A6,
TARSKI:def 1;
then
A18: (h
. J)
= (((((((((B
.--> B9)
+* (C
.--> C9))
+* (D
.--> D9))
+* (E
.--> E9))
+* (F
.--> F9))
+* (J
.--> J9))
+* (M
.--> M9))
+* (N
.--> N9))
. J) by
A11,
FUNCT_4: 11
.= J9 by
A9,
Th62;
not F
in (
dom (A
.--> A9)) by
A5,
TARSKI:def 1;
then
A19: (h
. F)
= (((((((((B
.--> B9)
+* (C
.--> C9))
+* (D
.--> D9))
+* (E
.--> E9))
+* (F
.--> F9))
+* (J
.--> J9))
+* (M
.--> M9))
+* (N
.--> N9))
. F) by
A11,
FUNCT_4: 11
.= F9 by
A9,
Th62;
not M
in (
dom (A
.--> A9)) by
A7,
TARSKI:def 1;
then
A20: (h
. M)
= (((((((((B
.--> B9)
+* (C
.--> C9))
+* (D
.--> D9))
+* (E
.--> E9))
+* (F
.--> F9))
+* (J
.--> J9))
+* (M
.--> M9))
+* (N
.--> N9))
. M) by
A11,
FUNCT_4: 11
.= M9 by
A10,
Lm1;
not B
in (
dom (A
.--> A9)) by
A1,
TARSKI:def 1;
then (h
. B)
= (((((((((B
.--> B9)
+* (C
.--> C9))
+* (D
.--> D9))
+* (E
.--> E9))
+* (F
.--> F9))
+* (J
.--> J9))
+* (M
.--> M9))
+* (N
.--> N9))
. B) by
A11,
FUNCT_4: 11
.= B9 by
A9,
Th62;
hence thesis by
A9,
A13,
A17,
A16,
A14,
A19,
A18,
A20,
A15,
Th62,
FUNCOP_1: 72;
end;
theorem ::
BVFUNC14:77
Th77: for A,B,C,D,E,F,J,M,N be
set, h be
Function, A9,B9,C9,D9,E9,F9,J9,M9,N9 be
set st h
= (((((((((B
.--> B9)
+* (C
.--> C9))
+* (D
.--> D9))
+* (E
.--> E9))
+* (F
.--> F9))
+* (J
.--> J9))
+* (M
.--> M9))
+* (N
.--> N9))
+* (A
.--> A9)) holds (
dom h)
=
{A, B, C, D, E, F, J, M, N}
proof
let A,B,C,D,E,F,J,M,N be
set;
let h be
Function;
let A9,B9,C9,D9,E9,F9,J9,M9,N9 be
set;
assume
A1: h
= (((((((((B
.--> B9)
+* (C
.--> C9))
+* (D
.--> D9))
+* (E
.--> E9))
+* (F
.--> F9))
+* (J
.--> J9))
+* (M
.--> M9))
+* (N
.--> N9))
+* (A
.--> A9));
A2: (
dom (A
.--> A9))
=
{A};
(
dom ((((((((B
.--> B9)
+* (C
.--> C9))
+* (D
.--> D9))
+* (E
.--> E9))
+* (F
.--> F9))
+* (J
.--> J9))
+* (M
.--> M9))
+* (N
.--> N9)))
=
{N, B, C, D, E, F, J, M} by
Th63
.= (
{N}
\/
{B, C, D, E, F, J, M}) by
ENUMSET1: 22
.=
{B, C, D, E, F, J, M, N} by
ENUMSET1: 28;
then (
dom (((((((((B
.--> B9)
+* (C
.--> C9))
+* (D
.--> D9))
+* (E
.--> E9))
+* (F
.--> F9))
+* (J
.--> J9))
+* (M
.--> M9))
+* (N
.--> N9))
+* (A
.--> A9)))
= (
{B, C, D, E, F, J, M, N}
\/
{A}) by
A2,
FUNCT_4:def 1
.=
{A, B, C, D, E, F, J, M, N} by
ENUMSET1: 77;
hence thesis by
A1;
end;
theorem ::
BVFUNC14:78
Th78: for A,B,C,D,E,F,J,M,N be
set, h be
Function, A9,B9,C9,D9,E9,F9,J9,M9,N9 be
set st h
= (((((((((B
.--> B9)
+* (C
.--> C9))
+* (D
.--> D9))
+* (E
.--> E9))
+* (F
.--> F9))
+* (J
.--> J9))
+* (M
.--> M9))
+* (N
.--> N9))
+* (A
.--> A9)) holds (
rng h)
=
{(h
. A), (h
. B), (h
. C), (h
. D), (h
. E), (h
. F), (h
. J), (h
. M), (h
. N)}
proof
