neckla_3.miz
begin
reserve A,B,a,b,c,d,e,f,g,h for
set;
theorem ::
NECKLA_3:1
Th1: ((
id A)
| B)
= ((
id A)
/\
[:B, B:])
proof
thus ((
id A)
| B)
c= ((
id A)
/\
[:B, B:])
proof
let a be
object;
assume
A1: a
in ((
id A)
| B);
((
id A)
| B) is
Relation of B, A by
RELSET_1: 18;
then
consider x,y be
object such that
A2: a
=
[x, y] and
A3: x
in B and y
in A by
A1,
RELSET_1: 2;
A4:
[x, y]
in (
id A) by
A1,
A2,
RELAT_1:def 11;
then x
= y by
RELAT_1:def 10;
then
[x, y]
in
[:B, B:] by
A3,
ZFMISC_1: 87;
hence thesis by
A2,
A4,
XBOOLE_0:def 4;
end;
let a be
object;
assume
A5: a
in ((
id A)
/\
[:B, B:]);
then a
in
[:B, B:] by
XBOOLE_0:def 4;
then
A6: ex x1,y1 be
object st x1
in B & y1
in B & a
=
[x1, y1] by
ZFMISC_1:def 2;
a
in (
id A) by
A5,
XBOOLE_0:def 4;
hence thesis by
A6,
RELAT_1:def 11;
end;
theorem ::
NECKLA_3:2
(
id
{a, b, c, d})
=
{
[a, a],
[b, b],
[c, c],
[d, d]}
proof
set X =
{a, b, c, d};
thus (
id X)
c=
{
[a, a],
[b, b],
[c, c],
[d, d]}
proof
let x be
object;
assume
A1: x
in (
id X);
then
consider x1,y1 be
object such that
A2: x
=
[x1, y1] and
A3: x1
in X and y1
in X by
RELSET_1: 2;
A4: x1
= y1 by
A1,
A2,
RELAT_1:def 10;
per cases by
A3,
ENUMSET1:def 2;
suppose x1
= a;
hence thesis by
A2,
A4,
ENUMSET1:def 2;
end;
suppose x1
= b;
hence thesis by
A2,
A4,
ENUMSET1:def 2;
end;
suppose x1
= c;
hence thesis by
A2,
A4,
ENUMSET1:def 2;
end;
suppose x1
= d;
hence thesis by
A2,
A4,
ENUMSET1:def 2;
end;
end;
let x be
object;
assume
A5: x
in
{
[a, a],
[b, b],
[c, c],
[d, d]};
per cases by
A5,
ENUMSET1:def 2;
suppose
A6: x
=
[a, a];
a
in X by
ENUMSET1:def 2;
hence thesis by
A6,
RELAT_1:def 10;
end;
suppose
A7: x
=
[b, b];
b
in X by
ENUMSET1:def 2;
hence thesis by
A7,
RELAT_1:def 10;
end;
suppose
A8: x
=
[c, c];
c
in X by
ENUMSET1:def 2;
hence thesis by
A8,
RELAT_1:def 10;
end;
suppose
A9: x
=
[d, d];
d
in X by
ENUMSET1:def 2;
hence thesis by
A9,
RELAT_1:def 10;
end;
end;
theorem ::
NECKLA_3:3
Th3:
[:
{a, b, c, d},
{e, f, g, h}:]
= (
{
[a, e],
[a, f],
[b, e],
[b, f],
[a, g],
[a, h],
[b, g],
[b, h]}
\/
{
[c, e],
[c, f],
[d, e],
[d, f],
[c, g],
[c, h],
[d, g],
[d, h]})
proof
set X1 =
{a, b, c, d}, Y1 =
{e, f, g, h}, X11 =
{a, b}, X12 =
{c, d}, Y11 =
{e, f}, Y12 =
{g, h};
A1:
[:X12, Y11:]
=
{
[c, e],
[c, f],
[d, e],
[d, f]} &
[:X12, Y12:]
=
{
[c, g],
[c, h],
[d, g],
[d, h]} by
MCART_1: 23;
X1
= (X11
\/ X12) & Y1
= (Y11
\/ Y12) by
ENUMSET1: 5;
then
A2:
[:X1, Y1:]
= (((
[:X11, Y11:]
\/
[:X11, Y12:])
\/
[:X12, Y11:])
\/
[:X12, Y12:]) by
ZFMISC_1: 98;
[:X11, Y11:]
=
{
[a, e],
[a, f],
[b, e],
[b, f]} &
[:X11, Y12:]
=
{
[a, g],
[a, h],
[b, g],
[b, h]} by
MCART_1: 23;
then
[:X1, Y1:]
= ((
{
[a, e],
[a, f],
[b, e],
[b, f],
[a, g],
[a, h],
[b, g],
[b, h]}
\/
{
[c, e],
[c, f],
[d, e],
[d, f]})
\/
{
[c, g],
[c, h],
[d, g],
[d, h]}) by
A1,
A2,
ENUMSET1: 25
.= (
{
[a, e],
[a, f],
[b, e],
[b, f],
[a, g],
[a, h],
[b, g],
[b, h]}
\/ (
{
[c, e],
[c, f],
[d, e],
[d, f]}
\/
{
[c, g],
[c, h],
[d, g],
[d, h]})) by
XBOOLE_1: 4
.= (
{
[a, e],
[a, f],
[b, e],
[b, f],
[a, g],
[a, h],
[b, g],
[b, h]}
\/
{
[c, e],
[c, f],
[d, e],
[d, f],
[c, g],
[c, h],
[d, g],
[d, h]}) by
ENUMSET1: 25;
hence thesis;
end;
registration
let X,Y be
trivial
set;
cluster ->
trivial for
Relation of X, Y;
correctness ;
end
theorem ::
NECKLA_3:4
Th4: for X be
trivial
set, R be
Relation of X st R is non
empty holds ex x be
object st R
=
{
[x, x]}
proof
let X be
trivial
set;
let R be
Relation of X;
assume R is non
empty;
then
consider x be
object such that
A1: x
in R;
consider y,z be
object such that
A2: x
=
[y, z] and
A3: y
in X and
A4: z
in X by
A1,
RELSET_1: 2;
consider a be
object such that
A5: X
=
{a} by
A3,
ZFMISC_1: 131;
A6: y
= a & z
= a by
A3,
A4,
A5,
TARSKI:def 1;
R
=
{
[a, a]}
proof
thus R
c=
{
[a, a]}
proof
let r be
object;
assume r
in R;
then
consider y,z be
object such that
A7: r
=
[y, z] and
A8: y
in X & z
in X by
RELSET_1: 2;
y
= a & z
= a by
A5,
A8,
TARSKI:def 1;
hence thesis by
A7,
TARSKI:def 1;
end;
let z be
object;
assume z
in
{
[a, a]};
hence thesis by
A1,
A2,
A6,
TARSKI:def 1;
end;
hence thesis;
end;
registration
let X be
trivial
set;
cluster ->
trivial
reflexive
symmetric
transitive
strongly_connected for
Relation of X;
correctness
proof
let R be
Relation of X, X;
A1: R
is_reflexive_in (
field R)
proof
per cases ;
suppose
A2: R is
empty;
let x be
object;
assume x
in (
field R);
hence thesis by
A2,
RELAT_1: 40;
end;
suppose R is non
empty;
then
consider z be
object such that
A3: R
=
{
[z, z]} by
Th4;
let x be
object;
assume x
in (
field R);
then
A4: x
in ((
dom R)
\/ (
rng R)) by
RELAT_1:def 6;
(
dom R)
=
{z} & (
rng R)
=
{z} by
A3,
RELAT_1: 9;
then x
= z by
A4,
TARSKI:def 1;
hence thesis by
A3,
TARSKI:def 1;
end;
end;
A5: R
is_transitive_in (
field R)
proof
per cases ;
suppose
A6: R is
empty;
let x,y,z be
object;
assume that x
in (
field R) and y
in (
field R) and z
in (
field R) and
A7:
[x, y]
in R and
[y, z]
in R;
thus thesis by
A6,
A7;
end;
suppose
A8: R is non
empty;
let x,y,z be
object;
assume that x
in (
field R) and y
in (
field R) and z
in (
field R) and
A9:
[x, y]
in R and
A10:
[y, z]
in R;
consider a be
object such that
A11: R
=
{
[a, a]} by
A8,
Th4;
[y, z]
=
[a, a] by
A11,
A10,
TARSKI:def 1;
then
A12: z
= a by
XTUPLE_0: 1;
[x, y]
=
[a, a] by
A11,
A9,
TARSKI:def 1;
then x
= a by
XTUPLE_0: 1;
hence thesis by
A11,
A12,
TARSKI:def 1;
end;
end;
A13: R
is_strongly_connected_in (
field R)
proof
per cases ;
suppose
A14: R is
empty;
let x,y be
object;
assume that
A15: x
in (
field R) and y
in (
field R);
thus thesis by
A14,
A15,
RELAT_1: 40;
end;
suppose
A16: R is non
empty;
let x,y be
object;
assume that
A17: x
in (
field R) and
A18: y
in (
field R);
consider a be
object such that
A19: R
=
{
[a, a]} by
A16,
Th4;
A20: (
dom R)
=
{a} & (
rng R)
=
{a} by
A19,
RELAT_1: 9;
y
in ((
dom R)
\/ (
rng R)) by
A18,
RELAT_1:def 6;
then
A21: y
= a by
A20,
TARSKI:def 1;
x
in ((
dom R)
\/ (
rng R)) by
A17,
RELAT_1:def 6;
then x
= a by
A20,
TARSKI:def 1;
hence thesis by
A19,
A21,
TARSKI:def 1;
end;
end;
R
is_symmetric_in (
field R)
proof
per cases ;
suppose
A22: R is
empty;
let x,y be
object;
assume that x
in (
field R) and y
in (
field R) and
A23:
[x, y]
in R;
thus thesis by
A22,
A23;
end;
suppose
A24: R is non
empty;
let x,y be
object;
assume that x
in (
field R) and y
in (
field R) and
A25:
[x, y]
in R;
consider a be
object such that
A26: R
=
{
[a, a]} by
A24,
Th4;
[x, y]
=
[a, a] by
A26,
A25,
TARSKI:def 1;
then x
= a & y
= a by
XTUPLE_0: 1;
hence thesis by
A26,
TARSKI:def 1;
end;
end;
hence thesis by
A1,
A5,
A13;
end;
end
theorem ::
NECKLA_3:5
Th5: for X be 1
-element
set, R be
Relation of X holds R
is_symmetric_in X
proof
let X be 1
-element
set;
let R be
Relation of X;
consider x be
object such that
A1: X
=
{x} by
ZFMISC_1: 131;
let a,b be
object;
assume that
A2: a
in X and
A3: b
in X &
[a, b]
in R;
a
= x by
A1,
A2,
TARSKI:def 1;
hence thesis by
A1,
A3,
TARSKI:def 1;
end;
registration
cluster non
empty
strict
finite
irreflexive
symmetric for
RelStr;
correctness
proof
set X =
{
0 , 1}, r =
{
[
0 , 1],
[1,
0 ]};
0
in X & 1
in X by
TARSKI:def 2;
then
A1:
[
0 , 1]
in
[:X, X:] &
[1,
0 ]
in
[:X, X:] by
ZFMISC_1:def 2;
r
c=
[:X, X:] by
A1,
TARSKI:def 2;
then
reconsider r as
Relation of X, X;
take R =
RelStr (# X, r #);
A2: for x be
set st x
in the
carrier of R holds not
[x, x]
in the
InternalRel of R
proof
let x be
set;
A3: not
[
0 ,
0 ]
in r
proof
assume
[
0 ,
0 ]
in r;
then
[
0 ,
0 ]
=
[
0 , 1] or
[
0 ,
0 ]
=
[1,
0 ] by
TARSKI:def 2;
hence contradiction by
XTUPLE_0: 1;
end;
A4: not
[1, 1]
in r
proof
assume
[1, 1]
in r;
then
[1, 1]
=
[
0 , 1] or
[1, 1]
=
[1,
0 ] by
TARSKI:def 2;
hence contradiction by
XTUPLE_0: 1;
end;
assume x
in the
carrier of R;
then x
=
0 or x
= 1 by
TARSKI:def 2;
hence thesis by
A3,
A4;
end;
for x,y be
object st x
in X & y
in X &
[x, y]
in r holds
[y, x]
in r
proof
let x,y be
object;
assume that x
in X and y
in X and
A5:
[x, y]
in r;
per cases by
A5,
TARSKI:def 2;
suppose
[x, y]
=
[
0 , 1];
then x
=
0 & y
= 1 by
XTUPLE_0: 1;
hence thesis by
TARSKI:def 2;
end;
suppose
[x, y]
=
[1,
0 ];
then x
= 1 & y
=
0 by
XTUPLE_0: 1;
hence thesis by
TARSKI:def 2;
end;
end;
then r
is_symmetric_in X;
hence thesis by
A2;
end;
end
registration
let L be
irreflexive
RelStr;
cluster ->
irreflexive for
full
SubRelStr of L;
correctness
proof
let S be
full
SubRelStr of L;
let x be
set;
assume
A1: x
in the
carrier of S;
the
carrier of S
c= the
carrier of L by
YELLOW_0:def 13;
then the
InternalRel of S
= (the
InternalRel of L
|_2 the
carrier of S) & not
[x, x]
in the
InternalRel of L by
A1,
NECKLACE:def 5,
YELLOW_0:def 14;
hence thesis by
XBOOLE_0:def 4;
end;
end
registration
let L be
symmetric
RelStr;
cluster ->
symmetric for
full
SubRelStr of L;
correctness
proof
let S be
full
SubRelStr of L;
let x,y be
object;
assume that
A1: x
in the
carrier of S & y
in the
carrier of S and
A2:
[x, y]
in the
InternalRel of S;
A3:
[y, x]
in
[:the
carrier of S, the
carrier of S:] by
A1,
ZFMISC_1: 87;
A4: the
carrier of S
c= the
carrier of L & the
InternalRel of L
is_symmetric_in the
carrier of L by
NECKLACE:def 3,
YELLOW_0:def 13;
A5: the
InternalRel of S
= (the
InternalRel of L
|_2 the
carrier of S) by
YELLOW_0:def 14;
then
[x, y]
in the
InternalRel of L by
A2,
XBOOLE_0:def 4;
then
[y, x]
in the
InternalRel of L by
A1,
A4;
hence thesis by
A5,
A3,
XBOOLE_0:def 4;
end;
end
theorem ::
NECKLA_3:6
Th6: for R be
irreflexive
symmetric
RelStr st (
card the
carrier of R)
= 2 holds ex a,b be
object st the
carrier of R
=
{a, b} & (the
InternalRel of R
=
{
[a, b],
[b, a]} or the
InternalRel of R
=
{} )
proof
let R be
irreflexive
symmetric
RelStr;
set Q = the
InternalRel of R;
assume
A1: (
card the
carrier of R)
= 2;
then
reconsider X = the
carrier of R as
finite
set;
consider a,b be
object such that
A2: a
<> b and
A3: X
=
{a, b} by
A1,
CARD_2: 60;
A4: the
InternalRel of R
c=
{
[a, b],
[b, a]}
proof
let x be
object;
assume
A5: x
in the
InternalRel of R;
then
consider x1,x2 be
object such that
A6: x
=
[x1, x2] and
A7: x1
in X and
A8: x2
in X by
RELSET_1: 2;
A9: x1
= a or x1
= b by
A3,
A7,
TARSKI:def 2;
per cases by
A3,
A6,
A8,
A9,
TARSKI:def 2;
suppose
A10: x
=
[a, a];
a
in the
carrier of R by
A3,
TARSKI:def 2;
hence thesis by
A5,
A10,
NECKLACE:def 5;
end;
suppose x
=
[a, b];
hence thesis by
TARSKI:def 2;
end;
suppose x
=
[b, a];
hence thesis by
TARSKI:def 2;
end;
suppose
A11: x
=
[b, b];
b
in the
carrier of R by
A3,
TARSKI:def 2;
hence thesis by
A5,
A11,
NECKLACE:def 5;
end;
end;
per cases by
A4,
ZFMISC_1: 36;
suppose Q
=
{} ;
hence thesis by
A3;
end;
suppose
A12: Q
=
{
[a, b]};
A13: a
in X & b
in X by
A3,
TARSKI:def 2;
A14: Q
is_symmetric_in X by
NECKLACE:def 3;
[a, b]
in Q by
A12,
TARSKI:def 1;
then
[b, a]
in Q by
A13,
A14;
then
[b, a]
=
[a, b] by
A12,
TARSKI:def 1;
hence thesis by
A2,
XTUPLE_0: 1;
end;
suppose
A15: Q
=
{
[b, a]};
A16: a
in X & b
in X by
A3,
TARSKI:def 2;
A17: Q
is_symmetric_in X by
NECKLACE:def 3;
[b, a]
in Q by
A15,
TARSKI:def 1;
then
[a, b]
in Q by
A16,
A17;
then
[b, a]
=
[a, b] by
A15,
TARSKI:def 1;
hence thesis by
A2,
XTUPLE_0: 1;
end;
suppose Q
=
{
[a, b],
[b, a]};
hence thesis by
A3;
end;
end;
begin
registration
let R be non
empty
RelStr, S be
RelStr;
cluster (
union_of (R,S)) -> non
empty;
correctness
proof
(the
carrier of R
\/ the
carrier of S) is non
empty;
hence thesis by
NECKLA_2:def 2;
end;
cluster (
sum_of (R,S)) -> non
empty;
correctness
proof
(the
carrier of R
\/ the
carrier of S) is non
empty;
hence thesis by
NECKLA_2:def 3;
end;
end
registration
let R be
RelStr, S be non
empty
RelStr;
cluster (
union_of (R,S)) -> non
empty;
correctness
proof
(the
carrier of R
\/ the
carrier of S) is non
empty;
hence thesis by
NECKLA_2:def 2;
end;
cluster (
sum_of (R,S)) -> non
empty;
correctness
proof
(the
carrier of R
\/ the
carrier of S) is non
empty;
hence thesis by
NECKLA_2:def 3;
end;
end
registration
let R,S be
finite
RelStr;
cluster (
union_of (R,S)) ->
finite;
correctness
proof
(the
carrier of R
\/ the
carrier of S) is
finite;
hence thesis by
NECKLA_2:def 2;
end;
cluster (
sum_of (R,S)) ->
finite;
correctness
proof
(the
carrier of R
\/ the
carrier of S) is
finite;
hence thesis by
NECKLA_2:def 3;
end;
end
registration
let R,S be
symmetric
RelStr;
cluster (
union_of (R,S)) ->
symmetric;
correctness
proof
let x,y be
object;
set U = (
union_of (R,S)), cU = the
carrier of U, IU = the
InternalRel of U, cR = the
carrier of R, cS = the
carrier of S;
assume that x
in cU and y
in cU and
A1:
[x, y]
in IU;
A2:
[x, y]
in (the
InternalRel of R
\/ the
InternalRel of S) by
A1,
NECKLA_2:def 2;
per cases by
A2,
XBOOLE_0:def 3;
suppose
A3:
[x, y]
in the
InternalRel of R;
A4: the
InternalRel of R
is_symmetric_in cR by
NECKLACE:def 3;
x
in cR & y
in cR by
A3,
ZFMISC_1: 87;
then
[y, x]
in the
InternalRel of R by
A3,
A4;
then
[y, x]
in (the
InternalRel of R
\/ the
InternalRel of S) by