let A,B,C,D,E,F,J,M,N be
set;
let h be
Function;
let A9,B9,C9,D9,E9,F9,J9,M9,N9 be
set;
assume h
= (((((((((B
.--> B9)
+* (C
.--> C9))
+* (D
.--> D9))
+* (E
.--> E9))
+* (F
.--> F9))
+* (J
.--> J9))
+* (M
.--> M9))
+* (N
.--> N9))
+* (A
.--> A9));
then
A1: (
dom h)
=
{A, B, C, D, E, F, J, M, N} by
Th77;
then
A2: B
in (
dom h) by
ENUMSET1:def 7;
A3: M
in (
dom h) by
A1,
ENUMSET1:def 7;
A4: J
in (
dom h) by
A1,
ENUMSET1:def 7;
A5: N
in (
dom h) by
A1,
ENUMSET1:def 7;
A6: D
in (
dom h) by
A1,
ENUMSET1:def 7;
A7: C
in (
dom h) by
A1,
ENUMSET1:def 7;
A8: (
rng h)
c=
{(h
. A), (h
. B), (h
. C), (h
. D), (h
. E), (h
. F), (h
. J), (h
. M), (h
. N)}
proof
let t be
object;
assume t
in (
rng h);
then
consider x1 be
object such that
A9: x1
in (
dom h) and
A10: t
= (h
. x1) by
FUNCT_1:def 3;
now
per cases by
A1,
A9,
ENUMSET1:def 7;
case x1
= A;
hence thesis by
A10,
ENUMSET1:def 7;
end;
case x1
= B;
hence thesis by
A10,
ENUMSET1:def 7;
end;
case x1
= C;
hence thesis by
A10,
ENUMSET1:def 7;
end;
case x1
= D;
hence thesis by
A10,
ENUMSET1:def 7;
end;
case x1
= E;
hence thesis by
A10,
ENUMSET1:def 7;
end;
case x1
= F;
hence thesis by
A10,
ENUMSET1:def 7;
end;
case x1
= J;
hence thesis by
A10,
ENUMSET1:def 7;
end;
case x1
= M;
hence thesis by
A10,
ENUMSET1:def 7;
end;
case x1
= N;
hence thesis by
A10,
ENUMSET1:def 7;
end;
end;
hence thesis;
end;
A11: F
in (
dom h) by
A1,
ENUMSET1:def 7;
A12: E
in (
dom h) by
A1,
ENUMSET1:def 7;
A13: A
in (
dom h) by
A1,
ENUMSET1:def 7;
{(h
. A), (h
. B), (h
. C), (h
. D), (h
. E), (h
. F), (h
. J), (h
. M), (h
. N)}
c= (
rng h)
proof
let t be
object;
assume
A14: t
in
{(h
. A), (h
. B), (h
. C), (h
. D), (h
. E), (h
. F), (h
. J), (h
. M), (h
. N)};
now
per cases by
A14,
ENUMSET1:def 7;
case t
= (h
. A);
hence thesis by
A13,
FUNCT_1:def 3;
end;
case t
= (h
. B);
hence thesis by
A2,
FUNCT_1:def 3;
end;
case t
= (h
. C);
hence thesis by
A7,
FUNCT_1:def 3;
end;
case t
= (h
. D);
hence thesis by
A6,
FUNCT_1:def 3;
end;
case t
= (h
. E);
hence thesis by
A12,
FUNCT_1:def 3;
end;
case t
= (h
. F);
hence thesis by
A11,
FUNCT_1:def 3;
end;
case t
= (h
. J);
hence thesis by
A4,
FUNCT_1:def 3;
end;
case t
= (h
. M);
hence thesis by
A3,
FUNCT_1:def 3;
end;
case t
= (h
. N);
hence thesis by
A5,
FUNCT_1:def 3;
end;
end;
hence thesis;
end;
hence thesis by
A8,
XBOOLE_0:def 10;
end;
theorem ::
BVFUNC14:79
for G be
Subset of (
PARTITIONS Y), A,B,C,D,E,F,J,M,N be
a_partition of Y, z,u be
Element of Y st G is
independent & G
=
{A, B, C, D, E, F, J, M, N} & A
<> B & A
<> C & A
<> D & A
<> E & A
<> F & A
<> J & A