XBOOLE_0:def 3;
hence thesis by
NECKLA_2:def 2;
end;
suppose
A5:
[x, y]
in the
InternalRel of S;
A6: the
InternalRel of S
is_symmetric_in cS by
NECKLACE:def 3;
x
in cS & y
in cS by
A5,
ZFMISC_1: 87;
then
[y, x]
in the
InternalRel of S by
A5,
A6;
then
[y, x]
in (the
InternalRel of R
\/ the
InternalRel of S) by
XBOOLE_0:def 3;
hence thesis by
NECKLA_2:def 2;
end;
end;
cluster (
sum_of (R,S)) ->
symmetric;
correctness
proof
set SU = (
sum_of (R,S)), cSU = the
carrier of (
sum_of (R,S)), ISU = the
InternalRel of SU, cR = the
carrier of R, IR = the
InternalRel of R, cS = the
carrier of S, IS = the
InternalRel of S;
A7: IS
is_symmetric_in cS by
NECKLACE:def 3;
A8: IR
is_symmetric_in cR by
NECKLACE:def 3;
the
InternalRel of (
sum_of (R,S))
is_symmetric_in cSU
proof
let x,y be
object;
assume that x
in cSU and y
in cSU and
A9:
[x, y]
in ISU;
[x, y]
in (((IR
\/ IS)
\/
[:cR, cS:])
\/
[:cS, cR:]) by
A9,
NECKLA_2:def 3;
then
[x, y]
in ((IR
\/ IS)
\/
[:cR, cS:]) or
[x, y]
in
[:cS, cR:] by
XBOOLE_0:def 3;
then
A10:
[x, y]
in (IR
\/ IS) or
[x, y]
in
[:cR, cS:] or
[x, y]
in
[:cS, cR:] by
XBOOLE_0:def 3;
per cases by
A10,
XBOOLE_0:def 3;
suppose
A11:
[x, y]
in IR;
then x
in cR & y
in cR by
ZFMISC_1: 87;
then
[y, x]
in IR by
A8,
A11;
then
[y, x]
in (IR
\/ IS) by
XBOOLE_0:def 3;
then
[y, x]
in ((IR
\/ IS)
\/
[:cR, cS:]) by
XBOOLE_0:def 3;
then
[y, x]
in (((IR
\/ IS)
\/
[:cR, cS:])
\/
[:cS, cR:]) by
XBOOLE_0:def 3;
hence thesis by
NECKLA_2:def 3;
end;
suppose
A12:
[x, y]
in IS;
then x
in cS & y
in cS by
ZFMISC_1: 87;
then
[y, x]
in IS by
A7,
A12;
then
[y, x]
in (IR
\/ IS) by
XBOOLE_0:def 3;
then
[y, x]
in ((IR
\/ IS)
\/
[:cR, cS:]) by
XBOOLE_0:def 3;
then
[y, x]
in (((IR
\/ IS)
\/
[:cR, cS:])
\/
[:cS, cR:]) by
XBOOLE_0:def 3;
hence thesis by
NECKLA_2:def 3;
end;
suppose
[x, y]
in
[:cR, cS:];
then x
in cR & y
in cS by
ZFMISC_1: 87;
then
[y, x]
in
[:cS, cR:] by
ZFMISC_1: 87;
then
[y, x]
in (
[:cR, cS:]
\/
[:cS, cR:]) by
XBOOLE_0:def 3;
then
[y, x]
in (IS
\/ (
[:cR, cS:]
\/
[:cS, cR:])) by
XBOOLE_0:def 3;
then
[y, x]
in ((IS
\/
[:cR, cS:])
\/
[:cS, cR:]) by
XBOOLE_1: 4;
then
[y, x]
in (IR
\/ ((IS
\/
[:cR, cS:])
\/
[:cS, cR:])) by
XBOOLE_0:def 3;
then
[y, x]
in (IR
\/ (IS
\/ (
[:cR, cS:]
\/
[:cS, cR:]))) by
XBOOLE_1: 4;
then
[y, x]
in ((IR
\/ IS)
\/ (
[:cR, cS:]
\/
[:cS, cR:])) by
XBOOLE_1: 4;
then
[y, x]
in (((IR
\/ IS)
\/
[:cR, cS:])
\/
[:cS, cR:]) by
XBOOLE_1: 4;
hence thesis by
NECKLA_2:def 3;
end;
suppose
[x, y]
in
[:cS, cR:];
then x
in cS & y
in cR by
ZFMISC_1: 87;
then
[y, x]
in
[:cR, cS:] by
ZFMISC_1: 87;
then
[y, x]
in (
[:cR, cS:]
\/
[:cS, cR:]) by
XBOOLE_0:def 3;
then
[y, x]
in (IS
\/ (
[:cR, cS:]
\/
[:cS, cR:])) by
XBOOLE_0:def 3;
then
[y, x]
in ((IS
\/
[:cR, cS:])
\/
[:cS, cR:]) by
XBOOLE_1: 4;
then
[y, x]
in (IR
\/ ((IS
\/
[:cR, cS:])
\/
[:cS, cR:])) by
XBOOLE_0:def 3;
then
[y, x]
in (IR
\/ (IS
\/ (
[:cR, cS:]
\/
[:cS, cR:]))) by
XBOOLE_1: 4;
then
[y, x]
in ((IR
\/ IS)
\/ (
[:cR, cS:]
\/
[:cS, cR:])) by
XBOOLE_1: 4;
then
[y, x]
in (((IR
\/ IS)
\/
[:cR, cS:])
\/
[:cS, cR:]) by
XBOOLE_1: 4;
hence thesis by
NECKLA_2:def 3;
end;
end;
hence thesis;
end;
end
registration
let R,S be
irreflexive
RelStr;
cluster (
union_of (R,S)) ->
irreflexive;
correctness
proof
set U = (
union_of (R,S)), cU = the
carrier of U, IU = the
InternalRel of U, cR = the
carrier of R, cS = the
carrier of S;
for x be
set st x
in cU holds not
[x, x]
in IU
proof
let x be
set such that x
in cU;
assume not thesis;
then
A1:
[x, x]
in (the
InternalRel of R
\/ the
InternalRel of S) by
NECKLA_2:def 2;
per cases by
A1,
XBOOLE_0:def 3;
suppose
A2:
[x, x]
in the
InternalRel of R;
then x
in cR by
ZFMISC_1: 87;
hence thesis by
A2,
NECKLACE:def 5;
end;
suppose
A3:
[x, x]
in the
InternalRel of S;
then x
in cS by
ZFMISC_1: 87;
hence thesis by
A3,
NECKLACE:def 5;
end;
end;
hence thesis;
end;
end
theorem ::
NECKLA_3:7
for R,S be
irreflexive
RelStr st the
carrier of R
misses the
carrier of S holds (
sum_of (R,S)) is
irreflexive
proof
let R,S be
irreflexive
RelStr such that
A1: the
carrier of R
misses the
carrier of S;
for x be
set st x
in the
carrier of (
sum_of (R,S)) holds not
[x, x]
in the
InternalRel of (
sum_of (R,S))
proof
set IR = the
InternalRel of R, IS = the
InternalRel of S, RS =
[:the
carrier of R, the
carrier of S:], SR =
[:the
carrier of S, the
carrier of R:];
let x be
set;
assume x
in the
carrier of (
sum_of (R,S));
assume not thesis;
then
[x, x]
in (((IR
\/ IS)
\/ RS)
\/ SR) by
NECKLA_2:def 3;
then
[x, x]
in ((IR
\/ IS)
\/ RS) or
[x, x]
in SR by
XBOOLE_0:def 3;
then
A2:
[x, x]
in (IR
\/ IS) or
[x, x]
in RS or
[x, x]
in SR by
XBOOLE_0:def 3;
per cases by
A2,
XBOOLE_0:def 3;
suppose
A3:
[x, x]
in IR;
then x
in the
carrier of R by
ZFMISC_1: 87;
hence thesis by
A3,
NECKLACE:def 5;
end;
suppose
A4:
[x, x]
in IS;
then x
in the
carrier of S by
ZFMISC_1: 87;
hence thesis by
A4,
NECKLACE:def 5;
end;
suppose
[x, x]
in RS;
then x
in the
carrier of R & x
in the
carrier of S by
ZFMISC_1: 87;
hence thesis by
A1,
XBOOLE_0: 3;
end;
suppose
[x, x]
in SR;
then x
in the
carrier of S & x
in the
carrier of R by
ZFMISC_1: 87;
hence thesis by
A1,
XBOOLE_0: 3;
end;
end;
hence thesis;
end;
theorem ::
NECKLA_3:8
Th8: for R1,R2 be
RelStr holds (
union_of (R1,R2))
= (
union_of (R2,R1)) & (
sum_of (R1,R2))
= (
sum_of (R2,R1))
proof
let R1,R2 be
RelStr;
set U1 = (
union_of (R1,R2)), S1 = (
sum_of (R1,R2));
A1: the
carrier of S1
= (the
carrier of R2
\/ the
carrier of R1) by
NECKLA_2:def 3;
A2: the
InternalRel of S1
= (((the
InternalRel of R1
\/ the
InternalRel of R2)
\/
[:the
carrier of R1, the
carrier of R2:])
\/
[:the
carrier of R2, the
carrier of R1:]) by
NECKLA_2:def 3
.= (((the
InternalRel of R2
\/ the
InternalRel of R1)
\/
[:the
carrier of R2, the
carrier of R1:])
\/
[:the
carrier of R1, the
carrier of R2:]) by
XBOOLE_1: 4;
the
carrier of U1
= (the
carrier of R2
\/ the
carrier of R1) & the
InternalRel of U1
= (the
InternalRel of R2
\/ the
InternalRel of R1) by
NECKLA_2:def 2;
hence thesis by
A1,
A2,
NECKLA_2:def 2,
NECKLA_2:def 3;
end;
theorem ::
NECKLA_3:9
Th9: for G be
irreflexive
RelStr, G1,G2 be
RelStr st (G
= (
union_of (G1,G2)) or G
= (
sum_of (G1,G2))) holds G1 is
irreflexive & G2 is
irreflexive
proof
let G be
irreflexive
RelStr, G1,G2 be
RelStr;
assume
A1: G
= (
union_of (G1,G2)) or G
= (
sum_of (G1,G2));
per cases by
A1;
suppose
A2: G
= (
union_of (G1,G2));
assume
A3: not thesis;
thus thesis
proof
per cases by
A3;
suppose not G1 is
irreflexive;
then
consider x be
set such that
A4: x
in the
carrier of G1 and
A5:
[x, x]
in the
InternalRel of G1;
[x, x]
in (the
InternalRel of G1
\/ the
InternalRel of G2) by
A5,
XBOOLE_0:def 3;
then
A6:
[x, x]
in the
InternalRel of G by
A2,
NECKLA_2:def 2;
x
in (the
carrier of G1
\/ the
carrier of G2) by
A4,
XBOOLE_0:def 3;
then x
in the
carrier of G by
A2,
NECKLA_2:def 2;
hence thesis by
A6,
NECKLACE:def 5;
end;
suppose not G2 is
irreflexive;
then
consider x be
set such that
A7: x
in the
carrier of G2 and
A8:
[x, x]
in the
InternalRel of G2;
[x, x]
in (the
InternalRel of G1
\/ the
InternalRel of G2) by
A8,
XBOOLE_0:def 3;
then
A9:
[x, x]
in the
InternalRel of G by
A2,
NECKLA_2:def 2;
x
in (the
carrier of G1
\/ the
carrier of G2) by
A7,
XBOOLE_0:def 3;
then x
in the
carrier of G by
A2,
NECKLA_2:def 2;
hence thesis by
A9,
NECKLACE:def 5;
end;
end;
end;
suppose
A10: G
= (
sum_of (G1,G2));
assume
A11: not thesis;
thus thesis
proof
per cases by
A11;
suppose not G1 is
irreflexive;
then
consider x be
set such that
A12: x
in the
carrier of G1 and
A13:
[x, x]
in the
InternalRel of G1;
[x, x]
in (the
InternalRel of G1
\/ the
InternalRel of G2) by
A13,
XBOOLE_0:def 3;
then
[x, x]
in ((the
InternalRel of G1
\/ the
InternalRel of G2)
\/
[:the
carrier of G1, the
carrier of G2:]) by
XBOOLE_0:def 3;
then
[x, x]
in (((the
InternalRel of G1
\/ the
InternalRel of G2)
\/
[:the
carrier of G1, the
carrier of G2:])
\/
[:the
carrier of G2, the
carrier of G1:]) by
XBOOLE_0:def 3;
then
A14:
[x, x]
in the
InternalRel of G by
A10,
NECKLA_2:def 3;
x
in (the
carrier of G1
\/ the
carrier of G2) by
A12,
XBOOLE_0:def 3;
then x
in the
carrier of G by
A10,
NECKLA_2:def 3;
hence thesis by
A14,
NECKLACE:def 5;
end;
suppose not G2 is
irreflexive;
then
consider x be
set such that
A15: x
in the
carrier of G2 and
A16:
[x, x]
in the
InternalRel of G2;
[x, x]
in (the
InternalRel of G1
\/ the
InternalRel of G2) by
A16,
XBOOLE_0:def 3;
then
[x, x]
in ((the
InternalRel of G1
\/ the
InternalRel of G2)
\/
[:the
carrier of G1, the
carrier of G2:]) by
XBOOLE_0:def 3;
then
[x, x]
in (((the
InternalRel of G1
\/ the
InternalRel of G2)
\/
[:the
carrier of G1, the
carrier of G2:])
\/
[:the
carrier of G2, the
carrier of G1:]) by
XBOOLE_0:def 3;
then
A17:
[x, x]
in the
InternalRel of G by
A10,
NECKLA_2:def 3;
x
in (the
carrier of G1
\/ the
carrier of G2) by
A15,
XBOOLE_0:def 3;
then x
in the
carrier of G by
A10,
NECKLA_2:def 3;
hence thesis by
A17,
NECKLACE:def 5;
end;
end;
end;
end;
theorem ::
NECKLA_3:10
Th10: for G be non
empty
RelStr, H1,H2 be
RelStr st the
carrier of H1
misses the
carrier of H2 & ( the RelStr of G
= (
union_of (H1,H2)) or the RelStr of G
= (
sum_of (H1,H2))) holds H1 is
full
SubRelStr of G & H2 is
full
SubRelStr of G
proof
let G be non
empty
RelStr;
let H1,H2 be
RelStr;
assume that
A1: the
carrier of H1
misses the
carrier of H2 and
A2: the RelStr of G
= (
union_of (H1,H2)) or the RelStr of G
= (
sum_of (H1,H2));
set cH1 = the
carrier of H1, cH2 = the
carrier of H2, IH1 = the
InternalRel of H1, IH2 = the
InternalRel of H2, H1H2 =
[:cH1, cH2:], H2H1 =
[:cH2, cH1:];
per cases by
A2;
suppose
A3: the RelStr of G
= (
union_of (H1,H2));
A4: IH2
= (the
InternalRel of G
|_2 cH2)
proof
thus IH2
c= (the
InternalRel of G
|_2 cH2)
proof
let a be
object;
the
InternalRel of G
= (IH1
\/ IH2) by
A3,
NECKLA_2:def 2;
then
A5: IH2
c= the
InternalRel of G by
XBOOLE_1: 7;
assume a
in IH2;
hence thesis by
A5,
XBOOLE_0:def 4;
end;
let a be
object;
assume
A6: a
in (the
InternalRel of G
|_2 cH2);
then
A7: a
in
[:cH2, cH2:] by
XBOOLE_0:def 4;
a
in the
InternalRel of G by
A6,
XBOOLE_0:def 4;
then
A8: a
in (IH1
\/ IH2) by
A3,
NECKLA_2:def 2;
per cases by
A8,
XBOOLE_0:def 3;
suppose a
in IH1;
then
consider x,y be
object such that
A9: a
=
[x, y] and
A10: x
in cH1 and y
in cH1 by
RELSET_1: 2;
consider x1,y1 be
object such that
A11: x1
in cH2 and y1
in cH2 and
A12: a
=
[x1, y1] by
A7,
ZFMISC_1:def 2;
x
= x1 by
A9,
A12,
XTUPLE_0: 1;
then (cH1
/\ cH2)
<>
{} by
A10,
A11,
XBOOLE_0:def 4;
hence thesis by
A1;
end;
suppose a
in IH2;
hence thesis;
end;
end;
A13: IH1
= (the
InternalRel of G
|_2 cH1)
proof
thus IH1
c= (the
InternalRel of G
|_2 cH1)
proof
let a be
object;
the
InternalRel of G
= (IH1
\/ IH2) by
A3,
NECKLA_2:def 2;
then
A14: IH1
c= the
InternalRel of G by
XBOOLE_1: 7;
assume a
in IH1;
hence thesis by
A14,
XBOOLE_0:def 4;
end;
let a be
object;
assume
A15: a
in (the
InternalRel of G
|_2 cH1);
then
A16: a
in
[:cH1, cH1:] by
XBOOLE_0:def 4;
a
in the
InternalRel of G by
A15,
XBOOLE_0:def 4;
then
A17: a
in (IH1
\/ IH2) by
A3,
NECKLA_2:def 2;
per cases by
A17,
XBOOLE_0:def 3;
suppose a
in IH1;
hence thesis;
end;
suppose a
in IH2;
then
consider x,y be
object such that
A18: a
=
[x, y] and
A19: x
in cH2 and y
in cH2 by
RELSET_1: 2;
ex x1,y1 be
object st x1
in cH1 & y1
in cH1 & a
=
[x1, y1] by
A16,
ZFMISC_1:def 2;
then x
in cH1 by
A18,
XTUPLE_0: 1;
hence thesis by
A1,
A19,
XBOOLE_0: 3;
end;
end;
the
carrier of G
= (the
carrier of H1
\/ the
carrier of H2) by
A3,
NECKLA_2:def 2;
then
A20: the
carrier of H1
c= the
carrier of G & the
carrier of H2
c= the
carrier of G by
XBOOLE_1: 7;
the
InternalRel of G
= (IH1
\/ IH2) by
A3,
NECKLA_2:def 2;
then IH1
c= the
InternalRel of G & the
InternalRel of H2
c= the
InternalRel of G by
XBOOLE_1: 7;
hence thesis by
A20,
A13,
A4,
YELLOW_0:def 13,
YELLOW_0:def 14;
end;
suppose
A21: the RelStr of G
= (
sum_of (H1,H2));
A22: IH2
= (the
InternalRel of G
|_2 cH2)
proof
thus IH2
c= (the
InternalRel of G
|_2 cH2)
proof
let a be
object;
the
InternalRel of G
= (((IH1
\/ IH2)
\/ H1H2)
\/ H2H1) by
A21,
NECKLA_2:def 3;
then the
InternalRel of G
= (IH2
\/ ((IH1
\/ H1H2)
\/ H2H1)) by
XBOOLE_1: 113;
then
A23: IH2
c= the
InternalRel of G by
XBOOLE_1: 7;
assume a
in IH2;
hence thesis by
A23,
XBOOLE_0:def 4;
end;
let a be
object;
assume
A24: a
in (the
InternalRel of G
|_2 cH2);
then
A25: a
in
[:cH2, cH2:] by
XBOOLE_0:def 4;
a
in the
InternalRel of G by
A24,
XBOOLE_0:def 4;
then a
in (((IH1
\/ IH2)
\/ H1H2)
\/ H2H1) by
A21,
NECKLA_2:def 3;
then a
in (IH1
\/ ((IH2
\/ H1H2)
\/ H2H1)) by
XBOOLE_1: 113;
then a
in IH1 or a
in ((IH2
\/ H1H2)
\/ H2H1) by
XBOOLE_0:def 3;
then a
in IH1 or a
in (IH2
\/ (H1H2
\/ H2H1)) by
XBOOLE_1: 4;
then
A26: a
in IH1 or a
in IH2 or a
in (H1H2
\/ H2H1) by
XBOOLE_0:def 3;
per cases by
A26,
XBOOLE_0:def 3;
suppose a
in IH1;
then
consider x,y be
object such that
A27: a
=
[x, y] and
A28: x
in cH1 and y
in cH1 by
RELSET_1: 2;
consider x1,y1 be
object such that
A29: x1
in cH2 and y1
in cH2 and
A30: a
=
[x1, y1] by
A25,
ZFMISC_1:def 2;
x
= x1 by
A27,
A30,
XTUPLE_0: 1;
then (cH1
/\ cH2)
<>
{} by
A28,
A29,
XBOOLE_0:def 4;
hence thesis by
A1;
end;
suppose a
in IH2;
hence thesis;
end;
suppose a
in H1H2;
then
consider x,y be
object such that
A31: x
in cH1 and y
in cH2 and
A32: a
=
[x, y] by
ZFMISC_1:def 2;
consider x1,y1 be
object such that
A33: x1
in cH2 and y1
in cH2 and
A34: a
=
[x1, y1] by
A25,
ZFMISC_1:def 2;
x
= x1 by
A32,
A34,
XTUPLE_0: 1;
then (cH1
/\ cH2)
<>
{} by
A31,
A33,
XBOOLE_0:def 4;
hence thesis by
A1;
end;
suppose a
in H2H1;
then
consider x,y be
object such that x
in cH2 and
A35: y
in cH1 and
A36: a
=
[x, y] by
ZFMISC_1:def 2;
consider x1,y1 be
object such that x1
in cH2 and
A37: y1
in cH2 and
A38: a
=
[x1, y1] by
A25,
ZFMISC_1:def 2;
y
= y1 by
A36,
A38,
XTUPLE_0: 1;
then (cH1
/\ cH2)
<>
{} by
A35,
A37,
XBOOLE_0:def 4;
hence thesis by
A1;
end;
end;
IH2
c= ((IH1
\/ IH2)
\/
[:cH1, cH2:]) by
XBOOLE_1: 7,
XBOOLE_1: 10;
then
A39: IH2
c= (((IH1
\/ IH2)
\/
[:cH1, cH2:])
\/
[:cH2, cH1:]) by
XBOOLE_1: 10;
A40: IH1
= (the
InternalRel of G
|_2 cH1)
proof
thus IH1
c= (the
InternalRel of G
|_2 cH1)
proof
let a be
object;
the
InternalRel of G
= (((IH1
\/ IH2)
\/ H1H2)
\/ H2H1) by
A21,
NECKLA_2:def 3
.= (IH1
\/ ((IH2
\/ H1H2)
\/ H2H1)) by
XBOOLE_1: 113;
then
A41: IH1
c= the
InternalRel of G by
XBOOLE_1: 7;
assume a
in IH1;
hence thesis by
A41,
XBOOLE_0:def 4;
end;
let a be
object;
assume
A42: a
in (the
InternalRel of G
|_2 cH1);
then
A43: a
in
[:cH1, cH1:] by
XBOOLE_0:def 4;
a
in the
InternalRel of G by
A42,
XBOOLE_0:def 4;
then a
in (((IH1
\/ IH2)
\/ H1H2)
\/ H2H1) by
A21,
NECKLA_2:def 3;
then a
in (IH1
\/ ((IH2
\/ H1H2)
\/ H2H1)) by
XBOOLE_1: 113;
then a
in IH1 or a
in ((IH2
\/ H1H2)
\/ H2H1) by
XBOOLE_0:def 3;
then a
in IH1 or a
in (IH2
\/ (H1H2
\/ H2H1)) by
XBOOLE_1: 4;
then
A44: a
in IH1 or a
in IH2 or a
in (H1H2
\/ H2H1) by
XBOOLE_0:def 3;
per cases by
A44,
XBOOLE_0:def 3;
suppose a
in IH1;
hence thesis;
end;
suppose a
in IH2;
then
consider x,y be
object such that
A45: a
=
[x, y] and
A46: x
in cH2 and y
in cH2 by
RELSET_1: 2;
consider x1,y1 be
object such that
A47: x1
in cH1 and y1
in cH1 and
A48: a
=
[x1, y1] by
A43,
ZFMISC_1:def 2;
x
= x1 by
A45,
A48,
XTUPLE_0: 1;
then (cH1
/\ cH2)
<>
{} by
A46,
A47,
XBOOLE_0:def 4;
hence thesis by
A1;
end;
suppose a
in H1H2;
then
consider x,y be
object such that x
in cH1 and
A49: y
in cH2 and
A50: a
=
[x, y] by
ZFMISC_1:def 2;
consider x1,y1 be
object such that x1
in cH1 and
A51: y1
in cH1 and
A52: a
=
[x1, y1] by
A43,
ZFMISC_1:def 2;
y
= y1 by
A50,
A52,
XTUPLE_0: 1;
then (cH1
/\ cH2)
<>
{} by
A49,
A51,
XBOOLE_0:def 4;
hence thesis by
A1;
end;
suppose a
in H2H1;
then
consider x,y be
object such that
A53: x
in cH2 and y
in cH1 and
A54: a
=
[x, y] by
ZFMISC_1:def 2;
consider x1,y1 be
object such that
A55: x1
in cH1 and y1
in cH1 and
A56: a
=
[x1, y1] by
A43,
ZFMISC_1:def 2;
x
= x1 by
A54,
A56,
XTUPLE_0: 1;
then (cH1
/\ cH2)
<>
{} by
A53,
A55,
XBOOLE_0:def 4;
hence thesis by
A1;
end;
end;
IH1
c= (IH1
\/ (IH2
\/
[:cH1, cH2:])) by
XBOOLE_1: 7;
then
A57: IH1
c= ((IH1
\/ IH2)
\/
[:cH1, cH2:]) by
XBOOLE_1: 4;
the
carrier of G
= (the
carrier of H1
\/ the
carrier of H2) by
A21,
NECKLA_2:def 3;
then
A58: the
carrier of H1
c= the
carrier of G & the
carrier of H2
c= the
carrier of G by
XBOOLE_1: 7;
A59: the
InternalRel of G
= (((IH1
\/ IH2)
\/
[:cH1, cH2:])
\/
[:cH2, cH1:]) by
A21,
NECKLA_2:def 3;
then ((IH1
\/ IH2)
\/
[:cH1, cH2:])
c= the
InternalRel of G by
XBOOLE_1: 7;
then IH1
c= the
InternalRel of G by
A57;
hence thesis by
A59,
A58,
A39,
A40,
A22,
YELLOW_0:def 13,
YELLOW_0:def 14;
end;
end;
begin
theorem ::
NECKLA_3:11
Th11: the
InternalRel of (
ComplRelStr (
Necklace 4))
=
{
[
0 , 2],
[2,
0 ],
[
0 , 3],
[3,
0 ],
[1, 3],
[3, 1]}
proof
set N4 = (
Necklace 4), cN4 = the
carrier of N4, CmpN4 = (
ComplRelStr N4);
A1: the
carrier of (
Necklace 4)
=
{
0 , 1, 2, 3} by
NECKLACE: 1,
NECKLACE: 20;
thus the
InternalRel of CmpN4
c=
{
[
0 , 2],
[2,
0 ],
[
0 , 3],
[3,
0 ],
[1, 3],
[3, 1]}
proof
let x be
object;
assume x
in the
InternalRel of CmpN4;
then
A2: x
in ((the
InternalRel of N4
` )
\ (
id cN4)) by
NECKLACE:def 8;
then
A3: not x
in (
id cN4) by
XBOOLE_0:def 5;
x
in (the
InternalRel of N4
` ) by
A2,
XBOOLE_0:def 5;
then
A4: x
in (
[:cN4, cN4:]
\ the
InternalRel of N4) by
SUBSET_1:def 4;
consider a1,b1 be
object such that
A5: a1
in cN4 and
A6: b1
in cN4 and
A7: x
=
[a1, b1] by
A2,
ZFMISC_1:def 2;
per cases by
A1,
A5,
A6,
ENUMSET1:def 2;
suppose a1
=
0 & b1
=
0 ;
hence thesis by
A3,
A5,
A7,
RELAT_1:def 10;
end;
suppose a1
=
0 & b1
= 1;
then x
in the
InternalRel of N4 by
A7,
ENUMSET1:def 4,
NECKLA_2: 2;
hence thesis by
A4,
XBOOLE_0:def 5;
end;
suppose a1
=
0 & b1
= 2;
hence thesis by
A7,
ENUMSET1:def 4;
end;
suppose a1
=
0 & b1
= 3;
hence thesis by
A7,
ENUMSET1:def 4;
end;
suppose a1
= 1 & b1
=
0 ;
then x
in the
InternalRel of N4 by
A7,
ENUMSET1:def 4,
NECKLA_2: 2;
hence thesis by
A4,
XBOOLE_0:def 5;
end;
suppose a1
= 2 & b1
=
0 ;
hence thesis by
A7,
ENUMSET1:def 4;
end;
suppose a1
= 3 & b1
=
0 ;
hence thesis by
A7,
ENUMSET1:def 4;
end;
suppose a1
= 1 & b1
= 1;
hence thesis by
A3,
A5,
A7,
RELAT_1:def 10;
end;
suppose a1
= 1 & b1
= 2;
then x
in the
InternalRel of N4 by
A7,
ENUMSET1:def 4,
NECKLA_2: 2;
hence thesis by
A4,
XBOOLE_0:def 5;
end;
suppose a1
= 1 & b1
= 3;
hence thesis by
A7,
ENUMSET1:def 4;
end;
suppose a1
= 2 & b1
= 2;
hence thesis by
A3,
A5,
A7,
RELAT_1:def 10;
end;
suppose a1
= 2 & b1
= 1;
then x
in the
InternalRel of N4 by
A7,
ENUMSET1:def 4,
NECKLA_2: 2;
hence thesis by
A4,
XBOOLE_0:def 5;
end;
suppose a1
= 2 & b1
= 3;
then x
in the
InternalRel of N4 by
A7,
ENUMSET1:def 4,
NECKLA_2: 2;
hence thesis by
A4,
XBOOLE_0:def 5;
end;
suppose a1
= 3 & b1
= 3;
hence thesis by
A3,
A5,
A7,
RELAT_1:def 10;
end;
suppose a1
= 3 & b1
= 1;
hence thesis by
A7,
ENUMSET1:def 4;
end;
suppose a1
= 3 & b1
= 2;
then x
in the
InternalRel of N4 by
A7,
ENUMSET1:def 4,
NECKLA_2: 2;
hence thesis by
A4,
XBOOLE_0:def 5;
end;
end;
let a be
object;
assume
A8: a
in
{
[
0 , 2],
[2,
0 ],
[
0 , 3],
[3,
0 ],
[1, 3],
[3, 1]};
per cases by
A8,
ENUMSET1:def 4;
suppose
A9: a
=
[
0 , 2];
A10: not a
in the
InternalRel of N4
proof
assume
A11: not thesis;
per cases by
A11,
ENUMSET1:def 4,
NECKLA_2: 2;
suppose a
=
[
0 , 1];
hence contradiction by
A9,
XTUPLE_0: 1;
end;
suppose a
=
[1,
0 ];
hence contradiction by
A9,
XTUPLE_0: 1;
end;
suppose a
=
[1, 2];
hence contradiction by
A9,
XTUPLE_0: 1;
end;
suppose a
=
[2, 1];
hence contradiction by
A9,
XTUPLE_0: 1;
end;
suppose a
=
[2, 3];
hence contradiction by
A9,
XTUPLE_0: 1;
end;
suppose a
=
[3, 2];
hence contradiction by
A9,
XTUPLE_0: 1;
end;
end;
0
in cN4 & 2
in cN4 by
A1,
ENUMSET1:def 2;
then a
in
[:cN4, cN4:] by
A9,
ZFMISC_1: 87;
then a
in (
[:cN4, cN4:]
\ the
InternalRel of N4) by
A10,
XBOOLE_0:def 5;
then
A12: a
in (the
InternalRel of N4
` ) by
SUBSET_1:def 4;
not a
in (
id cN4) by
A9,
RELAT_1:def 10;
then a
in ((the
InternalRel of N4
` )
\ (
id cN4)) by
A12,
XBOOLE_0:def 5;
hence thesis by
NECKLACE:def 8;
end;
suppose
A13: a
=
[2,
0 ];
A14: not a
in the
InternalRel of N4
proof
assume
A15: not thesis;
per cases by
A15,
ENUMSET1:def 4,
NECKLA_2: 2;
suppose a
=
[
0 , 1];
hence contradiction by
A13,
XTUPLE_0: 1;
end;
suppose a
=
[1,
0 ];
hence contradiction by
A13,
XTUPLE_0: 1;
end;
suppose a
=
[1, 2];
hence contradiction by
A13,
XTUPLE_0: 1;
end;
suppose a
=
[2, 1];
hence contradiction by
A13,
XTUPLE_0: 1;
end;
suppose a
=
[2, 3];
hence contradiction by
A13,
XTUPLE_0: 1;
end;
suppose a
=
[3, 2];
hence contradiction by
A13,
XTUPLE_0: 1;
end;
end;
0
in cN4 & 2
in cN4 by
A1,
ENUMSET1:def 2;
then a
in
[:cN4, cN4:] by
A13,
ZFMISC_1: 87;
then a
in (
[:cN4, cN4:]
\ the
InternalRel of N4) by
A14,
XBOOLE_0:def 5;
then
A16: a
in (the
InternalRel of N4
` ) by
SUBSET_1:def 4;
not a
in (
id cN4) by
A13,
RELAT_1:def 10;
then a
in ((the
InternalRel of N4
` )
\ (
id cN4)) by
A16,
XBOOLE_0:def 5;
hence thesis by
NECKLACE:def 8;
end;
suppose
A17: a
=
[
0 , 3];
A18: not a
in the
InternalRel of N4
proof
assume
A19: not thesis;
per cases by
A19,
ENUMSET1:def 4,
NECKLA_2: 2;
suppose a
=
[
0 , 1];
hence contradiction by
A17,
XTUPLE_0: 1;
end;
suppose a
=
[1,
0 ];
hence contradiction by
A17,
XTUPLE_0: 1;
end;
suppose a
=
[1, 2];
hence contradiction by
A17,
XTUPLE_0: 1;
end;
suppose a
=
[2, 1];
hence contradiction by
A17,
XTUPLE_0: 1;
end;
suppose a
=
[2, 3];
hence contradiction by
A17,
XTUPLE_0: 1;
end;
suppose a
=
[3, 2];
hence contradiction by
A17,
XTUPLE_0: 1;
end;
end;
0
in cN4 & 3
in cN4 by
A1,
ENUMSET1:def 2;
then a
in
[:cN4, cN4:] by
A17,
ZFMISC_1: 87;
then a
in (
[:cN4, cN4:]
\ the
InternalRel of N4) by
A18,
XBOOLE_0:def 5;
then
A20: a
in (the
InternalRel of N4
` ) by
SUBSET_1:def 4;
not a
in (
id cN4) by
A17,
RELAT_1:def 10;
then a
in ((the
InternalRel of N4
` )
\ (
id cN4)) by
A20,
XBOOLE_0:def 5;
hence thesis by
NECKLACE:def 8;
end;
suppose
A21: a
=
[3,
0 ];
A22: not a
in the
InternalRel of N4
proof
assume
A23: not thesis;
per cases by
A23,
ENUMSET1:def 4,
NECKLA_2: 2;
suppose a
=
[
0 , 1];
hence contradiction by
A21,
XTUPLE_0: 1;
end;
suppose a
=
[1,
0 ];
hence contradiction by
A21,
XTUPLE_0: 1;
end;
suppose a
=
[1, 2];
hence contradiction by
A21,
XTUPLE_0: 1;
end;
suppose a
=
[2, 1];
hence contradiction by
A21,
XTUPLE_0: 1;
end;
suppose a
=
[2, 3];
hence contradiction by
A21,
XTUPLE_0: 1;
end;
suppose a
=
[3, 2];
hence contradiction by
A21,
XTUPLE_0: 1;
end;
end;
0
in cN4 & 3
in cN4 by
A1,
ENUMSET1:def 2;
then a
in
[:cN4, cN4:] by
A21,
ZFMISC_1: 87;
then a
in (
[:cN4, cN4:]
\ the
InternalRel of N4) by
A22,
XBOOLE_0:def 5;
then
A24: a
in (the
InternalRel of N4
` ) by
SUBSET_1:def 4;
not a
in (
id cN4) by
A21,
RELAT_1:def 10;
then a
in ((the
InternalRel of N4
` )
\ (
id cN4)) by
A24,
XBOOLE_0:def 5;
hence thesis by
NECKLACE:def 8;
end;
suppose
A25: a
=
[1, 3];
A26: not a
in the
InternalRel of N4
proof
assume
A27: not thesis;
per cases by
A27,
ENUMSET1:def 4,
NECKLA_2: 2;
suppose a
=
[
0 , 1];
hence contradiction by
A25,
XTUPLE_0: 1;
end;
suppose a
=
[1,
0 ];
hence contradiction by
A25,
XTUPLE_0: 1;
end;
suppose a
=
[1, 2];
hence contradiction by
A25,
XTUPLE_0: 1;
end;
suppose a
=
[2, 1];
hence contradiction by
A25,
XTUPLE_0: 1;
end;
suppose a
=
[2, 3];
hence contradiction by
A25,
XTUPLE_0: 1;
end;
suppose a
=
[3, 2];
hence contradiction by
A25,
XTUPLE_0: 1;
end;
end;
1
in cN4 & 3
in cN4 by
A1,
ENUMSET1:def 2;
then a
in
[:cN4, cN4:] by
A25,
ZFMISC_1: 87;
then a
in (
[:cN4, cN4:]
\ the
InternalRel of N4) by
A26,
XBOOLE_0:def 5;
then
A28: a
in (the
InternalRel of N4
` ) by
SUBSET_1:def 4;
not a
in (
id cN4) by
A25,
RELAT_1:def 10;
then a
in ((the
InternalRel of N4
` )
\ (
id cN4)) by
A28,
XBOOLE_0:def 5;
hence thesis by
NECKLACE:def 8;
end;
suppose
A29: a
=
[3, 1];
A30: not a
in the
InternalRel of N4
proof
assume
A31: not thesis;
per cases by
A31,
ENUMSET1:def 4,
NECKLA_2: 2;
suppose a
=
[
0 , 1];
hence contradiction by
A29,
XTUPLE_0: 1;
end;
suppose a
=
[1,
0 ];
hence contradiction by
A29,
XTUPLE_0: 1;
end;
suppose a
=
[1, 2];
hence contradiction by
A29,
XTUPLE_0: 1;
end;
suppose a
=
[2, 1];
hence contradiction by
A29,
XTUPLE_0: 1;
end;
suppose a
=
[2, 3];
hence contradiction by
A29,
XTUPLE_0: 1;
end;
suppose a
=
[3, 2];
hence contradiction by
A29,
XTUPLE_0: 1;
end;
end;
1
in cN4 & 3
in cN4 by
A1,
ENUMSET1:def 2;
then a
in
[:cN4, cN4:] by
A29,
ZFMISC_1: 87;
then a
in (
[:cN4, cN4:]
\ the
InternalRel of N4) by
A30,
XBOOLE_0:def 5;
then
A32: a
in (the
InternalRel of N4
` ) by
SUBSET_1:def 4;
not a
in (
id cN4) by
A29,
RELAT_1:def 10;
then a
in ((the
InternalRel of N4
` )
\ (
id cN4)) by
A32,
XBOOLE_0:def 5;
hence thesis by
NECKLACE:def 8;
end;
end;
registration
let R be
RelStr;
cluster (
ComplRelStr R) ->
irreflexive;
correctness
proof
set R1 = (
ComplRelStr R);
for x be
set st x
in the
carrier of R1 holds not
[x, x]
in the
InternalRel of R1
proof
let x be
set;
assume x
in the
carrier of R1;
then
A1: x
in the
carrier of R by
NECKLACE:def 8;
not
[x, x]
in the
InternalRel of R1
proof
assume
[x, x]
in the
InternalRel of R1;
then
[x, x]
in ((the
InternalRel of R
` )
\ (
id the
carrier of R)) by
NECKLACE:def 8;
then not
[x, x]
in (
id the
carrier of R) by
XBOOLE_0:def 5;
hence contradiction by
A1,
RELAT_1:def 10;
end;
hence thesis;
end;
hence thesis;
end;
end
registration
let R be
symmetric
RelStr;
cluster (
ComplRelStr R) ->
symmetric;
correctness
proof
let x,y be
object;
set S = (
ComplRelStr R);
assume that
A1: x
in the
carrier of S & y
in the
carrier of S and
A2:
[x, y]
in the
InternalRel of S;
per cases ;
suppose x
= y;
hence thesis by
A2;
end;
suppose
A3: x
<> y;
A4: x
in the
carrier of R & y
in the
carrier of R by
A1,
NECKLACE:def 8;
then
A5:
[y, x]