<> M & A
<> N & B
<> C & B
<> D & B
<> E & B
<> F & B
<> J & B
<> M & B
<> N & C
<> D & C
<> E & C
<> F & C
<> J & C
<> M & C
<> N & D
<> E & D
<> F & D
<> J & D
<> M & D
<> N & E
<> F & E
<> J & E
<> M & E
<> N & F
<> J & F
<> M & F
<> N & J
<> M & J
<> N & M
<> N holds ((
EqClass (u,(((((((B
'/\' C)
'/\' D)
'/\' E)
'/\' F)
'/\' J)
'/\' M)
'/\' N)))
/\ (
EqClass (z,A)))
<>
{}
proof
let G be
Subset of (
PARTITIONS Y);
let A,B,C,D,E,F,J,M,N be
a_partition of Y;
let z,u be
Element of Y;
assume that
A1: G is
independent and
A2: G
=
{A, B, C, D, E, F, J, M, N} and
A3: A
<> B & A
<> C & A
<> D & A
<> E & A
<> F & A
<> J & A
<> M & A
<> N & B
<> C & B
<> D & B
<> E & B
<> F & B
<> J & B
<> M & B
<> N & C
<> D & C
<> E & C
<> F & C
<> J & C
<> M & C
<> N & D
<> E & D
<> F & D
<> J & D
<> M & D
<> N & E
<> F & E
<> J & E
<> M & E
<> N & F
<> J & F
<> M & F
<> N & J
<> M & J
<> N & M
<> N;
set h = (((((((((B
.--> (
EqClass (u,B)))
+* (C
.--> (
EqClass (u,C))))
+* (D
.--> (
EqClass (u,D))))
+* (E
.--> (
EqClass (u,E))))
+* (F
.--> (
EqClass (u,F))))
+* (J
.--> (
EqClass (u,J))))
+* (M
.--> (
EqClass (u,M))))
+* (N
.--> (
EqClass (u,N))))
+* (A
.--> (
EqClass (z,A))));
A4: (h
. A)
= (
EqClass (z,A)) by
A3,
Th76;
set GG = (
EqClass (u,(((((((B
'/\' C)
'/\' D)
'/\' E)
'/\' F)
'/\' J)
'/\' M)
'/\' N)));
GG
= ((
EqClass (u,((((((B
'/\' C)
'/\' D)
'/\' E)
'/\' F)
'/\' J)
'/\' M)))
/\ (
EqClass (u,N))) by
Th1;
then GG
= (((
EqClass (u,(((((B
'/\' C)
'/\' D)
'/\' E)
'/\' F)
'/\' J)))
/\ (
EqClass (u,M)))
/\ (
EqClass (u,N))) by
Th1;
then GG
= ((((
EqClass (u,((((B
'/\' C)
'/\' D)
'/\' E)
'/\' F)))
/\ (
EqClass (u,J)))
/\ (
EqClass (u,M)))
/\ (
EqClass (u,N))) by
Th1;
then GG
= (((((
EqClass (u,(((B
'/\' C)
'/\' D)
'/\' E)))
/\ (
EqClass (u,F)))
/\ (
EqClass (u,J)))
/\ (
EqClass (u,M)))
/\ (
EqClass (u,N))) by
Th1;
then GG
= ((((((
EqClass (u,((B
'/\' C)
'/\' D)))
/\ (
EqClass (u,E)))
/\ (
EqClass (u,F)))
/\ (
EqClass (u,J)))
/\ (
EqClass (u,M)))
/\ (
EqClass (u,N))) by
Th1;
then GG
= (((((((
EqClass (u,(B
'/\' C)))
/\ (
EqClass (u,D)))
/\ (
EqClass (u,E)))
/\ (
EqClass (u,F)))
/\ (
EqClass (u,J)))
/\ (
EqClass (u,M)))
/\ (
EqClass (u,N))) by
Th1;
then
A5: (GG
/\ (
EqClass (z,A)))
= (((((((((
EqClass (u,B))
/\ (
EqClass (u,C)))
/\ (
EqClass (u,D)))
/\ (
EqClass (u,E)))
/\ (
EqClass (u,F)))
/\ (
EqClass (u,J)))
/\ (
EqClass (u,M)))
/\ (
EqClass (u,N)))
/\ (
EqClass (z,A))) by
Th1;
A6: (h
. B)
= (
EqClass (u,B)) by
A3,
Th76;
A7: (h
. F)
= (
EqClass (u,F)) by
A3,
Th76;
A8: (h
. E)
= (
EqClass (u,E)) by
A3,
Th76;
A9: (h
. M)