in
[:the
carrier of R, the
carrier of R:] by
ZFMISC_1: 87;
[x, y]
in ((the
InternalRel of R
` )
\ (
id the
carrier of R)) by
A2,
NECKLACE:def 8;
then
[x, y]
in (the
InternalRel of R
` ) by
XBOOLE_0:def 5;
then
[x, y]
in (
[:the
carrier of R, the
carrier of R:]
\ the
InternalRel of R) by
SUBSET_1:def 4;
then
A6: not
[x, y]
in the
InternalRel of R by
XBOOLE_0:def 5;
the
InternalRel of R
is_symmetric_in the
carrier of R by
NECKLACE:def 3;
then not
[y, x]
in the
InternalRel of R by
A4,
A6;
then
[y, x]
in (
[:the
carrier of R, the
carrier of R:]
\ the
InternalRel of R) by
A5,
XBOOLE_0:def 5;
then
A7:
[y, x]
in (the
InternalRel of R
` ) by
SUBSET_1:def 4;
not
[y, x]
in (
id the
carrier of R) by
A3,
RELAT_1:def 10;
then
[y, x]
in ((the
InternalRel of R
` )
\ (
id the
carrier of R)) by
A7,
XBOOLE_0:def 5;
hence thesis by
NECKLACE:def 8;
end;
end;
end
theorem ::
NECKLA_3:12
Th12: for R be
RelStr holds the
InternalRel of R
misses the
InternalRel of (
ComplRelStr R)
proof
let R be
RelStr;
assume not thesis;
then (the
InternalRel of R
/\ the
InternalRel of (
ComplRelStr R))
<>
{} ;
then
consider a be
object such that
A1: a
in (the
InternalRel of R
/\ the
InternalRel of (
ComplRelStr R)) by
XBOOLE_0:def 1;
a
in the
InternalRel of (
ComplRelStr R) by
A1,
XBOOLE_0:def 4;
then a
in ((the
InternalRel of R
` )
\ (
id the
carrier of R)) by
NECKLACE:def 8;
then a
in (the
InternalRel of R
` ) by
XBOOLE_0:def 5;
then a
in (
[:the
carrier of R, the
carrier of R:]
\ the
InternalRel of R) by
SUBSET_1:def 4;
then not a
in the
InternalRel of R by
XBOOLE_0:def 5;
hence thesis by
A1,
XBOOLE_0:def 4;
end;
theorem ::
NECKLA_3:13
Th13: for R be
RelStr holds (
id the
carrier of R)
misses the
InternalRel of (
ComplRelStr R)
proof
let R be
RelStr;
assume not thesis;
then ((
id the
carrier of R)
/\ the
InternalRel of (
ComplRelStr R))
<>
{} ;
then
consider a be
object such that
A1: a
in ((
id the
carrier of R)
/\ the
InternalRel of (
ComplRelStr R)) by
XBOOLE_0:def 1;
a
in the
InternalRel of (
ComplRelStr R) by
A1,
XBOOLE_0:def 4;
then
A2: a
in ((the
InternalRel of R
` )
\ (
id the
carrier of R)) by
NECKLACE:def 8;
a
in (
id the
carrier of R) by
A1,
XBOOLE_0:def 4;
hence contradiction by
A2,
XBOOLE_0:def 5;
end;
theorem ::
NECKLA_3:14
Th14: for G be
RelStr holds
[:the
carrier of G, the
carrier of G:]
= (((
id the
carrier of G)
\/ the
InternalRel of G)
\/ the
InternalRel of (
ComplRelStr G))
proof
let G be
RelStr;
set idcG = (
id the
carrier of G), IG = the
InternalRel of G, ICmpG = the
InternalRel of (
ComplRelStr G), cG = the
carrier of G;
thus
[:cG, cG:]
c= ((idcG
\/ IG)
\/ ICmpG)
proof
let a be
object;
assume
A1: a
in
[:cG, cG:];
then
consider x,y be
object such that
A2: x
in cG and y
in cG and
A3: a
=
[x, y] by
ZFMISC_1:def 2;
per cases ;
suppose
A4: x
= y;
[x, x]
in (
id cG) by
A2,
RELAT_1:def 10;
then a
in ((
id cG)
\/ IG) by
A3,
A4,
XBOOLE_0:def 3;
hence thesis by
XBOOLE_0:def 3;
end;
suppose x
<> y;
then
A5: not a
in (
id cG) by
A3,
RELAT_1:def 10;
thus thesis
proof
per cases ;
suppose a
in IG;
then a
in ((
id cG)
\/ IG) by
XBOOLE_0:def 3;
hence thesis by
XBOOLE_0:def 3;
end;
suppose not a
in IG;
then a
in (
[:cG, cG:]
\ IG) by
A1,
XBOOLE_0:def 5;
then a
in (IG
` ) by
SUBSET_1:def 4;
then a
in ((IG
` )
\ (
id cG)) by
A5,
XBOOLE_0:def 5;
then a
in ICmpG by
NECKLACE:def 8;
then a
in (IG
\/ ICmpG) by
XBOOLE_0:def 3;
then a
in ((
id cG)
\/ (IG
\/ ICmpG)) by
XBOOLE_0:def 3;
hence thesis by
XBOOLE_1: 4;
end;
end;
end;
end;
let a be
object;
assume a
in ((idcG
\/ IG)
\/ ICmpG);
then
A6: a
in ((
id cG)
\/ IG) or a
in ICmpG by
XBOOLE_0:def 3;
per cases by
A6,
XBOOLE_0:def 3;
suppose a
in (
id cG);
hence thesis;
end;
suppose a
in IG;
hence thesis;
end;
suppose a
in ICmpG;
then a
in ((IG
` )
\ (
id cG)) by
NECKLACE:def 8;
hence thesis;
end;
end;
theorem ::
NECKLA_3:15
Th15: for G be
strict
irreflexive
RelStr st G is
trivial holds (
ComplRelStr G)
= G
proof
let G be
strict
irreflexive
RelStr;
set CG = (
ComplRelStr G);
assume
A1: G is
trivial;
per cases by
A1,
ZFMISC_1: 131;
suppose
A2: the
carrier of G is
empty;
the
InternalRel of CG
= ((the
InternalRel of G
` )
\ (
id the
carrier of G)) by
NECKLACE:def 8;
then
A3: the
InternalRel of CG
= ((
{}
\
{} )
\ (
id
{} )) by
A2;
the
InternalRel of G
=
{} by
A2;
hence thesis by
A3,
NECKLACE:def 8;
end;
suppose ex x be
object st the
carrier of G
=
{x};
then
consider x be
object such that
A4: the
carrier of G
=
{x};
A5: the
carrier of CG
=
{x} by
A4,
NECKLACE:def 8;
the
InternalRel of G
c=
[:
{x},
{x}:] by
A4;
then the
InternalRel of G
c=
{
[x, x]} by
ZFMISC_1: 29;
then
A6: the
InternalRel of G
=
{} or the
InternalRel of G
=
{
[x, x]} by
ZFMISC_1: 33;
A7: the
InternalRel of G
<>
{
[x, x]}
proof
assume not thesis;
then
A8:
[x, x]
in the
InternalRel of G by
TARSKI:def 1;
x
in the
carrier of G by
A4,
TARSKI:def 1;
hence contradiction by
A8,
NECKLACE:def 5;
end;
the
InternalRel of CG
= ((the
InternalRel of G
` )
\ (
id the
carrier of G)) by
NECKLACE:def 8;
then the
InternalRel of CG
= ((
[:
{x},
{x}:]
\
{} )
\ (
id
{x})) by
A4,
A6,
A7,
SUBSET_1:def 4;
then the
InternalRel of CG
= (
{
[x, x]}
\ (
id
{x})) by
ZFMISC_1: 29;
then the
InternalRel of CG
= (
{
[x, x]}
\
{
[x, x]}) by
SYSREL: 13;
hence thesis by
A4,
A6,
A7,
A5,
XBOOLE_1: 37;
end;
end;
theorem ::
NECKLA_3:16
Th16: for G be
strict
irreflexive
RelStr holds (
ComplRelStr (
ComplRelStr G))
= G
proof
let G be
strict
irreflexive
RelStr;
set CCmpG = (
ComplRelStr (
ComplRelStr G)), CmpG = (
ComplRelStr G), cG = the
carrier of G, IG = the
InternalRel of G, ICmpG = the
InternalRel of CmpG, ICCmpG = the
InternalRel of CCmpG;
A1: cG
= the
carrier of CmpG by
NECKLACE:def 8
.= the
carrier of CCmpG by
NECKLACE:def 8;
A2: cG
= the
carrier of CmpG by
NECKLACE:def 8;
A3: (
id cG)
misses IG
proof
assume not thesis;
then ((
id cG)
/\ IG)
<>
{} ;
then
consider a be
object such that
A4: a
in ((
id cG)
/\ IG) by
XBOOLE_0:def 1;
A5: a
in IG by
A4,
XBOOLE_0:def 4;
consider x,y be
object such that
A6: a
=
[x, y] and
A7: x
in cG and y
in cG by
A4,
RELSET_1: 2;
a
in (
id cG) by
A4,
XBOOLE_0:def 4;
then x
= y by
A6,
RELAT_1:def 10;
hence contradiction by
A5,
A6,
A7,
NECKLACE:def 5;
end;
ICCmpG
= ((ICmpG
` )
\ (
id the
carrier of CmpG)) by
NECKLACE:def 8
.= ((
[:the
carrier of CmpG, the
carrier of CmpG:]
\ ICmpG)
\ (
id the
carrier of CmpG)) by
SUBSET_1:def 4
.= ((
[:cG, cG:]
\ ((IG
` )
\ (
id cG)))
\ (
id cG)) by
A2,
NECKLACE:def 8
.= (((
[:cG, cG:]
\ (IG
` ))
\/ (
[:cG, cG:]
/\ (
id cG)))
\ (
id cG)) by
XBOOLE_1: 52
.= (((
[:cG, cG:]
\ (IG
` ))
\/ (
id cG))
\ (
id cG)) by
XBOOLE_1: 28
.= ((
[:cG, cG:]
\ (IG
` ))
\ (
id cG)) by
XBOOLE_1: 40
.= ((
[:cG, cG:]
\ (
[:cG, cG:]
\ IG))
\ (
id cG)) by
SUBSET_1:def 4
.= ((
[:cG, cG:]
/\ IG)
\ (
id cG)) by
XBOOLE_1: 48
.= (IG
\ (
id cG)) by
XBOOLE_1: 28
.= IG by
A3,
XBOOLE_1: 83;
hence thesis by
A1;
end;
theorem ::
NECKLA_3:17
Th17: for G1,G2 be
RelStr st the
carrier of G1
misses the
carrier of G2 holds (
ComplRelStr (
union_of (G1,G2)))
= (
sum_of ((
ComplRelStr G1),(
ComplRelStr G2)))
proof
let G1,G2 be
RelStr;
A1: the
carrier of (
sum_of ((
ComplRelStr G1),(
ComplRelStr G2)))
= (the
carrier of (
ComplRelStr G1)
\/ the
carrier of (
ComplRelStr G2)) by
NECKLA_2:def 3
.= (the
carrier of G1
\/ the
carrier of (
ComplRelStr G2)) by
NECKLACE:def 8
.= (the
carrier of G1
\/ the
carrier of G2) by
NECKLACE:def 8;
set P = the
InternalRel of (
ComplRelStr (
union_of (G1,G2))), R = the
InternalRel of (
sum_of ((
ComplRelStr G1),(
ComplRelStr G2))), X1 = the
InternalRel of (
ComplRelStr G1), X2 = the
InternalRel of (
ComplRelStr G2), X3 =
[:the
carrier of (
ComplRelStr G1), the
carrier of (
ComplRelStr G2):], X4 =
[:the
carrier of (
ComplRelStr G2), the
carrier of (
ComplRelStr G1):], X5 =
[:the
carrier of G1, the
carrier of G1:], X6 =
[:the
carrier of G2, the
carrier of G2:], X7 =
[:the
carrier of G1, the
carrier of G2:], X8 =
[:the
carrier of G2, the
carrier of G1:];
assume
A2: the
carrier of G1
misses the
carrier of G2;
A3: for a,b be
object holds
[a, b]
in P iff
[a, b]
in R
proof
let a,b be
object;
set x =
[a, b];
thus x
in P implies x
in R
proof
assume x
in P;
then
A4: x
in ((the
InternalRel of (
union_of (G1,G2))
` )
\ (
id the
carrier of (
union_of (G1,G2)))) by
NECKLACE:def 8;
then x
in
[:the
carrier of (
union_of (G1,G2)), the
carrier of (
union_of (G1,G2)):];
then x
in
[:(the
carrier of G1
\/ the
carrier of G2), the
carrier of (
union_of (G1,G2)):] by
NECKLA_2:def 2;
then
A5: x
in
[:(the
carrier of G1
\/ the
carrier of G2), (the
carrier of G1
\/ the
carrier of G2):] by
NECKLA_2:def 2;
not x
in (
id the
carrier of (
union_of (G1,G2))) by
A4,
XBOOLE_0:def 5;
then
A6: not x
in (
id (the
carrier of G1
\/ the
carrier of G2)) by
NECKLA_2:def 2;
A7: not x
in (
id the
carrier of G1) & not x
in (
id the
carrier of G2)
proof
assume not thesis;
then x
in ((
id the
carrier of G1)
\/ (
id the
carrier of G2)) by
XBOOLE_0:def 3;
hence contradiction by
A6,
SYSREL: 14;
end;
the
carrier of G1
= the
carrier of (
ComplRelStr G1) & the
carrier of G2
= the
carrier of (
ComplRelStr G2) by
NECKLACE:def 8;
then x
in (((X5
\/ X3)
\/ X4)
\/ X6) by
A5,
ZFMISC_1: 98;
then x
in (X5
\/ ((X3
\/ X4)
\/ X6)) by
XBOOLE_1: 113;
then x
in X5 or x
in ((X3
\/ X4)
\/ X6) by
XBOOLE_0:def 3;
then x
in X5 or x
in (X3
\/ (X4
\/ X6)) by
XBOOLE_1: 4;
then
A8: x
in X5 or x
in X3 or x
in (X4
\/ X6) by
XBOOLE_0:def 3;
x
in (the
InternalRel of (
union_of (G1,G2))
` ) by
A4,
XBOOLE_0:def 5;
then x
in (
[:the
carrier of (
union_of (G1,G2)), the
carrier of (
union_of (G1,G2)):]
\ the
InternalRel of (
union_of (G1,G2))) by
SUBSET_1:def 4;
then not x
in the
InternalRel of (
union_of (G1,G2)) by
XBOOLE_0:def 5;
then
A9: not x
in (the
InternalRel of G1
\/ the
InternalRel of G2) by
NECKLA_2:def 2;
then
A10: not x
in the
InternalRel of G1 by
XBOOLE_0:def 3;
A11: not x
in the
InternalRel of G2 by
A9,
XBOOLE_0:def 3;
per cases by
A8,
XBOOLE_0:def 3;
suppose x
in
[:the
carrier of G1, the
carrier of G1:];
then x
in (
[:the
carrier of G1, the
carrier of G1:]
\ the
InternalRel of G1) by
A10,
XBOOLE_0:def 5;
then x
in (the
InternalRel of G1
` ) by
SUBSET_1:def 4;
then x
in ((the
InternalRel of G1
` )
\ (
id the
carrier of G1)) by
A7,
XBOOLE_0:def 5;
then x
in X1 by
NECKLACE:def 8;
then x
in (X1
\/ ((X2
\/ X3)
\/ X4)) by
XBOOLE_0:def 3;
then x
in (((X1
\/ X2)
\/ X3)
\/ X4) by
XBOOLE_1: 113;
hence thesis by
NECKLA_2:def 3;
end;
suppose x
in X3;
then x
in (X2
\/ X3) by
XBOOLE_0:def 3;
then x
in ((X2
\/ X3)
\/ X4) by
XBOOLE_0:def 3;
then x
in (X1
\/ ((X2
\/ X3)
\/ X4)) by
XBOOLE_0:def 3;
then x
in (((X1
\/ X2)
\/ X3)
\/ X4) by
XBOOLE_1: 113;
hence thesis by
NECKLA_2:def 3;
end;
suppose x
in X4;
then x
in (X3
\/ X4) by
XBOOLE_0:def 3;
then x
in (X2
\/ (X3
\/ X4)) by
XBOOLE_0:def 3;
then x
in ((X2
\/ X3)
\/ X4) by
XBOOLE_1: 4;
then x
in (X1
\/ ((X2
\/ X3)
\/ X4)) by
XBOOLE_0:def 3;
then x
in (((X1
\/ X2)
\/ X3)
\/ X4) by
XBOOLE_1: 113;
hence thesis by
NECKLA_2:def 3;
end;
suppose x
in
[:the
carrier of G2, the
carrier of G2:];
then x
in (
[:the
carrier of G2, the
carrier of G2:]
\ the
InternalRel of G2) by
A11,
XBOOLE_0:def 5;
then x
in (the
InternalRel of G2
` ) by
SUBSET_1:def 4;
then x
in ((the
InternalRel of G2
` )
\ (
id the
carrier of G2)) by
A7,
XBOOLE_0:def 5;
then x
in the
InternalRel of (
ComplRelStr G2) by
NECKLACE:def 8;
then x
in (the
InternalRel of (
ComplRelStr G1)
\/ the
InternalRel of (
ComplRelStr G2)) by
XBOOLE_0:def 3;
then x
in ((the
InternalRel of (
ComplRelStr G1)
\/ the
InternalRel of (
ComplRelStr G2))
\/
[:the
carrier of (
ComplRelStr G1), the
carrier of (
ComplRelStr G2):]) by
XBOOLE_0:def 3;
then x
in (((the
InternalRel of (
ComplRelStr G1)
\/ the
InternalRel of (
ComplRelStr G2))
\/
[:the
carrier of (
ComplRelStr G1), the
carrier of (
ComplRelStr G2):])
\/
[:the
carrier of (
ComplRelStr G2), the
carrier of (
ComplRelStr G1):]) by
XBOOLE_0:def 3;
hence thesis by
NECKLA_2:def 3;
end;
end;
assume x
in R;
then x
in (((X1
\/ X2)
\/ X3)
\/ X4) by
NECKLA_2:def 3;
then x
in ((X1
\/ X2)
\/ X3) or x
in X4 by
XBOOLE_0:def 3;
then
A12: x
in (X1
\/ X2) or x
in X3 or x
in X4 by
XBOOLE_0:def 3;
per cases by
A12,
XBOOLE_0:def 3;
suppose x
in X1;
then
A13: x
in ((the
InternalRel of G1
` )
\ (
id the
carrier of G1)) by
NECKLACE:def 8;
then x
in (the
InternalRel of G1
` ) by
XBOOLE_0:def 5;
then x
in (
[:the
carrier of G1, the
carrier of G1:]
\ the
InternalRel of G1) by
SUBSET_1:def 4;
then
A14: not x
in the
InternalRel of G1 by
XBOOLE_0:def 5;
A15: not x
in the
InternalRel of (
union_of (G1,G2))
proof
assume not thesis;
then x
in (the
InternalRel of G1
\/ the
InternalRel of G2) by
NECKLA_2:def 2;
then x
in the
InternalRel of G2 by
A14,
XBOOLE_0:def 3;
then (
[:the
carrier of G1, the
carrier of G1:]
/\
[:the
carrier of G2, the
carrier of G2:]) is non
empty by
A13,
XBOOLE_0:def 4;
then
[:the
carrier of G1, the
carrier of G1:]
meets
[:the
carrier of G2, the
carrier of G2:];
hence contradiction by
A2,
ZFMISC_1: 104;
end;
A16: not x
in (
id the
carrier of (
union_of (G1,G2)))
proof
assume not thesis;
then x
in (
id (the
carrier of G1
\/ the
carrier of G2)) by
NECKLA_2:def 2;
then
A17: x
in ((
id the
carrier of G1)
\/ (
id the
carrier of G2)) by
SYSREL: 14;
thus thesis
proof
per cases by
A17,
XBOOLE_0:def 3;
suppose x
in (
id the
carrier of G1);
hence contradiction by
A13,
XBOOLE_0:def 5;
end;
suppose x
in (
id the
carrier of G2);
then (
[:the
carrier of G1, the
carrier of G1:]
/\
[:the
carrier of G2, the
carrier of G2:]) is non
empty by
A13,
XBOOLE_0:def 4;
then
[:the
carrier of G1, the
carrier of G1:]
meets
[:the
carrier of G2, the
carrier of G2:];
hence contradiction by
A2,
ZFMISC_1: 104;
end;
end;
end;
x
in (X5
\/ X7) by
A13,
XBOOLE_0:def 3;
then x
in ((X5
\/ X7)
\/ X8) by
XBOOLE_0:def 3;
then x
in (((X5
\/ X7)
\/ X8)
\/ X6) by
XBOOLE_0:def 3;
then x
in
[:(the
carrier of G1
\/ the
carrier of G2), (the
carrier of G1
\/ the
carrier of G2):] by
ZFMISC_1: 98;
then x
in
[:(the
carrier of G1
\/ the
carrier of G2), the
carrier of (
union_of (G1,G2)):] by
NECKLA_2:def 2;
then x
in
[:the
carrier of (
union_of (G1,G2)), the
carrier of (
union_of (G1,G2)):] by
NECKLA_2:def 2;
then x
in (
[:the
carrier of (
union_of (G1,G2)), the
carrier of (
union_of (G1,G2)):]
\ the
InternalRel of (
union_of (G1,G2))) by
A15,
XBOOLE_0:def 5;
then x
in (the
InternalRel of (
union_of (G1,G2))
` ) by
SUBSET_1:def 4;
then x
in ((the
InternalRel of (
union_of (G1,G2))
` )
\ (
id the
carrier of (
union_of (G1,G2)))) by
A16,
XBOOLE_0:def 5;
hence thesis by
NECKLACE:def 8;
end;
suppose x
in X2;
then
A18: x
in ((the
InternalRel of G2
` )
\ (
id the
carrier of G2)) by
NECKLACE:def 8;
then x
in (the
InternalRel of G2
` ) by
XBOOLE_0:def 5;
then x
in (
[:the
carrier of G2, the
carrier of G2:]
\ the
InternalRel of G2) by
SUBSET_1:def 4;
then
A19: not x
in the
InternalRel of G2 by
XBOOLE_0:def 5;
A20: not x
in the
InternalRel of (
union_of (G1,G2))
proof
assume not thesis;
then x
in (the
InternalRel of G1
\/ the
InternalRel of G2) by
NECKLA_2:def 2;
then x
in the
InternalRel of G1 by
A19,
XBOOLE_0:def 3;
then (
[:the
carrier of G1, the
carrier of G1:]
/\
[:the
carrier of G2, the
carrier of G2:]) is non
empty by
A18,
XBOOLE_0:def 4;
then
[:the
carrier of G1, the
carrier of G1:]
meets
[:the
carrier of G2, the
carrier of G2:];
hence contradiction by
A2,
ZFMISC_1: 104;
end;
A21: not x
in (
id the
carrier of (
union_of (G1,G2)))
proof
assume not thesis;
then x
in (
id (the
carrier of G1
\/ the
carrier of G2)) by
NECKLA_2:def 2;
then
A22: x
in ((
id the
carrier of G1)
\/ (
id the
carrier of G2)) by
SYSREL: 14;
per cases by
A22,
XBOOLE_0:def 3;
suppose x
in (
id the
carrier of G2);
hence contradiction by
A18,
XBOOLE_0:def 5;
end;
suppose x
in (
id the
carrier of G1);
then (
[:the
carrier of G1, the
carrier of G1:]
/\
[:the
carrier of G2, the
carrier of G2:]) is non
empty by
A18,
XBOOLE_0:def 4;
then
[:the
carrier of G1, the
carrier of G1:]
meets
[:the
carrier of G2, the
carrier of G2:];
hence contradiction by
A2,
ZFMISC_1: 104;
end;
end;
x
in (X8
\/ X6) by
A18,
XBOOLE_0:def 3;
then x
in (X7
\/ (X8
\/ X6)) by
XBOOLE_0:def 3;
then x
in ((X7
\/ X8)
\/ X6) by
XBOOLE_1: 4;
then x
in (X5
\/ ((X7
\/ X8)
\/ X6)) by
XBOOLE_0:def 3;
then x
in (((X5
\/ X7)
\/ X8)
\/ X6) by
XBOOLE_1: 113;
then x
in
[:(the
carrier of G1
\/ the
carrier of G2), (the
carrier of G1
\/ the
carrier of G2):] by
ZFMISC_1: 98;
then x
in
[:(the
carrier of G1
\/ the
carrier of G2), the
carrier of (
union_of (G1,G2)):] by
NECKLA_2:def 2;
then x
in
[:the
carrier of (
union_of (G1,G2)), the
carrier of (
union_of (G1,G2)):] by
NECKLA_2:def 2;
then x
in (
[:the
carrier of (
union_of (G1,G2)), the
carrier of (
union_of (G1,G2)):]
\ the
InternalRel of (
union_of (G1,G2))) by
A20,
XBOOLE_0:def 5;
then x
in (the
InternalRel of (
union_of (G1,G2))
` ) by
SUBSET_1:def 4;
then x
in ((the
InternalRel of (
union_of (G1,G2))
` )
\ (
id the
carrier of (
union_of (G1,G2)))) by
A21,
XBOOLE_0:def 5;
hence thesis by
NECKLACE:def 8;
end;
suppose x
in X3;
then
A23: x
in
[:the
carrier of G1, the
carrier of (
ComplRelStr G2):] by
NECKLACE:def 8;
then
A24: x
in
[:the
carrier of G1, the
carrier of G2:] by
NECKLACE:def 8;
A25: not x
in the
InternalRel of (
union_of (G1,G2))
proof
assume not thesis;
then
A26: x
in (the
InternalRel of G1
\/ the
InternalRel of G2) by
NECKLA_2:def 2;
per cases by
A26,
XBOOLE_0:def 3;
suppose
A27: x
in the
InternalRel of G1;
A28: b
in the
carrier of G2 by
A24,
ZFMISC_1: 87;
b
in the
carrier of G1 by
A27,
ZFMISC_1: 87;
then b
in (the
carrier of G1
/\ the
carrier of G2) by
A28,
XBOOLE_0:def 4;
hence contradiction by
A2;
end;
suppose
A29: x
in the
InternalRel of G2;
A30: a
in the
carrier of G1 by
A23,
ZFMISC_1: 87;
a
in the
carrier of G2 by
A29,
ZFMISC_1: 87;
then a
in (the
carrier of G1
/\ the
carrier of G2) by
A30,
XBOOLE_0:def 4;
hence contradiction by
A2;
end;
end;
A31: not x
in (
id the
carrier of (
union_of (G1,G2)))
proof
assume not thesis;
then x
in (
id (the
carrier of G1
\/ the
carrier of G2)) by
NECKLA_2:def 2;
then
A32: x
in ((
id the
carrier of G1)
\/ (
id the
carrier of G2)) by
SYSREL: 14;
per cases by
A32,
XBOOLE_0:def 3;
suppose
A33: x
in (
id the
carrier of G1);
A34: b
in the
carrier of G2 by
A24,
ZFMISC_1: 87;
b
in the
carrier of G1 by
A33,
ZFMISC_1: 87;
then b
in (the
carrier of G1
/\ the
carrier of G2) by
A34,
XBOOLE_0:def 4;
hence contradiction by
A2;
end;
suppose
A35: x
in (
id the
carrier of G2);
A36: a
in the
carrier of G1 by
A23,
ZFMISC_1: 87;
a
in the
carrier of G2 by
A35,
ZFMISC_1: 87;
then a
in (the
carrier of G1
/\ the
carrier of G2) by
A36,
XBOOLE_0:def 4;
hence contradiction by
A2;
end;
end;
x
in (X7
\/ X8) by
A24,
XBOOLE_0:def 3;
then x
in (X5
\/ (X7
\/ X8)) by
XBOOLE_0:def 3;
then x
in ((X5
\/ X7)
\/ X8) by
XBOOLE_1: 4;
then x
in (((X5
\/ X7)
\/ X8)
\/ X6) by
XBOOLE_0:def 3;
then x
in
[:(the
carrier of G1
\/ the
carrier of G2), (the
carrier of G1
\/ the
carrier of G2):] by
ZFMISC_1: 98;
then x
in
[:(the
carrier of G1
\/ the
carrier of G2), the
carrier of (
union_of (G1,G2)):] by
NECKLA_2:def 2;
then x
in
[:the
carrier of (
union_of (G1,G2)), the
carrier of (
union_of (G1,G2)):] by
NECKLA_2:def 2;
then x
in (
[:the
carrier of (
union_of (G1,G2)), the
carrier of (
union_of (G1,G2)):]
\ the
InternalRel of (
union_of (G1,G2))) by
A25,
XBOOLE_0:def 5;
then x
in (the
InternalRel of (
union_of (G1,G2))
` ) by
SUBSET_1:def 4;
then x
in ((the
InternalRel of (
union_of (G1,G2))
` )
\ (
id the
carrier of (
union_of (G1,G2)))) by
A31,
XBOOLE_0:def 5;
hence thesis by
NECKLACE:def 8;
end;
suppose x
in X4;
then
A37: x
in
[:the
carrier of G2, the
carrier of (
ComplRelStr G1):] by
NECKLACE:def 8;
then
A38: x
in
[:the
carrier of G2, the
carrier of G1:] by
NECKLACE:def 8;
A39: not x
in the
InternalRel of (
union_of (G1,G2))
proof
assume not thesis;
then
A40: x
in (the
InternalRel of G1
\/ the
InternalRel of G2) by
NECKLA_2:def 2;
per cases by
A40,
XBOOLE_0:def 3;
suppose
A41: x
in the
InternalRel of G1;
A42: a
in the
carrier of G2 by
A37,
ZFMISC_1: 87;
a
in the
carrier of G1 by
A41,
ZFMISC_1: 87;
then a
in (the
carrier of G1
/\ the
carrier of G2) by
A42,
XBOOLE_0:def 4;
hence contradiction by
A2;
end;
suppose
A43: x
in the
InternalRel of G2;
A44: b
in the
carrier of G1 by
A38,
ZFMISC_1: 87;
b
in the
carrier of G2 by
A43,
ZFMISC_1: 87;
then b
in (the
carrier of G1
/\ the
carrier of G2) by
A44,
XBOOLE_0:def 4;
hence contradiction by
A2;
end;
end;
A45: not x
in (
id the
carrier of (
union_of (G1,G2)))
proof
assume not thesis;
then x
in (
id (the
carrier of G1
\/ the
carrier of G2)) by
NECKLA_2:def 2;
then
A46: x
in ((
id the
carrier of G1)
\/ (
id the
carrier of G2)) by
SYSREL: 14;
per cases by
A46,
XBOOLE_0:def 3;
suppose
A47: x
in (
id the
carrier of G1);
A48: a
in the
carrier of G2 by
A37,
ZFMISC_1: 87;
a
in the
carrier of G1 by
A47,
ZFMISC_1: 87;
then a
in (the
carrier of G1
/\ the
carrier of G2) by
A48,
XBOOLE_0:def 4;
hence contradiction by
A2;
end;
suppose
A49: x
in (
id the
carrier of G2);
A50: b
in the
carrier of G1 by
A38,
ZFMISC_1: 87;
b
in the
carrier of G2 by
A49,
ZFMISC_1: 87;
then b
in (the
carrier of G1
/\ the
carrier of G2) by
A50,
XBOOLE_0:def 4;
hence contradiction by
A2;
end;
end;
x
in (X7
\/ X8) by
A38,
XBOOLE_0:def 3;
then x
in (X5
\/ (X7
\/ X8)) by
XBOOLE_0:def 3;
then x
in ((X5
\/ X7)
\/ X8) by
XBOOLE_1: 4;
then x
in (((X5
\/ X7)
\/ X8)
\/ X6) by
XBOOLE_0:def 3;
then x
in
[:(the
carrier of G1
\/ the
carrier of G2), (the
carrier of G1
\/ the
carrier of G2):] by
ZFMISC_1: 98;
then x
in
[:(the
carrier of G1
\/ the
carrier of G2), the
carrier of (
union_of (G1,G2)):] by
NECKLA_2:def 2;
then x
in
[:the
carrier of (
union_of (G1,G2)), the
carrier of (
union_of (G1,G2)):] by
NECKLA_2:def 2;
then x
in (
[:the
carrier of (
union_of (G1,G2)), the
carrier of (
union_of (G1,G2)):]
\ the
InternalRel of (
union_of (G1,G2))) by
A39,
XBOOLE_0:def 5;
then x
in (the
InternalRel of (
union_of (G1,G2))
` ) by
SUBSET_1:def 4;
then x
in ((the
InternalRel of (
union_of (G1,G2))
` )
\ (
id the
carrier of (
union_of (G1,G2)))) by
A45,
XBOOLE_0:def 5;
hence thesis by
NECKLACE:def 8;
end;
end;
the
carrier of (
ComplRelStr (
union_of (G1,G2)))
= the
carrier of (
union_of (G1,G2)) by
NECKLACE:def 8
.= (the
carrier of G1
\/ the
carrier of G2) by
NECKLA_2:def 2;
hence thesis by
A1,
A3,
RELAT_1:def 2;
end;
theorem ::
NECKLA_3:18
Th18: for G1,G2 be
RelStr st the
carrier of G1
misses the
carrier of G2 holds (
ComplRelStr (
sum_of (G1,G2)))
= (
union_of ((
ComplRelStr G1),(
ComplRelStr G2)))
proof
let G1,G2 be
RelStr;
assume
A1: the
carrier of G1
misses the
carrier of G2;
set P = the
InternalRel of (
ComplRelStr (
sum_of (G1,G2))), R = the
InternalRel of (
union_of ((
ComplRelStr G1),(
ComplRelStr G2))), X1 = the
InternalRel of (
ComplRelStr G1), X2 = the
InternalRel of (
ComplRelStr G2), X5 =
[:the
carrier of G1, the
carrier of G1:], X6 =
[:the
carrier of G2, the
carrier of G2:], X7 =
[:the
carrier of G1, the
carrier of G2:], X8 =
[:the
carrier of G2, the
carrier of G1:];
A2:
[:the
carrier of (
sum_of (G1,G2)), the
carrier of (
sum_of (G1,G2)):]
=
[:(the
carrier of G1
\/ the
carrier of G2), the
carrier of (
sum_of (G1,G2)):] by
NECKLA_2:def 3
.=
[:(the
carrier of G1
\/ the
carrier of G2), (the
carrier of G1
\/ the
carrier of G2):] by
NECKLA_2:def 3
.= (((X5
\/ X7)
\/ X8)
\/ X6) by
ZFMISC_1: 98;
A3: for a,b be
object holds
[a, b]
in P iff
[a, b]
in R
proof
let a,b be
object;
set x =
[a, b];
thus x
in P implies x
in R
proof
assume x
in P;
then
A4: x
in ((the
InternalRel of (
sum_of (G1,G2))
` )
\ (
id the
carrier of (
sum_of (G1,G2)))) by
NECKLACE:def 8;
then x
in ((X5
\/ X7)
\/ X8) or x
in X6 by
A2,
XBOOLE_0:def 3;
then
A5: x
in (X5
\/ X7) or x
in X8 or x
in X6 by
XBOOLE_0:def 3;
x
in (the
InternalRel of (
sum_of (G1,G2))
` ) by
A4,
XBOOLE_0:def 5;
then x
in (
[:the
carrier of (
sum_of (G1,G2)), the
carrier of (
sum_of (G1,G2)):]
\ the
InternalRel of (
sum_of (G1,G2))) by
SUBSET_1:def 4;
then not x
in the
InternalRel of (
sum_of (G1,G2)) by
XBOOLE_0:def 5;
then
A6: not x
in (((the
InternalRel of G1
\/ the
InternalRel of G2)
\/
[:the
carrier of G1, the
carrier of G2:])
\/
[:the
carrier of G2, the
carrier of G1:]) by
NECKLA_2:def 3;
A7: not x
in the
InternalRel of G1 & not x
in the
InternalRel of G2 & not x
in
[:the
carrier of G1, the
carrier of G2:] & not x
in
[:the
carrier of G2, the
carrier of G1:]
proof
assume not thesis;
then x
in (the
InternalRel of G1
\/ the
InternalRel of G2) or x
in
[:the
carrier of G1, the
carrier of G2:] or x
in
[:the
carrier of G2, the
carrier of G1:] by
XBOOLE_0:def 3;
then x
in ((the
InternalRel of G1
\/ the
InternalRel of G2)
\/
[:the
carrier of G1, the
carrier of G2:]) or x
in
[:the
carrier of G2, the
carrier of G1:] by
XBOOLE_0:def 3;
hence contradiction by
A6,
XBOOLE_0:def 3;
end;
not x
in (
id the
carrier of (
sum_of (G1,G2))) by
A4,
XBOOLE_0:def 5;
then not x
in (
id (the
carrier of G1
\/ the
carrier of G2)) by
NECKLA_2:def 3;
then
A8: not x
in ((
id the
carrier of G1)
\/ (
id the
carrier of G2)) by
SYSREL: 14;
then
A9: not x
in (
id the
carrier of G1) by
XBOOLE_0:def 3;
A10: not x
in (
id the
carrier of G2) by
A8,
XBOOLE_0:def 3;
per cases by
A5,
XBOOLE_0:def 3;
suppose x
in X5;
then x
in (X5
\ the
InternalRel of G1) by
A7,
XBOOLE_0:def 5;
then x
in (the
InternalRel of G1
` ) by
SUBSET_1:def 4;
then x
in ((the
InternalRel of G1
` )
\ (
id the
carrier of G1)) by
A9,
XBOOLE_0:def 5;
then x
in X1 by
NECKLACE:def 8;
then x
in (X1
\/ X2) by
XBOOLE_0:def 3;
hence thesis by
NECKLA_2:def 2;
end;
suppose x
in X7;
hence thesis by
A7;
end;
suppose x
in X8;
hence thesis by
A7;
end;
suppose x
in X6;
then x
in (X6
\ the
InternalRel of G2) by
A7,
XBOOLE_0:def 5;