= (
EqClass (u,M)) by
A3,
Th76;
A10: (h
. J)
= (
EqClass (u,J)) by
A3,
Th76;
A11: (h
. N)
= (
EqClass (u,N)) by
A3,
Th76;
A12: (h
. D)
= (
EqClass (u,D)) by
A3,
Th76;
A13: (h
. C)
= (
EqClass (u,C)) by
A3,
Th76;
A14: (
rng h)
=
{(h
. A), (h
. B), (h
. C), (h
. D), (h
. E), (h
. F), (h
. J), (h
. M), (h
. N)} by
Th78;
(
rng h)
c= (
bool Y)
proof
let t be
object;
assume t
in (
rng h);
then t
= (h
. A) or t
= (h
. B) or t
= (h
. C) or t
= (h
. D) or t
= (h
. E) or t
= (h
. F) or t
= (h
. J) or t
= (h
. M) or t
= (h
. N) by
A14,
ENUMSET1:def 7;
hence thesis by
A4,
A6,
A13,
A12,
A8,
A7,
A10,
A9,
A11;
end;
then
reconsider FF = (
rng h) as
Subset-Family of Y;
A15: (
dom h)
= G by
A2,
Th77;
then A
in (
dom h) by
A2,
ENUMSET1:def 7;
then
A16: (h
. A)
in (
rng h) by
FUNCT_1:def 3;
then
A17: (
Intersect FF)
= (
meet (
rng h)) by
SETFAM_1:def 9;
for d be
set st d
in G holds (h
. d)
in d
proof
let d be
set;
assume d
in G;
then d
= A or d
= B or d
= C or d
= D or d
= E or d
= F or d
= J or d
= M or d
= N by
A2,
ENUMSET1:def 7;
hence thesis by
A4,
A6,
A13,
A12,
A8,
A7,
A10,
A9,
A11;
end;
then (
Intersect FF)
<>
{} by
A1,
A15,
BVFUNC_2:def 5;
then
consider m be
object such that
A18: m
in (
Intersect FF) by
XBOOLE_0:def 1;
C
in (
dom h) by
A2,
A15,
ENUMSET1:def 7;
then (h
. C)
in (
rng h) by
FUNCT_1:def 3;
then
A19: m
in (
EqClass (u,C)) by
A13,
A17,
A18,
SETFAM_1:def 1;
B
in (
dom h) by
A2,
A15,
ENUMSET1:def 7;
then (h
. B)
in (
rng h) by
FUNCT_1:def 3;
then m
in (
EqClass (u,B)) by
A6,
A17,
A18,
SETFAM_1:def 1;
then
A20: m
in ((
EqClass (u,B))
/\ (
EqClass (u,C))) by
A19,
XBOOLE_0:def 4;
D
in (
dom h) by
A2,
A15,
ENUMSET1:def 7;
then (h
. D)
in (
rng h) by
FUNCT_1:def 3;
then m
in (
EqClass (u,D)) by
A12,
A17,
A18,
SETFAM_1:def 1;
then
A21: m
in (((
EqClass (u,B))
/\ (
EqClass (u,C)))
/\ (
EqClass (u,D))) by
A20,
XBOOLE_0:def 4;
E
in (
dom h) by
A2,
A15,
ENUMSET1:def 7;
then (h
. E)
in (
rng h) by
FUNCT_1:def 3;
then m
in (
EqClass (u,E)) by
A8,
A17,
A18,
SETFAM_1:def 1;
then
A22: m
in ((((
EqClass (u,B))
/\ (
EqClass (u,C)))
/\ (
EqClass (u,D)))
/\ (
EqClass (u,E))) by
A21,
XBOOLE_0:def 4;
F
in (
dom h) by
A2,
A15,
ENUMSET1:def 7;
then (h
. F)
in (
rng h) by
FUNCT_1:def 3;
then m
in (
EqClass (u,F)) by
A7,
A17,
A18,
SETFAM_1:def 1;
then
A23: m
in (((((
EqClass (u,B))
/\ (
EqClass (u,C)))
/\ (
EqClass (u,D)))
/\ (
EqClass (u,E)))
/\ (
EqClass (u,F))) by
A22,
XBOOLE_0:def 4;
J
in (
dom h) by
A2,
A15,
ENUMSET1:def 7;
then (h
. J)
in (
rng h) by
FUNCT_1:def 3;
then m
in (
EqClass (u,J)) by
A10,
A17,
A18,
SETFAM_1:def 1;
then