then x
in (the
InternalRel of G2
` ) by
SUBSET_1:def 4;
then x
in ((the
InternalRel of G2
` )
\ (
id the
carrier of G2)) by
A10,
XBOOLE_0:def 5;
then x
in X2 by
NECKLACE:def 8;
then x
in (X1
\/ X2) by
XBOOLE_0:def 3;
hence thesis by
NECKLA_2:def 2;
end;
end;
assume x
in R;
then
A11: x
in (the
InternalRel of (
ComplRelStr G1)
\/ the
InternalRel of (
ComplRelStr G2)) by
NECKLA_2:def 2;
per cases by
A11,
XBOOLE_0:def 3;
suppose x
in the
InternalRel of (
ComplRelStr G1);
then
A12: x
in ((the
InternalRel of G1
` )
\ (
id the
carrier of G1)) by
NECKLACE:def 8;
then
A13: not x
in (
id the
carrier of G1) by
XBOOLE_0:def 5;
A14: not x
in (
id the
carrier of (
sum_of (G1,G2)))
proof
assume not thesis;
then x
in (
id (the
carrier of G1
\/ the
carrier of G2)) by
NECKLA_2:def 3;
then x
in ((
id the
carrier of G1)
\/ (
id the
carrier of G2)) by
SYSREL: 14;
then x
in (
id the
carrier of G2) by
A13,
XBOOLE_0:def 3;
then
A15: a
in the
carrier of G2 by
ZFMISC_1: 87;
a
in the
carrier of G1 by
A12,
ZFMISC_1: 87;
then (the
carrier of G1
/\ the
carrier of G2) is non
empty by
A15,
XBOOLE_0:def 4;
hence contradiction by
A1;
end;
x
in (the
InternalRel of G1
` ) by
A12,
XBOOLE_0:def 5;
then x
in (
[:the
carrier of G1, the
carrier of G1:]
\ the
InternalRel of G1) by
SUBSET_1:def 4;
then
A16: not x
in the
InternalRel of G1 by
XBOOLE_0:def 5;
A17: not x
in the
InternalRel of (
sum_of (G1,G2))
proof
assume not thesis;
then x
in (((the
InternalRel of G1
\/ the
InternalRel of G2)
\/ X7)
\/ X8) by
NECKLA_2:def 3;
then x
in ((the
InternalRel of G1
\/ the
InternalRel of G2)
\/ X7) or x
in X8 by
XBOOLE_0:def 3;
then
A18: x
in (the
InternalRel of G1
\/ the
InternalRel of G2) or x
in X7 or x
in X8 by
XBOOLE_0:def 3;
per cases by
A16,
A18,
XBOOLE_0:def 3;
suppose
A19: x
in the
InternalRel of G2;
A20: a
in the
carrier of G1 by
A12,
ZFMISC_1: 87;
a
in the
carrier of G2 by
A19,
ZFMISC_1: 87;
then (the
carrier of G1
/\ the
carrier of G2) is non
empty by
A20,
XBOOLE_0:def 4;
hence contradiction by
A1;
end;
suppose
A21: x
in X7;
A22: b
in the
carrier of G1 by
A12,
ZFMISC_1: 87;
b
in the
carrier of G2 by
A21,
ZFMISC_1: 87;
then (the
carrier of G1
/\ the
carrier of G2) is non
empty by
A22,
XBOOLE_0:def 4;
hence contradiction by
A1;
end;
suppose
A23: x
in X8;
A24: a
in the
carrier of G1 by
A12,
ZFMISC_1: 87;
a
in the
carrier of G2 by
A23,
ZFMISC_1: 87;
then (the
carrier of G1
/\ the
carrier of G2) is non
empty by
A24,
XBOOLE_0:def 4;
hence contradiction by
A1;
end;
end;
x
in (X5
\/ ((X7
\/ X8)
\/ X6)) by
A12,
XBOOLE_0:def 3;
then x
in
[:the
carrier of (
sum_of (G1,G2)), the
carrier of (
sum_of (G1,G2)):] by
A2,
XBOOLE_1: 113;
then x
in (
[:the
carrier of (
sum_of (G1,G2)), the
carrier of (
sum_of (G1,G2)):]
\ the
InternalRel of (
sum_of (G1,G2))) by
A17,
XBOOLE_0:def 5;
then x
in (the
InternalRel of (
sum_of (G1,G2))
` ) by
SUBSET_1:def 4;
then x
in ((the
InternalRel of (
sum_of (G1,G2))
` )
\ (
id the
carrier of (
sum_of (G1,G2)))) by
A14,
XBOOLE_0:def 5;
hence thesis by
NECKLACE:def 8;
end;
suppose x
in the
InternalRel of (
ComplRelStr G2);
then
A25: x
in ((the
InternalRel of G2
` )
\ (
id the
carrier of G2)) by
NECKLACE:def 8;
then
A26: not x
in (
id the
carrier of G2) by
XBOOLE_0:def 5;
A27: not x
in (
id the
carrier of (
sum_of (G1,G2)))
proof
assume not thesis;
then x
in (
id (the
carrier of G1
\/ the
carrier of G2)) by
NECKLA_2:def 3;
then x
in ((
id the
carrier of G1)
\/ (
id the
carrier of G2)) by
SYSREL: 14;
then x
in (
id the
carrier of G1) by
A26,
XBOOLE_0:def 3;
then
A28: b
in the
carrier of G1 by
ZFMISC_1: 87;
b
in the
carrier of G2 by
A25,
ZFMISC_1: 87;
then (the
carrier of G1
/\ the
carrier of G2) is non
empty by
A28,
XBOOLE_0:def 4;
hence contradiction by
A1;
end;
x
in (the
InternalRel of G2
` ) by
A25,
XBOOLE_0:def 5;
then x
in (
[:the
carrier of G2, the
carrier of G2:]
\ the
InternalRel of G2) by
SUBSET_1:def 4;
then
A29: not x
in the
InternalRel of G2 by
XBOOLE_0:def 5;
A30: not x
in the
InternalRel of (
sum_of (G1,G2))
proof
assume not thesis;
then x
in (((the
InternalRel of G1
\/ the
InternalRel of G2)
\/ X7)
\/ X8) by
NECKLA_2:def 3;
then x
in ((the
InternalRel of G1
\/ the
InternalRel of G2)
\/ X7) or x
in X8 by
XBOOLE_0:def 3;
then
A31: x
in (the
InternalRel of G1
\/ the
InternalRel of G2) or x
in X7 or x
in X8 by
XBOOLE_0:def 3;
per cases by
A29,
A31,
XBOOLE_0:def 3;
suppose
A32: x
in the
InternalRel of G1;
A33: a
in the
carrier of G2 by
A25,
ZFMISC_1: 87;
a
in the
carrier of G1 by
A32,
ZFMISC_1: 87;
then (the
carrier of G1
/\ the
carrier of G2) is non
empty by
A33,
XBOOLE_0:def 4;
hence contradiction by
A1;
end;
suppose
A34: x
in X7;
A35: a
in the
carrier of G2 by
A25,
ZFMISC_1: 87;
a
in the
carrier of G1 by
A34,
ZFMISC_1: 87;
then (the
carrier of G1
/\ the
carrier of G2) is non
empty by
A35,
XBOOLE_0:def 4;
hence contradiction by
A1;
end;
suppose
A36: x
in X8;
A37: b
in the
carrier of G2 by
A25,
ZFMISC_1: 87;
b
in the
carrier of G1 by
A36,
ZFMISC_1: 87;
then (the
carrier of G1
/\ the
carrier of G2) is non
empty by
A37,
XBOOLE_0:def 4;
hence contradiction by
A1;
end;
end;
x
in
[:the
carrier of (
sum_of (G1,G2)), the
carrier of (
sum_of (G1,G2)):] by
A2,
A25,
XBOOLE_0:def 3;
then x
in (
[:the
carrier of (
sum_of (G1,G2)), the
carrier of (
sum_of (G1,G2)):]
\ the
InternalRel of (
sum_of (G1,G2))) by
A30,
XBOOLE_0:def 5;
then x
in (the
InternalRel of (
sum_of (G1,G2))
` ) by
SUBSET_1:def 4;
then x
in ((the
InternalRel of (
sum_of (G1,G2))
` )
\ (
id the
carrier of (
sum_of (G1,G2)))) by
A27,
XBOOLE_0:def 5;
hence thesis by
NECKLACE:def 8;
end;
end;
A38: the
carrier of (
union_of ((
ComplRelStr G1),(
ComplRelStr G2)))
= (the
carrier of (
ComplRelStr G1)
\/ the
carrier of (
ComplRelStr G2)) by
NECKLA_2:def 2
.= (the
carrier of G1
\/ the
carrier of (
ComplRelStr G2)) by
NECKLACE:def 8
.= (the
carrier of G1
\/ the
carrier of G2) by
NECKLACE:def 8;
the
carrier of (
ComplRelStr (
sum_of (G1,G2)))
= the
carrier of (
sum_of (G1,G2)) by
NECKLACE:def 8
.= (the
carrier of G1
\/ the
carrier of G2) by
NECKLA_2:def 3;
hence thesis by
A38,
A3,
RELAT_1:def 2;
end;
theorem ::
NECKLA_3:19
for G be
RelStr, H be
full
SubRelStr of G holds the
InternalRel of (
ComplRelStr H)
= (the
InternalRel of (
ComplRelStr G)
|_2 the
carrier of (
ComplRelStr H))
proof
let G be
RelStr, H be
full
SubRelStr of G;
set IH = the
InternalRel of H, ICmpH = the
InternalRel of (
ComplRelStr H), cH = the
carrier of H, IG = the
InternalRel of G, cG = the
carrier of G, ICmpG = the
InternalRel of (
ComplRelStr G);
A1: ICmpH
= ((IH
` )
\ (
id cH)) by
NECKLACE:def 8
.= ((
[:cH, cH:]
\ IH)
\ (
id cH)) by
SUBSET_1:def 4;
A2: ICmpG
= ((IG
` )
\ (
id cG)) by
NECKLACE:def 8
.= ((
[:cG, cG:]
\ IG)
\ (
id cG)) by
SUBSET_1:def 4;
A3: cH
c= cG by
YELLOW_0:def 13;
(ICmpG
|_2 the
carrier of (
ComplRelStr H))
= (ICmpG
|_2 cH) by
NECKLACE:def 8
.= (((
[:cG, cG:]
\ IG)
/\
[:cH, cH:])
\ ((
id cG)
/\
[:cH, cH:])) by
A2,
XBOOLE_1: 50
.= (((
[:cG, cG:]
/\
[:cH, cH:])
\ (IG
/\
[:cH, cH:]))
\ ((
id cG)
/\
[:cH, cH:])) by
XBOOLE_1: 50
.= (((
[:cG, cG:]
/\
[:cH, cH:])
\ (IG
/\
[:cH, cH:]))
\ ((
id cG)
| cH)) by
Th1
.= (((
[:cG, cG:]
/\
[:cH, cH:])
\ (IG
|_2 cH))
\ (
id cH)) by
A3,
FUNCT_3: 1
.= (((
[:cG, cG:]
/\
[:cH, cH:])
\ IH)
\ (
id cH)) by
YELLOW_0:def 14
.= ((
[:(cG
/\ cH), (cG
/\ cH):]
\ IH)
\ (
id cH)) by
ZFMISC_1: 100
.= ((
[:cH, (cG
/\ cH):]
\ IH)
\ (
id cH)) by
A3,
XBOOLE_1: 28
.= ICmpH by
A1,
A3,
XBOOLE_1: 28;
hence thesis;
end;
theorem ::
NECKLA_3:20
Th20: for G be non
empty
irreflexive
RelStr, x be
Element of G, x9 be
Element of (
ComplRelStr G) st x
= x9 holds (
ComplRelStr (
subrelstr ((
[#] G)
\
{x})))
= (
subrelstr ((
[#] (
ComplRelStr G))
\
{x9}))
proof
let G be non
empty
irreflexive
RelStr, x be
Element of G, x9 be
Element of (
ComplRelStr G);
assume
A1: x
= x9;
set R = (
subrelstr ((
[#] G)
\
{x})), cR = the
carrier of R, cG = the
carrier of G;
A2: (
[#] (
ComplRelStr G))
= cG by
NECKLACE:def 8;
A3:
[:(cG
\
{x}), (cG
\
{x}):]
=
[:cR, ((
[#] G)
\
{x}):] by
YELLOW_0:def 15
.=
[:cR, cR:] by
YELLOW_0:def 15;
A4: cR
c= cG by
YELLOW_0:def 13;
A5: the
InternalRel of (
subrelstr ((
[#] (
ComplRelStr G))
\
{x9}))
= (the
InternalRel of (
ComplRelStr G)
|_2 the
carrier of (
subrelstr ((
[#] (
ComplRelStr G))
\
{x9}))) by
YELLOW_0:def 14
.= (the
InternalRel of (
ComplRelStr G)
|_2 (cG
\
{x})) by
A1,
A2,
YELLOW_0:def 15
.= (((the
InternalRel of G
` )
\ (
id cG))
/\
[:(cG
\
{x}), (cG
\
{x}):]) by
NECKLACE:def 8
.= ((
[:cR, cR:]
/\ (the
InternalRel of G
` ))
\ (
id cG)) by
A3,
XBOOLE_1: 49
.= ((
[:cR, cR:]
/\ (
[:cG, cG:]
\ the
InternalRel of G))
\ (
id cG)) by
SUBSET_1:def 4
.= (((
[:cR, cR:]
/\
[:cG, cG:])
\ the
InternalRel of G)
\ (
id cG)) by
XBOOLE_1: 49
.= ((
[:cR, cR:]
\ the
InternalRel of G)
\ (
id cG)) by
A4,
XBOOLE_1: 28,
ZFMISC_1: 96;
A6: the
InternalRel of (
ComplRelStr R)
= ((the
InternalRel of R
` )
\ (
id cR)) by
NECKLACE:def 8
.= ((
[:cR, cR:]
\ the
InternalRel of R)
\ (
id cR)) by
SUBSET_1:def 4
.= ((
[:cR, cR:]
\ (the
InternalRel of G
|_2 cR))
\ (
id cR)) by
YELLOW_0:def 14
.= (((
[:cR, cR:]
\ the
InternalRel of G)
\/ (
[:cR, cR:]
\
[:cR, cR:]))
\ (
id cR)) by
XBOOLE_1: 54
.= (((
[:cR, cR:]
\ the
InternalRel of G)
\/
{} )
\ (
id cR)) by
XBOOLE_1: 37
.= ((
[:cR, cR:]
\ the
InternalRel of G)
\ (
id cR));
A7:
[:cR, cR:]
= (
[:(
[#] G), ((
[#] G)
\
{x}):]
\
[:
{x}, ((
[#] G)
\
{x}):]) by
A3,
ZFMISC_1: 102
.= ((
[:(
[#] G), (
[#] G):]
\
[:(
[#] G),
{x}:])
\
[:
{x}, ((
[#] G)
\
{x}):]) by
ZFMISC_1: 102
.= ((
[:cG, cG:]
\
[:cG,
{x}:])
\ (
[:
{x}, cG:]
\
[:
{x},
{x}:])) by
ZFMISC_1: 102
.= (((
[:cG, cG:]
\
[:cG,
{x}:])
\
[:
{x}, cG:])
\/ ((
[:cG, cG:]
\
[:cG,
{x}:])
/\
[:
{x},
{x}:])) by
XBOOLE_1: 52
.= ((
[:cG, cG:]
\ (
[:cG,
{x}:]
\/
[:
{x}, cG:]))
\/ ((
[:cG, cG:]
\
[:cG,
{x}:])
/\
[:
{x},
{x}:])) by
XBOOLE_1: 41;
A8: the
InternalRel of (
subrelstr ((
[#] (
ComplRelStr G))
\
{x9}))
= the
InternalRel of (
ComplRelStr R)
proof
thus the
InternalRel of (
subrelstr ((
[#] (
ComplRelStr G))
\
{x9}))
c= the
InternalRel of (
ComplRelStr R)
proof
let a be
object;
assume
A9: a
in the
InternalRel of (
subrelstr ((
[#] (
ComplRelStr G))
\
{x9}));
then
A10: not a
in (
id cG) by
A5,
XBOOLE_0:def 5;
A11: not a
in (
id cR)
proof
assume
A12: not thesis;
then
consider x2,y2 be
object such that
A13: a
=
[x2, y2] and
A14: x2
in cR and y2
in cR by
RELSET_1: 2;
A15: x2
in (cG
\
{x}) by
A14,
YELLOW_0:def 15;
x2
= y2 by
A12,
A13,
RELAT_1:def 10;
hence contradiction by
A10,
A13,
A15,
RELAT_1:def 10;
end;
a
in (
[:cR, cR:]
\ the
InternalRel of G) by
A5,
A9,
XBOOLE_0:def 5;
hence thesis by
A6,
A11,
XBOOLE_0:def 5;
end;
let a be
object;
assume
A16: a
in the
InternalRel of (
ComplRelStr R);
then not a
in (
id cR) by
A6,
XBOOLE_0:def 5;
then not a
in (
id (cG
\
{x})) by
YELLOW_0:def 15;
then
A17: not a
in ((
id cG)
\ (
id
{x})) by
SYSREL: 14;
per cases by
A17,
XBOOLE_0:def 5;
suppose
A18: not a
in (
id cG);
a
in (
[:cR, cR:]
\ the
InternalRel of G) by
A6,
A16,
XBOOLE_0:def 5;
hence thesis by
A5,
A18,
XBOOLE_0:def 5;
end;
suppose a
in (
id
{x});
then
A19: a
in
{
[x, x]} by
SYSREL: 13;
thus thesis
proof
per cases by
A7,
A6,
A16,
XBOOLE_0:def 3;
suppose
A20: a
in (
[:cG, cG:]
\ (
[:cG,
{x}:]
\/
[:
{x}, cG:]));
x
in
{x} by
TARSKI:def 1;
then
A21:
[x, x]
in
[:
{x}, cG:] by
ZFMISC_1: 87;
not a
in (
[:cG,
{x}:]
\/
[:
{x}, cG:]) by
A20,
XBOOLE_0:def 5;
then not a
in
[:
{x}, cG:] by
XBOOLE_0:def 3;
hence thesis by
A19,
A21,
TARSKI:def 1;
end;
suppose
A22: a
in ((
[:cG, cG:]
\
[:cG,
{x}:])
/\
[:
{x},
{x}:]);
x
in
{x} by
TARSKI:def 1;
then
A23:
[x, x]
in
[:cG,
{x}:] by
ZFMISC_1: 87;
a
in (
[:cG, cG:]
\
[:cG,
{x}:]) by
A22,
XBOOLE_0:def 4;
then not a
in
[:cG,
{x}:] by
XBOOLE_0:def 5;
hence thesis by
A19,
A23,
TARSKI:def 1;
end;
end;
end;
end;
the
carrier of (
ComplRelStr (
subrelstr ((
[#] G)
\
{x})))
= the
carrier of (
subrelstr ((
[#] G)
\
{x})) by
NECKLACE:def 8
.= (the
carrier of G
\
{x}) by
YELLOW_0:def 15
.= ((
[#] (
ComplRelStr G))
\
{x9}) by
A1,
NECKLACE:def 8
.= the
carrier of (
subrelstr ((
[#] (
ComplRelStr G))
\
{x9})) by
YELLOW_0:def 15;
hence thesis by
A8;
end;
begin
registration
cluster
trivial
strict ->
N-free for non
empty
RelStr;
correctness
proof
set Y = (
Necklace 4);
let R be non
empty
RelStr;
assume R is
trivial
strict;
then
consider y be
object such that
A1: the
carrier of R
=
{y} by
GROUP_6:def 2;
assume not R is
N-free;
then R
embeds Y by
NECKLA_2:def 1;
then
consider f be
Function of Y, R such that
A2: f is
one-to-one and for x,y be
Element of Y holds
[x, y]
in the
InternalRel of Y iff
[(f
. x), (f
. y)]
in the
InternalRel of R;
A3: (
dom f)
= the
carrier of Y by
FUNCT_2:def 1
.=
{
0 , 1, 2, 3} by
NECKLACE: 1,
NECKLACE: 20;
then
A4: 1
in (
dom f) by
ENUMSET1:def 2;
then (f
. 1)
in
{y} by
A1,
PARTFUN1: 4;
then
A5: (f
. 1)
= y by
TARSKI:def 1;
A6:
0
in (
dom f) by
A3,
ENUMSET1:def 2;
then (f
.