A24: m
in ((((((
EqClass (u,B))
/\ (
EqClass (u,C)))
/\ (
EqClass (u,D)))
/\ (
EqClass (u,E)))
/\ (
EqClass (u,F)))
/\ (
EqClass (u,J))) by
A23,
XBOOLE_0:def 4;
M
in (
dom h) by
A2,
A15,
ENUMSET1:def 7;
then (h
. M)
in (
rng h) by
FUNCT_1:def 3;
then m
in (
EqClass (u,M)) by
A9,
A17,
A18,
SETFAM_1:def 1;
then
A25: m
in (((((((
EqClass (u,B))
/\ (
EqClass (u,C)))
/\ (
EqClass (u,D)))
/\ (
EqClass (u,E)))
/\ (
EqClass (u,F)))
/\ (
EqClass (u,J)))
/\ (
EqClass (u,M))) by
A24,
XBOOLE_0:def 4;
N
in (
dom h) by
A2,
A15,
ENUMSET1:def 7;
then (h
. N)
in (
rng h) by
FUNCT_1:def 3;
then m
in (
EqClass (u,N)) by
A11,
A17,
A18,
SETFAM_1:def 1;
then
A26: m
in ((((((((
EqClass (u,B))
/\ (
EqClass (u,C)))
/\ (
EqClass (u,D)))
/\ (
EqClass (u,E)))
/\ (
EqClass (u,F)))
/\ (
EqClass (u,J)))
/\ (
EqClass (u,M)))
/\ (
EqClass (u,N))) by
A25,
XBOOLE_0:def 4;
m
in (
EqClass (z,A)) by
A4,
A16,
A17,
A18,
SETFAM_1:def 1;
hence thesis by
A5,
A26,
XBOOLE_0:def 4;
end;
theorem ::
BVFUNC14:80
for G be
Subset of (
PARTITIONS Y), A,B,C,D,E,F,J,M,N be
a_partition of Y, z,u be
Element of Y st G is
independent & G
=
{A, B, C, D, E, F, J, M, N} & A
<> B & A
<> C & A
<> D & A
<> E & A
<> F & A
<> J & A
<> M & A
<> N & B
<> C & B
<> D & B
<> E & B
<> F & B
<> J & B
<> M & B
<> N & C
<> D & C
<> E & C
<> F & C
<> J & C
<> M & C
<> N & D
<> E & D
<> F & D
<> J & D
<> M & D
<> N & E
<> F & E
<> J & E
<> M & E
<> N & F
<> J & F
<> M & F
<> N & J
<> M & J
<> N & M
<> N & (
EqClass (z,((((((C
'/\' D)
'/\' E)
'/\' F)
'/\' J)
'/\' M)
'/\' N)))
= (
EqClass (u,((((((C
'/\' D)
'/\' E)
'/\' F)
'/\' J)
'/\' M)
'/\' N))) holds (
EqClass (u,(
CompF (A,G))))
meets (
EqClass (z,(
CompF (B,G))))
proof
let G be
Subset of (
PARTITIONS Y);
let A,B,C,D,E,F,J,M,N be
a_partition of Y;
let z,u be
Element of Y;
assume that
A1: G is
independent and
A2: G
=
{A, B, C, D, E, F, J, M, N} and
A3: A
<> B & A
<> C & A
<> D & A
<> E & A
<> F & A
<> J & A
<> M & A
<> N & B
<> C & B
<> D & B
<> E & B
<> F & B
<> J & B
<> M & B
<> N & C
<> D & C
<> E & C
<> F & C
<> J & C
<> M & C
<> N & D
<> E & D
<> F & D
<> J & D
<> M & D
<> N & E
<> F & E
<> J & E
<> M & E
<> N & F
<> J & F
<> M & F
<> N & J
<> M & J
<> N & M
<> N and
A4: (
EqClass (z,((((((C
'/\' D)
'/\' E)
'/\' F)
'/\' J)
'/\' M)
'/\' N)))
= (
EqClass (u,((((((C
'/\' D)
'/\' E)
'/\' F)
'/\' J)
'/\' M)
'/\' N)));
set h = (((((((((B
.--> (
EqClass (u,B)))
+* (C
.--> (
EqClass (u,C))))
+* (D
.--> (
EqClass (u,D))))
+* (E
.--> (
EqClass (u,E))))
+* (F
.--> (
EqClass (u,F))))
+* (J
.--> (
EqClass (u,J))))
+* (M
.--> (
EqClass (u,M))))
+* (N
.--> (
EqClass (u,N))))
+* (A
.--> (
EqClass (z,A))));
A5: (h
. A)
= (
EqClass (z,A)) by
A3,
Th76;
set L = (
EqClass (z,((((((C
'/\' D)
'/\' E)
'/\' F)
'/\' J)
'/\' M)
'/\' N)));
set GG = (
EqClass (u,(((((((B
'/\' C)
'/\' D)
'/\' E)
'/\' F)
'/\' J)
'/\' M)
'/\' N)));
reconsider I = (
EqClass (z,A)) as
set;
GG
= ((
EqClass (u,((((((B
'/\' C)
'/\' D)
'/\' E)
'/\' F)
'/\' J)
'/\' M)))
/\ (
EqClass (u,N))) by
Th1;
then GG
= (((
EqClass (u,(((((B
'/\' C)
'/\' D)
'/\' E)
'/\' F)
'/\' J)))
/\ (
EqClass (u,M)))
/\ (
EqClass (u,N))) by
Th1;
then GG
= ((((
EqClass (u,((((B
'/\' C)
'/\' D)
'/\' E)
'/\' F)))
/\ (
EqClass (u,J)))
/\ (
EqClass (u,M)))
/\ (
EqClass (u,N))) by
Th1;
then GG
= (((((
EqClass (u,(((B
'/\' C)
'/\' D)
'/\' E)))
/\ (
EqClass (u,F)))
/\ (
EqClass (u,J)))
/\ (
EqClass (u,M)))
/\ (
EqClass (u,N))) by
Th1;
then GG
= ((((((
EqClass (u,((B
'/\' C)
'/\' D)))
/\ (
EqClass (u,E)))
/\ (
EqClass (u,F)))
/\ (
EqClass (u,J)))
/\ (
EqClass (u,M)))
/\ (
EqClass (u,N))) by
Th1;
then GG
= (((((((
EqClass (u,(B
'/\' C)))
/\ (
EqClass (u,D)))
/\ (
EqClass (u,E)))
/\ (
EqClass (u,F)))
/\ (
EqClass (u,J)))
/\ (
EqClass (u,M)))
/\ (
EqClass (u,N))) by
Th1;
then
A6: (GG
/\ I)
= (((((((((
EqClass (u,B))
/\ (
EqClass (u,C)))
/\ (
EqClass (u,D)))
/\ (
EqClass (u,E)))
/\ (
EqClass (u,F)))
/\ (
EqClass (u,J)))
/\ (
EqClass (u,M)))
/\ (
EqClass (u,N)))
/\ (
EqClass (z,A))) by
Th1;
A7: (
CompF (A,G))
= (((((((B
'/\' C)
'/\' D)
'/\' E)
'/\' F)
'/\' J)
'/\' M)
'/\' N) by
A2,
A3,
Th67;
reconsider HH = (
EqClass (z,(
CompF (B,G)))) as
set;
A8: z
in HH by
EQREL_1:def 6;
A9: (A
'/\' ((((((C
'/\' D)
'/\' E)
'/\' F)
'/\' J)
'/\' M)
'/\' N))
= ((A
'/\' (((((C
'/\' D)
'/\' E)
'/\' F)
'/\' J)
'/\' M))
'/\' N) by
PARTIT1: 14
.= (((A
'/\' ((((C
'/\' D)
'/\' E)
'/\' F)
'/\' J))
'/\' M)
'/\' N) by
PARTIT1: 14
.= ((((A
'/\' (((C
'/\' D)
'/\' E)
'/\' F))
'/\' J)
'/\' M)
'/\' N) by
PARTIT1: 14
.= (((((A
'/\' ((C
'/\' D)
'/\' E))
'/\' F)
'/\' J)
'/\' M)
'/\' N) by
PARTIT1: 14
.= ((((((A
'/\' (C
'/\' D))
'/\' E)
'/\' F)
'/\' J)
'/\' M)
'/\' N) by
PARTIT1: 14
.= (((((((A
'/\' C)
'/\' D)
'/\' E)
'/\' F)
'/\' J)
'/\' M)
'/\' N) by
PARTIT1: 14;
A10: (h
. B)
= (
EqClass (u,B)) by
A3,
Th76;
A11: (h
. N)
= (
EqClass (u,N)) by
A3,
Th76;
A12: (h
. D)
= (
EqClass (u,D)) by
A3,
Th76;
A13: (h
. C)
= (
EqClass (u,C)) by
A3,
Th76;
A14: (h
. M)
= (
EqClass (u,M)) by
A3,
Th76;
A15: (h
. J)
= (
EqClass (u,J)) by
A3,
Th76;
A16: (h
. F)
= (
EqClass (u,F)) by
A3,
Th76;
A17: (h
. E)
= (
EqClass (u,E)) by
A3,
Th76;
A18: (
rng h)
=
{(h
. A), (h
. B), (h