0 )
in
{y} by
A1,
PARTFUN1: 4;
then (f
.
0 )
= y by
TARSKI:def 1;
hence contradiction by
A2,
A6,
A4,
A5,
FUNCT_1:def 4;
end;
end
theorem ::
NECKLA_3:21
for R be
reflexive
antisymmetric
RelStr, S be
RelStr holds (ex f be
Function of R, S st for x,y be
Element of R holds
[x, y]
in the
InternalRel of R iff
[(f
. x), (f
. y)]
in the
InternalRel of S) iff S
embeds R
proof
let R be
reflexive
antisymmetric
RelStr, S be
RelStr;
A1:
now
assume ex f be
Function of R, S st for x,y be
Element of R holds
[x, y]
in the
InternalRel of R iff
[(f
. x), (f
. y)]
in the
InternalRel of S;
then
consider f be
Function of R, S such that
A2: for x,y be
Element of R holds
[x, y]
in the
InternalRel of R iff
[(f
. x), (f
. y)]
in the
InternalRel of S;
for x1,x2 be
object st x1
in (
dom f) & x2
in (
dom f) & (f
. x1)
= (f
. x2) holds x1
= x2
proof
let x1,x2 be
object;
assume that
A3: x1
in (
dom f) and
A4: x2
in (
dom f) and
A5: (f
. x1)
= (f
. x2);
reconsider x1, x2 as
Element of R by
A3,
A4;
A6: the
InternalRel of R
is_reflexive_in the
carrier of R by
ORDERS_2:def 2;
then
[x2, x2]
in the
InternalRel of R by
A3;
then
[(f
. x2), (f
. x1)]
in the
InternalRel of S by
A2,
A5;
then
[x2, x1]
in the
InternalRel of R by
A2;
then
A7: x2
<= x1 by
ORDERS_2:def 5;
[x1, x1]
in the
InternalRel of R by
A3,
A6;
then
[(f
. x1), (f
. x2)]
in the
InternalRel of S by
A2,
A5;
then
[x1, x2]
in the
InternalRel of R by
A2;
then x1
<= x2 by
ORDERS_2:def 5;
hence thesis by
A7,
ORDERS_2: 2;
end;
then f is
one-to-one by
FUNCT_1:def 4;
hence S
embeds R by
A2;
end;
thus thesis by
A1;
end;
theorem ::
NECKLA_3:22
Th22: for G be non
empty
RelStr, H be non
empty
full
SubRelStr of G holds G
embeds H
proof
let G be non
empty
RelStr;
let H be non
empty
full
SubRelStr of G;
reconsider f = (
id the
carrier of H) as
Function of the
carrier of H, the
carrier of H;
A1: (
dom f)
= the
carrier of H;
A2: the
carrier of H
c= the
carrier of G by
YELLOW_0:def 13;
for x be
object st x
in the
carrier of H holds (f
. x)
in the
carrier of G
proof
let x be
object;
assume x
in the
carrier of H;
then (f
. x)
in the
carrier of H by
FUNCT_1: 17;
hence thesis by
A2;
end;
then
reconsider f = (
id the
carrier of H) as
Function of the
carrier of H, the
carrier of G by
A1,
FUNCT_2: 3;
reconsider f as
Function of H, G;
for x,y be
Element of H holds
[x, y]
in the
InternalRel of H iff
[(f
. x), (f
. y)]
in the
InternalRel of G
proof
set IH = the
InternalRel of H, IG = the
InternalRel of G, cH = the
carrier of H;
let x,y be
Element of H;
thus
[x, y]
in IH implies
[(f
. x), (f
. y)]
in IG
proof
assume
[x, y]
in IH;
then
[x, y]
in (IG
|_2 cH) by
YELLOW_0:def 14;
hence thesis by
XBOOLE_0:def 4;
end;
assume
[(f
. x), (f
. y)]
in IG;
then
[x, y]
in (IG
|_2 cH) by
XBOOLE_0:def 4;
hence thesis by
YELLOW_0:def 14;
end;
hence thesis;
end;
theorem ::
NECKLA_3:23
Th23: for G be non
empty
RelStr, H be non
empty
full
SubRelStr of G st G is
N-free holds H is
N-free
proof
let G be non
empty
RelStr, H be non
empty
full
SubRelStr of G;
assume
A1: G is
N-free;
A2: G
embeds H by
Th22;
assume not thesis;
then H
embeds (
Necklace 4) by
NECKLA_2:def 1;
then G
embeds (
Necklace 4) by
A2,
NECKLACE: 12;
hence contradiction by
A1,
NECKLA_2:def 1;
end;
theorem ::
NECKLA_3:24
Th24: for G be non
empty
irreflexive
RelStr holds G
embeds (
Necklace 4) iff (
ComplRelStr G)
embeds (
Necklace 4)
proof
let G be non
empty
irreflexive
RelStr;
set N4 = (
Necklace 4), CmpN4 = (
ComplRelStr (
Necklace 4)), CmpG = (
ComplRelStr G);
A1: the
carrier of CmpG
= the
carrier of G by
NECKLACE:def 8;
A2: the
carrier of (
Necklace 4)
=
{
0 , 1, 2, 3} by
NECKLACE: 1,
NECKLACE: 20;
then
A3:
0
in the
carrier of N4 by
ENUMSET1:def 2;
A4: the
carrier of CmpN4
= the
carrier of N4 by
NECKLACE:def 8;
thus G
embeds N4 implies CmpG
embeds N4
proof
(CmpN4,N4)
are_isomorphic by
NECKLACE: 29,
WAYBEL_1: 6;
then
consider g be
Function of CmpN4, N4 such that
A5: g is
isomorphic by
WAYBEL_1:def 8;
assume G
embeds (
Necklace 4);
then
consider f be
Function of N4, G such that
A6: f is
one-to-one and
A7: for x,y be
Element of N4 holds
[x, y]
in the
InternalRel of N4 iff
[(f
. x), (f
. y)]
in the
InternalRel of G;
reconsider h = (f
* g) as
Function of CmpN4, G;
A8: g is
one-to-one
monotone by
A5,
WAYBEL_0:def 38;
A9: for x,y be
Element of CmpN4 holds
[x, y]
in the
InternalRel of CmpN4 iff
[(h
. x), (h
. y)]
in the
InternalRel of G
proof
let x,y be
Element of CmpN4;
thus
[x, y]
in the
InternalRel of CmpN4 implies
[(h
. x), (h
. y)]
in the
InternalRel of G
proof
assume
[x, y]
in the
InternalRel of CmpN4;
then x
<= y by
ORDERS_2:def 5;
then (g
. x)
<= (g
. y) by
A8,
WAYBEL_1:def 2;
then
[(g
. x), (g
. y)]
in the
InternalRel of N4 by
ORDERS_2:def 5;
then
[(f
. (g
. x)), (f
. (g
. y))]
in the
InternalRel of G by
A7;
then
[((f
* g)
. x), (f
. (g
. y))]
in the
InternalRel of G by
FUNCT_2: 15;
hence thesis by
FUNCT_2: 15;
end;
assume
[(h
. x), (h
. y)]
in the
InternalRel of G;
then
[(f
. (g
. x)), (h
. y)]
in the
InternalRel of G by
FUNCT_2: 15;
then
[(f
. (g
. x)), (f
. (g
. y))]
in the
InternalRel of G by
FUNCT_2: 15;
then
[(g
. x), (g
. y)]
in the
InternalRel of N4 by
A7;
then (g
. x)
<= (g
. y) by
ORDERS_2:def 5;
then x
<= y by
A5,
WAYBEL_0: 66;
hence thesis by
ORDERS_2:def 5;
end;
A10:
0
in the
carrier of CmpN4 by
A2,
A4,
ENUMSET1:def 2;
A11: 1
in the
carrier of CmpN4 by
A2,
A4,
ENUMSET1:def 2;
A12: (
dom h)
= the
carrier of CmpN4 by
FUNCT_2:def 1;
A13:
[(h
.
0 ), (h
. 1)]
in the
InternalRel of CmpG
proof
assume
A14: not thesis;
[(h
.
0 ), (h
. 1)]
in the
InternalRel of G
proof
(h
.
0 )
in the
carrier of G & (h
. 1)
in the
carrier of G by
A10,
A11,
FUNCT_2: 5;
then
[(h
.
0 ), (h
. 1)]
in
[:the
carrier of G, the
carrier of G:] by
ZFMISC_1: 87;
then
[(h
.
0 ), (h
. 1)]
in (((
id the
carrier of G)
\/ the
InternalRel of G)
\/ the
InternalRel of CmpG) by
Th14;
then
A15:
[(h
.
0 ), (h
. 1)]
in ((
id the
carrier of G)
\/ the
InternalRel of G) or
[(h
.
0 ), (h
. 1)]
in the
InternalRel of CmpG by
XBOOLE_0:def 3;
assume not thesis;
then
[(h
.
0 ), (h
. 1)]
in (
id the
carrier of G) by
A14,
A15,
XBOOLE_0:def 3;
then (h
.
0 )
= (h
. 1) by
RELAT_1:def 10;
hence contradiction by
A6,
A8,
A12,
A10,
A11,
FUNCT_1:def 4;
end;
then
A16:
[
0 , 1]
in the
InternalRel of CmpN4 by
A9,
A10,
A11;
[
0 , 1]
in the
InternalRel of N4 by
ENUMSET1:def 4,
NECKLA_2: 2;
then
[
0 , 1]
in (the
InternalRel of N4
/\ the
InternalRel of CmpN4) by
A16,
XBOOLE_0:def 4;
then the
InternalRel of N4
meets the
InternalRel of CmpN4;
hence thesis by
Th12;
end;
A17: 2
in the
carrier of CmpN4 by
A2,
A4,
ENUMSET1:def 2;
A18:
[(h
. 1), (h
. 2)]
in the
InternalRel of CmpG
proof
assume
A19: not thesis;
[(h
. 1), (h
. 2)]
in the
InternalRel of G
proof
(h
. 1)
in the
carrier of G & (h
. 2)
in the
carrier of G by
A11,
A17,
FUNCT_2: 5;
then
[(h
. 1), (h
. 2)]
in
[:the
carrier of G, the
carrier of G:] by
ZFMISC_1: 87;
then
[(h
. 1), (h
. 2)]
in (((
id the
carrier of G)
\/ the
InternalRel of G)
\/ the
InternalRel of CmpG) by
Th14;
then
A20:
[(h
. 1), (h
. 2)]
in ((
id the
carrier of G)
\/ the
InternalRel of G) or
[(h
. 1), (h
. 2)]
in the
InternalRel of CmpG by
XBOOLE_0:def 3;
assume not thesis;
then
[(h
. 1), (h
. 2)]
in (
id the
carrier of G) by
A19,
A20,
XBOOLE_0:def 3;
then (h
. 1)
= (h
. 2) by
RELAT_1:def 10;
hence contradiction by
A6,
A8,
A12,
A11,
A17,
FUNCT_1:def 4;
end;
then
A21:
[1, 2]
in the
InternalRel of CmpN4 by
A9,
A11,
A17;
[1, 2]
in the
InternalRel of N4 by
ENUMSET1:def 4,
NECKLA_2: 2;
then
[1, 2]
in (the
InternalRel of N4
/\ the
InternalRel of CmpN4) by
A21,
XBOOLE_0:def 4;
then the
InternalRel of N4
meets the
InternalRel of CmpN4;
hence thesis by
Th12;
end;
A22: 3
in the
carrier of CmpN4 by
A2,
A4,
ENUMSET1:def 2;
A23:
[(h
. 2), (h
. 3)]
in the
InternalRel of CmpG
proof
assume
A24: not thesis;
[(h
. 2), (h
. 3)]
in the
InternalRel of G
proof
(h
. 2)
in the
carrier of G & (h
. 3)
in the
carrier of G by
A17,
A22,
FUNCT_2: 5;
then
[(h
. 2), (h
. 3)]
in
[:the
carrier of G, the
carrier of G:] by
ZFMISC_1: 87;
then
[(h
. 2), (h
. 3)]
in (((
id the
carrier of G)
\/ the
InternalRel of G)
\/ the
InternalRel of CmpG) by
Th14;
then
A25:
[(h
. 2), (h
. 3)]
in ((
id the
carrier of G)
\/ the
InternalRel of G) or
[(h
. 2), (h
. 3)]
in the
InternalRel of CmpG by
XBOOLE_0:def 3;
assume not thesis;
then
[(h
. 2), (h
. 3)]
in (
id the
carrier of G) by
A24,
A25,
XBOOLE_0:def 3;
then (h
. 2)
= (h
. 3) by
RELAT_1:def 10;
hence contradiction by
A6,
A8,
A12,
A17,
A22,
FUNCT_1:def 4;
end;
then
A26:
[2, 3]
in the
InternalRel of CmpN4 by
A9,
A17,
A22;
[2, 3]
in the
InternalRel of N4 by
ENUMSET1:def 4,
NECKLA_2: 2;
then
[2, 3]
in (the
InternalRel of N4
/\ the
InternalRel of CmpN4) by
A26,
XBOOLE_0:def 4;
then the
InternalRel of N4
meets the
InternalRel of CmpN4;
hence thesis by
Th12;
end;
[3, 1]
in the
InternalRel of CmpN4 by
Th11,
ENUMSET1:def 4;
then
A27:
[(h
. 3), (h
. 1)]
in the
InternalRel of G by
A9,
A11,
A22;
[1, 3]
in the
InternalRel of CmpN4 by
Th11,
ENUMSET1:def 4;
then
A28:
[(h
. 1), (h
. 3)]
in the
InternalRel of G by
A9,
A11,
A22;
[3,
0 ]
in the
InternalRel of CmpN4 by
Th11,
ENUMSET1:def 4;
then
A29:
[(h
. 3), (h
.
0 )]
in the
InternalRel of G by
A9,
A10,
A22;
[
0 , 3]
in the
InternalRel of CmpN4 by
Th11,
ENUMSET1:def 4;
then
A30:
[(h
.
0 ), (h
. 3)]
in the
InternalRel of G by
A9,
A10,
A22;
A31:
[(h
. 1), (h
.
0 )]
in the
InternalRel of CmpG
proof
assume
A32: not thesis;
[(h
. 1), (h
.
0 )]
in the
InternalRel of G
proof
(h
.
0 )
in the
carrier of G & (h
. 1)
in the
carrier of G by
A10,
A11,
FUNCT_2: 5;
then
[(h
. 1), (h
.
0 )]
in
[:the
carrier of G, the
carrier of G:] by
ZFMISC_1: 87;
then
[(h
. 1), (h
.
0 )]
in (((
id the
carrier of G)
\/ the
InternalRel of G)
\/ the
InternalRel of CmpG) by
Th14;
then
A33:
[(h
. 1), (h
.
0 )]
in ((
id the
carrier of G)
\/ the
InternalRel of G) or
[(h
. 1), (h
.
0 )]
in the
InternalRel of CmpG by
XBOOLE_0:def 3;
assume not thesis;
then
[(h
. 1), (h
.
0 )]
in (
id the
carrier of G) by
A32,
A33,
XBOOLE_0:def 3;
then (h
.
0 )
= (h
. 1) by
RELAT_1:def 10;
hence contradiction by
A6,
A8,
A12,
A10,
A11,
FUNCT_1:def 4;
end;
then
A34:
[1,
0 ]
in the
InternalRel of CmpN4 by
A9,
A10,
A11;
[1,
0 ]
in the
InternalRel of N4 by
ENUMSET1:def 4,
NECKLA_2: 2;
then
[1,
0 ]
in (the
InternalRel of N4
/\ the
InternalRel of CmpN4) by
A34,
XBOOLE_0:def 4;
then the
InternalRel of N4
meets the
InternalRel of CmpN4;
hence thesis by
Th12;
end;
A35:
[(h
. 2), (h
. 1)]
in the
InternalRel of CmpG
proof
assume
A36: not thesis;
[(h
. 2), (h
. 1)]
in the
InternalRel of G
proof
(h
. 1)
in the
carrier of G & (h
. 2)
in the
carrier of G by
A11,
A17,
FUNCT_2: 5;
then
[(h
. 2), (h
. 1)]
in
[:the
carrier of G, the
carrier of G:] by
ZFMISC_1: 87;
then
[(h
. 2), (h
. 1)]
in (((
id the
carrier of G)
\/ the
InternalRel of G)
\/ the
InternalRel of CmpG) by
Th14;
then
A37:
[(h
. 2), (h
. 1)]
in ((
id the
carrier of G)
\/ the
InternalRel of G) or
[(h
. 2), (h
. 1)]
in the
InternalRel of CmpG by
XBOOLE_0:def 3;
assume not thesis;
then
[(h
. 2), (h
. 1)]
in (
id the
carrier of G) by
A36,
A37,
XBOOLE_0:def 3;
then (h
. 1)
= (h
. 2) by
RELAT_1:def 10;
hence contradiction by
A6,
A8,
A12,
A11,
A17,
FUNCT_1:def 4;
end;
then
A38:
[2, 1]
in the
InternalRel of CmpN4 by
A9,
A11,
A17;
[2, 1]
in the
InternalRel of N4 by
ENUMSET1:def 4,
NECKLA_2: 2;
then
[2, 1]
in (the
InternalRel of N4
/\ the
InternalRel of CmpN4) by
A38,
XBOOLE_0:def 4;
then the
InternalRel of N4
meets the
InternalRel of CmpN4;
hence thesis by
Th12;
end;
A39:
[(h
. 3), (h
. 2)]
in the
InternalRel of CmpG
proof
assume
A40: not thesis;
[(h
. 3), (h
. 2)]
in the
InternalRel of G
proof
(h
. 2)
in the
carrier of G & (h
. 3)
in the
carrier of G by
A17,
A22,
FUNCT_2: 5;
then
[(h
. 3), (h
. 2)]
in
[:the
carrier of G, the
carrier of G:] by
ZFMISC_1: 87;
then
[(h
. 3), (h
. 2)]
in (((
id the
carrier of G)
\/ the
InternalRel of G)
\/ the
InternalRel of CmpG) by
Th14;
then
A41:
[(h
. 3), (h
. 2)]
in ((
id the
carrier of G)
\/ the
InternalRel of G) or
[(h
. 3), (h
. 2)]
in the
InternalRel of CmpG by
XBOOLE_0:def 3;
assume not thesis;
then
[(h
. 3), (h
. 2)]
in (
id the
carrier of G) by
A40,
A41,
XBOOLE_0:def 3;
then (h
. 2)
= (h
. 3) by
RELAT_1:def 10;
hence contradiction by
A6,
A8,
A12,
A17,
A22,
FUNCT_1:def 4;
end;
then
A42:
[3, 2]
in the
InternalRel of CmpN4 by
A9,
A17,
A22;
[3, 2]
in the
InternalRel of N4 by
ENUMSET1:def 4,
NECKLA_2: 2;
then
[3, 2]
in (the
InternalRel of N4
/\ the
InternalRel of CmpN4) by
A42,
XBOOLE_0:def 4;
then the
InternalRel of N4
meets the
InternalRel of CmpN4;
hence thesis by
Th12;
end;
[2,
0 ]
in the
InternalRel of CmpN4 by
Th11,
ENUMSET1:def 4;
then
A43:
[(h
. 2), (h
.