. C), (h
. D), (h
. E), (h
. F), (h
. J), (h
. M), (h
. N)} by
Th78;
(
rng h)
c= (
bool Y)
proof
let t be
object;
assume t
in (
rng h);
then t
= (h
. A) or t
= (h
. B) or t
= (h
. C) or t
= (h
. D) or t
= (h
. E) or t
= (h
. F) or t
= (h
. J) or t
= (h
. M) or t
= (h
. N) by
A18,
ENUMSET1:def 7;
hence thesis by
A5,
A10,
A13,
A12,
A17,
A16,
A15,
A14,
A11;
end;
then
reconsider FF = (
rng h) as
Subset-Family of Y;
A19: (
dom h)
= G by
A2,
Th77;
then A
in (
dom h) by
A2,
ENUMSET1:def 7;
then
A20: (h
. A)
in (
rng h) by
FUNCT_1:def 3;
then
A21: (
Intersect FF)
= (
meet (
rng h)) by
SETFAM_1:def 9;
for d be
set st d
in G holds (h
. d)
in d
proof
let d be
set;
assume d
in G;
then d
= A or d
= B or d
= C or d
= D or d
= E or d
= F or d
= J or d
= M or d
= N by
A2,
ENUMSET1:def 7;
hence thesis by
A5,
A10,
A13,
A12,
A17,
A16,
A15,
A14,
A11;
end;
then (
Intersect FF)
<>
{} by
A1,
A19,
BVFUNC_2:def 5;
then
consider m be
object such that
A22: m
in (
Intersect FF) by
XBOOLE_0:def 1;
C
in (
dom h) by
A2,
A19,
ENUMSET1:def 7;
then (h
. C)
in (
rng h) by
FUNCT_1:def 3;
then
A23: m
in (
EqClass (u,C)) by
A13,
A21,
A22,
SETFAM_1:def 1;
B
in (
dom h) by
A2,
A19,
ENUMSET1:def 7;
then (h
. B)
in (
rng h) by
FUNCT_1:def 3;
then m
in (
EqClass (u,B)) by
A10,
A21,
A22,
SETFAM_1:def 1;
then
A24: m
in ((
EqClass (u,B))
/\ (
EqClass (u,C))) by
A23,
XBOOLE_0:def 4;
D
in (
dom h) by
A2,
A19,
ENUMSET1:def 7;
then (h
. D)
in (
rng h) by
FUNCT_1:def 3;
then m
in (
EqClass (u,D)) by
A12,
A21,
A22,
SETFAM_1:def 1;
then
A25: m
in (((
EqClass (u,B))
/\ (
EqClass (u,C)))
/\ (
EqClass (u,D))) by
A24,
XBOOLE_0:def 4;
E
in (
dom h) by
A2,
A19,
ENUMSET1:def 7;
then (h
. E)
in (
rng h) by
FUNCT_1:def 3;
then m
in (
EqClass (u,E)) by
A17,
A21,
A22,
SETFAM_1:def 1;
then
A26: m
in ((((
EqClass (u,B))
/\ (
EqClass (u,C)))
/\ (
EqClass (u,D)))
/\ (
EqClass (u,E))) by
A25,
XBOOLE_0:def 4;
F
in (
dom h) by
A2,
A19,
ENUMSET1:def 7;
then (h
. F)
in (
rng h) by
FUNCT_1:def 3;
then m
in (
EqClass (u,F)) by
A16,
A21,
A22,
SETFAM_1:def 1;
then
A27: m
in (((((
EqClass (u,B))
/\ (
EqClass (u,C)))
/\ (
EqClass (u,D)))
/\ (
EqClass (u,E)))
/\ (
EqClass (u,F))) by
A26,
XBOOLE_0:def 4;
J
in (
dom h) by
A2,
A19,
ENUMSET1:def 7;
then (h
. J)
in (
rng h) by
FUNCT_1:def 3;
then m
in (
EqClass (u,J)) by
A15,
A21,
A22,
SETFAM_1:def 1;
then
A28: m
in ((((((
EqClass (u,B))
/\ (
EqClass (u,C)))
/\ (
EqClass (u,D)))
/\ (
EqClass (u,E)))
/\ (
EqClass (u,F)))
/\ (
EqClass (u,J))) by
A27,
XBOOLE_0:def 4;
M
in (
dom h) by
A2,
A19,
ENUMSET1:def 7;
then (h
. M)
in (
rng h) by
FUNCT_1:def 3;
then m
in (
EqClass (u,M)) by
A14,
A21,
A22,
SETFAM_1:def 1;
then
A29: m
in (((((((
EqClass (u,B))
/\ (
EqClass (u,C)))
/\ (
EqClass (u,D)))
/\ (
EqClass (u,E)))
/\ (
EqClass (u,F)))
/\ (
EqClass (u,J)))
/\ (
EqClass (u,M))) by
A28,
XBOOLE_0:def 4;
N
in (
dom h) by
A2,
A19,
ENUMSET1:def 7;
then (h
. N)
in (
rng h) by
FUNCT_1:def 3;
then m
in (
EqClass (u,N)) by
A11,
A21,
A22,
SETFAM_1:def 1;
then
A30: m
in ((((((((
EqClass (u,B))
/\ (
EqClass (u,C)))
/\ (
EqClass (u,D)))
/\ (
EqClass (u,E)))
/\ (
EqClass (u,F)))
/\ (
EqClass (u,J)))
/\ (
EqClass (u,M)))
/\ (
EqClass (u,N))) by
A29,
XBOOLE_0:def 4;
m
in (
EqClass (z,A)) by
A5,
A20,
A21,
A22,
SETFAM_1:def 1;
then (GG
/\ I)
<>
{} by
A6,
A30,
XBOOLE_0:def 4;
then
consider p be
object such that
A31: p
in (GG
/\ I) by
XBOOLE_0:def 1;
reconsider p as
Element of Y by
A31;
set K = (
EqClass (p,((((((C
'/\' D)
'/\' E)
'/\' F)
'/\' J)
'/\' M)
'/\' N)));
A32: p
in GG by
A31,
XBOOLE_0:def 4;
A33: p
in (
EqClass (p,((((((C
'/\' D)
'/\' E)
'/\' F)
'/\' J)
'/\' M)
'/\' N))) by
EQREL_1:def 6;
GG
= (
EqClass (u,((((((B
'/\' (C
'/\' D))
'/\' E)
'/\' F)
'/\' J)
'/\' M)
'/\' N))) by
PARTIT1: 14;
then GG
= (
EqClass (u,(((((B
'/\' ((C
'/\' D)
'/\' E))
'/\' F)
'/\' J)
'/\' M)
'/\' N))) by
PARTIT1: 14;
then GG
= (
EqClass (u,((((B
'/\' (((C
'/\' D)
'/\' E)
'/\' F))
'/\' J)
'/\' M)
'/\' N))) by
PARTIT1: 14;
then GG
= (
EqClass (u,(((B
'/\' ((((C
'/\' D)
'/\' E)
'/\' F)
'/\' J))
'/\' M)
'/\' N))) by
PARTIT1: 14;
then GG
= (
EqClass (u,((B
'/\' (((((C
'/\' D)
'/\' E)
'/\' F)
'/\' J)
'/\' M))
'/\' N))) by
PARTIT1: 14;
then GG
= (
EqClass (u,(B
'/\' ((((((C
'/\' D)
'/\' E)
'/\' F)
'/\' J)
'/\' M)
'/\' N)))) by
PARTIT1: 14;
then GG
c= L by
A4,
BVFUNC11: 3;
then K
meets L by
A32,
A33,
XBOOLE_0: 3;
then K
= L by
EQREL_1: 41;
then
A34: z
in K by
EQREL_1:def 6;
p
in K & p
in I by
A31,
EQREL_1:def 6,
XBOOLE_0:def 4;
then
A35: p
in (I
/\ K) by
XBOOLE_0:def 4;
then (I
/\ K)
in (
INTERSECTION (A,((((((C
'/\' D)
'/\' E)
'/\' F)
'/\' J)
'/\' M)
'/\' N))) & not (I
/\ K)
in
{
{} } by
SETFAM_1:def 5,
TARSKI:def 1;
then
A36: (I
/\ K)
in ((
INTERSECTION (A,((((((C
'/\' D)
'/\' E)
'/\' F)
'/\' J)
'/\' M)
'/\' N)))
\
{
{} }) by
XBOOLE_0:def 5;
z
in I by
EQREL_1:def 6;
then z
in (I
/\ K) by
A34,
XBOOLE_0:def 4;
then
A37: (I
/\ K)
meets HH by
A8,
XBOOLE_0: 3;
(
CompF (B,G))
= (((((((A
'/\' C)
'/\' D)
'/\' E)
'/\' F)
'/\' J)
'/\' M)
'/\' N) by
A2,
A3,
Th68;
then (I
/\ K)
in (
CompF (B,G)) by
A36,
A9,
PARTIT1:def 4;
then p
in HH by
A35,
A37,
EQREL_1:def 4;
hence thesis by
A7,
A32,
XBOOLE_0: 3;
end;