0 )]
in the
InternalRel of G by
A9,
A10,
A17;
[
0 , 2]
in the
InternalRel of CmpN4 by
Th11,
ENUMSET1:def 4;
then
A44:
[(h
.
0 ), (h
. 2)]
in the
InternalRel of G by
A9,
A10,
A17;
for x,y be
Element of N4 holds
[x, y]
in the
InternalRel of N4 iff
[(h
. x), (h
. y)]
in the
InternalRel of CmpG
proof
let x,y be
Element of N4;
thus
[x, y]
in the
InternalRel of N4 implies
[(h
. x), (h
. y)]
in the
InternalRel of CmpG
proof
assume
A45:
[x, y]
in the
InternalRel of N4;
per cases by
A45,
ENUMSET1:def 4,
NECKLA_2: 2;
suppose
A46:
[x, y]
=
[
0 , 1];
then x
=
0 by
XTUPLE_0: 1;
hence thesis by
A13,
A46,
XTUPLE_0: 1;
end;
suppose
A47:
[x, y]
=
[1,
0 ];
then x
= 1 by
XTUPLE_0: 1;
hence thesis by
A31,
A47,
XTUPLE_0: 1;
end;
suppose
A48:
[x, y]
=
[1, 2];
then x
= 1 by
XTUPLE_0: 1;
hence thesis by
A18,
A48,
XTUPLE_0: 1;
end;
suppose
A49:
[x, y]
=
[2, 1];
then x
= 2 by
XTUPLE_0: 1;
hence thesis by
A35,
A49,
XTUPLE_0: 1;
end;
suppose
A50:
[x, y]
=
[2, 3];
then x
= 2 by
XTUPLE_0: 1;
hence thesis by
A23,
A50,
XTUPLE_0: 1;
end;
suppose
A51:
[x, y]
=
[3, 2];
then x
= 3 by
XTUPLE_0: 1;
hence thesis by
A39,
A51,
XTUPLE_0: 1;
end;
end;
assume
A52:
[(h
. x), (h
. y)]
in the
InternalRel of CmpG;
per cases by
A2,
ENUMSET1:def 2;
suppose
A53: x
=
0 & y
=
0 ;
then (h
.
0 )
in the
carrier of CmpG by
A52,
ZFMISC_1: 87;
hence thesis by
A52,
A53,
NECKLACE:def 5;
end;
suppose x
=
0 & y
= 1;
hence thesis by
ENUMSET1:def 4,
NECKLA_2: 2;
end;
suppose x
=
0 & y
= 2;
then
[(h
.
0 ), (h
. 2)]
in (the
InternalRel of G
/\ the
InternalRel of CmpG) by
A44,
A52,
XBOOLE_0:def 4;
then the
InternalRel of G
meets the
InternalRel of CmpG;
hence thesis by
Th12;
end;
suppose x
=
0 & y
= 3;
then
[(h
.
0 ), (h
. 3)]
in (the
InternalRel of G
/\ the
InternalRel of CmpG) by
A30,
A52,
XBOOLE_0:def 4;
then the
InternalRel of G
meets the
InternalRel of CmpG;
hence thesis by
Th12;
end;
suppose x
= 1 & y
=
0 ;
hence thesis by
ENUMSET1:def 4,
NECKLA_2: 2;
end;
suppose
A54: x
= 1 & y
= 1;
then (h
. 1)
in the
carrier of CmpG by
A52,
ZFMISC_1: 87;
hence thesis by
A52,
A54,
NECKLACE:def 5;
end;
suppose x
= 1 & y
= 2;
hence thesis by
ENUMSET1:def 4,
NECKLA_2: 2;
end;
suppose x
= 1 & y
= 3;
then
[(h
. 1), (h
. 3)]
in (the
InternalRel of G
/\ the
InternalRel of CmpG) by
A28,
A52,
XBOOLE_0:def 4;
then the
InternalRel of G
meets the
InternalRel of CmpG;
hence thesis by
Th12;
end;
suppose x
= 2 & y
=
0 ;
then
[(h
. 2), (h
.
0 )]
in (the
InternalRel of G
/\ the
InternalRel of CmpG) by
A43,
A52,
XBOOLE_0:def 4;
then the
InternalRel of G
meets the
InternalRel of CmpG;
hence thesis by
Th12;
end;
suppose x
= 2 & y
= 1;
hence thesis by
ENUMSET1:def 4,
NECKLA_2: 2;
end;
suppose
A55: x
= 2 & y
= 2;
then (h
. 2)
in the
carrier of CmpG by
A52,
ZFMISC_1: 87;
hence thesis by
A52,
A55,
NECKLACE:def 5;
end;
suppose x
= 2 & y
= 3;
hence thesis by
ENUMSET1:def 4,
NECKLA_2: 2;
end;
suppose x
= 3 & y
=
0 ;
then
[(h
. 3), (h
.
0 )]
in (the
InternalRel of G
/\ the
InternalRel of CmpG) by
A29,
A52,
XBOOLE_0:def 4;
then the
InternalRel of G
meets the
InternalRel of CmpG;
hence thesis by
Th12;
end;
suppose x
= 3 & y
= 1;
then
[(h
. 3), (h
. 1)]
in (the
InternalRel of G
/\ the
InternalRel of CmpG) by
A27,
A52,
XBOOLE_0:def 4;
then the
InternalRel of G
meets the
InternalRel of CmpG;
hence thesis by
Th12;
end;
suppose x
= 3 & y
= 2;
hence thesis by
ENUMSET1:def 4,
NECKLA_2: 2;
end;
suppose
A56: x
= 3 & y
= 3;
then (h
. 3)
in the
carrier of CmpG by
A52,
ZFMISC_1: 87;
hence thesis by
A52,
A56,
NECKLACE:def 5;
end;
end;
hence thesis by
A4,
A1,
A6,
A8;
end;
assume CmpG
embeds N4;
then
consider f be
Function of N4, CmpG such that
A57: f is
one-to-one and
A58: for x,y be
Element of N4 holds
[x, y]
in the
InternalRel of N4 iff
[(f
. x), (f
. y)]
in the
InternalRel of CmpG;
consider g be
Function of N4, CmpN4 such that
A59: g is
isomorphic by
NECKLACE: 29,
WAYBEL_1:def 8;
A60: 2
in the
carrier of N4 by
A2,
ENUMSET1:def 2;
A61: (
dom f)
= the
carrier of N4 by
FUNCT_2:def 1;
A62:
[(f
.
0 ), (f
. 2)]
in the
InternalRel of G
proof
assume
A63: not thesis;
[(f
.
0 ), (f
. 2)]
in the
InternalRel of CmpG
proof
(f
.
0 )
in the
carrier of CmpG & (f
. 2)
in the
carrier of CmpG by
A3,
A60,
FUNCT_2: 5;
then
[(f
.
0 ), (f
. 2)]
in
[:the
carrier of G, the
carrier of G:] by
A1,
ZFMISC_1: 87;
then
[(f
.
0 ), (f
. 2)]
in (((
id the
carrier of G)
\/ the
InternalRel of G)
\/ the
InternalRel of CmpG) by
Th14;
then
A64:
[(f
.
0 ), (f
. 2)]
in ((
id the
carrier of G)
\/ the
InternalRel of G) or
[(f
.
0 ), (f
. 2)]
in the
InternalRel of CmpG by
XBOOLE_0:def 3;
assume not thesis;
then
[(f
.
0 ), (f
. 2)]
in (
id the
carrier of G) by
A63,
A64,
XBOOLE_0:def 3;
then (f
.
0 )
= (f
. 2) by
RELAT_1:def 10;
hence contradiction by
A57,
A61,
A3,
A60,
FUNCT_1:def 4;
end;
then
A65:
[
0 , 2]
in the
InternalRel of N4 by
A58,
A3,
A60;
[
0 , 2]
in the
InternalRel of CmpN4 by
Th11,
ENUMSET1:def 4;
then
[
0 , 2]
in (the
InternalRel of N4
/\ the
InternalRel of CmpN4) by
A65,
XBOOLE_0:def 4;
then the
InternalRel of N4
meets the
InternalRel of CmpN4;
hence thesis by
Th12;
end;
A66: 3
in the
carrier of N4 by
A2,
ENUMSET1:def 2;
A67:
[(f
.
0 ), (f
. 3)]
in the
InternalRel of G
proof
assume
A68: not
[(f
.
0 ), (f
. 3)]
in the
InternalRel of G;
[(f
.
0 ), (f
. 3)]
in the
InternalRel of CmpG
proof
(f
.
0 )
in the
carrier of CmpG & (f
. 3)
in the
carrier of CmpG by
A3,
A66,
FUNCT_2: 5;
then
[(f
.
0 ), (f
. 3)]
in
[:the
carrier of G, the
carrier of G:] by
A1,
ZFMISC_1: 87;
then
[(f
.
0 ), (f
. 3)]
in (((
id the
carrier of G)
\/ the
InternalRel of G)
\/ the
InternalRel of CmpG) by
Th14;
then
A69:
[(f
.
0 ), (f
. 3)]
in ((
id the
carrier of G)
\/ the
InternalRel of G) or
[(f
.
0 ), (f
. 3)]
in the
InternalRel of CmpG by
XBOOLE_0:def 3;
assume not thesis;
then
[(f
.
0 ), (f
. 3)]
in (
id the
carrier of G) by
A68,
A69,
XBOOLE_0:def 3;
then (f
.
0 )
= (f
. 3) by
RELAT_1:def 10;
hence contradiction by
A57,
A61,
A3,
A66,
FUNCT_1:def 4;
end;
then
A70:
[
0 , 3]
in the
InternalRel of N4 by
A58,
A3,
A66;
[
0 , 3]
in the
InternalRel of CmpN4 by
Th11,
ENUMSET1:def 4;
then
[
0 , 3]
in (the
InternalRel of N4
/\ the
InternalRel of CmpN4) by
A70,
XBOOLE_0:def 4;
then the
InternalRel of N4
meets the
InternalRel of CmpN4;
hence thesis by
Th12;
end;
A71: 1
in the
carrier of N4 by
A2,
ENUMSET1:def 2;
A72:
[(f
. 1), (f
. 3)]
in the
InternalRel of G
proof
assume
A73: not
[(f
. 1), (f
. 3)]
in the
InternalRel of G;
[(f
. 1), (f
. 3)]
in the
InternalRel of CmpG
proof
(f
. 1)
in the
carrier of CmpG & (f
. 3)
in the
carrier of CmpG by
A71,
A66,
FUNCT_2: 5;
then
[(f
. 1), (f
. 3)]
in
[:the
carrier of G, the
carrier of G:] by
A1,
ZFMISC_1: 87;
then
[(f
. 1), (f
. 3)]
in (((
id the
carrier of G)
\/ the
InternalRel of G)
\/ the
InternalRel of CmpG) by
Th14;
then
A74:
[(f
. 1), (f
. 3)]
in ((
id the
carrier of G)
\/ the
InternalRel of G) or
[(f
. 1), (f
. 3)]
in the
InternalRel of CmpG by
XBOOLE_0:def 3;
assume not thesis;
then
[(f
. 1), (f
. 3)]
in (
id the
carrier of G) by
A73,
A74,
XBOOLE_0:def 3;
then (f
. 1)
= (f
. 3) by
RELAT_1:def 10;
hence contradiction by
A57,
A61,
A71,
A66,
FUNCT_1:def 4;
end;
then
A75:
[1, 3]
in the
InternalRel of N4 by
A58,
A71,
A66;
[1, 3]
in the
InternalRel of CmpN4 by
Th11,
ENUMSET1:def 4;
then
[1, 3]
in (the
InternalRel of N4
/\ the
InternalRel of CmpN4) by
A75,
XBOOLE_0:def 4;
then the
InternalRel of N4
meets the
InternalRel of CmpN4;
hence thesis by
Th12;
end;
[3, 2]
in the
InternalRel of N4 by
ENUMSET1:def 4,
NECKLA_2: 2;
then
A76:
[(f
. 3), (f
. 2)]
in the
InternalRel of CmpG by
A58,
A60,
A66;
[2, 3]
in the
InternalRel of N4 by
ENUMSET1:def 4,
NECKLA_2: 2;
then
A77:
[(f
. 2), (f
. 3)]
in the
InternalRel of CmpG by
A58,
A60,
A66;
[1, 2]
in the
InternalRel of N4 by
ENUMSET1:def 4,
NECKLA_2: 2;
then
A78:
[(f
. 1), (f
. 2)]
in the
InternalRel of CmpG by
A58,
A71,
A60;
[1,
0 ]
in the
InternalRel of N4 by
ENUMSET1:def 4,
NECKLA_2: 2;
then
A79:
[(f
. 1), (f
.
0 )]
in the
InternalRel of CmpG by
A58,
A3,
A71;
A80:
[(f
. 2), (f
.
0 )]
in the
InternalRel of G
proof
assume
A81: not
[(f
. 2), (f
.
0 )]
in the
InternalRel of G;
[(f
. 2), (f
.
0 )]
in the
InternalRel of CmpG
proof
(f
.
0 )
in the
carrier of CmpG & (f
. 2)
in the
carrier of CmpG by
A3,
A60,
FUNCT_2: 5;
then
[(f
. 2), (f
.
0 )]
in
[:the
carrier of G, the
carrier of G:] by
A1,
ZFMISC_1: 87;
then
[(f
. 2), (f
.
0 )]
in (((
id the
carrier of G)
\/ the
InternalRel of G)
\/ the
InternalRel of CmpG) by
Th14;
then
A82:
[(f
. 2), (f
.
0 )]
in ((
id the
carrier of G)
\/ the
InternalRel of G) or
[(f
. 2), (f
.
0 )]
in the
InternalRel of CmpG by
XBOOLE_0:def 3;
assume not thesis;
then
[(f
. 2), (f
.
0 )]
in (
id the
carrier of G) by
A81,
A82,
XBOOLE_0:def 3;
then (f
.
0 )
= (f
. 2) by
RELAT_1:def 10;
hence contradiction by
A57,
A61,
A3,
A60,
FUNCT_1:def 4;
end;
then
A83:
[2,
0 ]
in the
InternalRel of N4 by
A58,
A3,
A60;
[2,
0 ]
in the
InternalRel of CmpN4 by
Th11,
ENUMSET1:def 4;
then
[2,
0 ]
in (the
InternalRel of N4
/\ the
InternalRel of CmpN4) by
A83,
XBOOLE_0:def 4;
then the
InternalRel of N4
meets the
InternalRel of CmpN4;
hence thesis by
Th12;
end;
A84:
[(f
. 3), (f
.
0 )]
in the
InternalRel of G
proof
assume
A85: not
[(f
. 3), (f
.
0 )]
in the
InternalRel of G;
[(f
. 3), (f
.
0 )]
in the
InternalRel of CmpG
proof
(f
.
0 )
in the
carrier of CmpG & (f
. 3)
in the
carrier of CmpG by
A3,
A66,
FUNCT_2: 5;
then
[(f
. 3), (f
.
0 )]
in
[:the
carrier of G, the
carrier of G:] by
A1,
ZFMISC_1: 87;
then
[(f
. 3), (f
.
0 )]
in (((
id the
carrier of G)
\/ the
InternalRel of G)
\/ the
InternalRel of CmpG) by
Th14;
then
A86:
[(f
. 3), (f
.
0 )]
in ((
id the
carrier of G)
\/ the
InternalRel of G) or
[(f
. 3), (f
.
0 )]
in the
InternalRel of CmpG by
XBOOLE_0:def 3;
assume not thesis;
then
[(f
. 3), (f
.
0 )]
in (
id the
carrier of G) by
A85,
A86,
XBOOLE_0:def 3;
then (f
.
0 )
= (f
. 3) by
RELAT_1:def 10;
hence contradiction by
A57,
A61,
A3,
A66,
FUNCT_1:def 4;
end;
then
A87:
[3,
0 ]
in the
InternalRel of N4 by
A58,
A3,
A66;
[3,
0 ]
in the
InternalRel of CmpN4 by
Th11,
ENUMSET1:def 4;
then
[3,
0 ]
in (the
InternalRel of N4
/\ the
InternalRel of CmpN4) by
A87,
XBOOLE_0:def 4;
then the
InternalRel of N4
meets the
InternalRel of CmpN4;
hence thesis by
Th12;
end;
A88:
[(f
. 3), (f
. 1)]
in the
InternalRel of G
proof
assume
A89: not
[(f
. 3), (f
. 1)]
in the
InternalRel of G;
[(f
. 3), (f
. 1)]
in the
InternalRel of CmpG
proof
(f
. 1)
in the
carrier of CmpG & (f
. 3)
in the
carrier of CmpG by
A71,
A66,
FUNCT_2: 5;
then
[(f
. 3), (f
. 1)]
in
[:the
carrier of G, the
carrier of G:] by
A1,
ZFMISC_1: 87;
then
[(f
. 3), (f
. 1)]
in (((
id the
carrier of G)
\/ the
InternalRel of G)
\/ the
InternalRel of CmpG) by
Th14;
then
A90:
[(f
. 3), (f
. 1)]
in ((
id the
carrier of G)
\/ the
InternalRel of G) or
[(f
. 3), (f
. 1)]
in the
InternalRel of CmpG by
XBOOLE_0:def 3;
assume not thesis;
then
[(f
. 3), (f
. 1)]
in (
id the
carrier of G) by
A89,
A90,
XBOOLE_0:def 3;
then (f
. 1)
= (f
. 3) by
RELAT_1:def 10;
hence contradiction by
A57,
A61,
A71,
A66,
FUNCT_1:def 4;
end;
then
A91:
[3, 1]
in the
InternalRel of N4 by
A58,
A71,
A66;
[3, 1]
in the
InternalRel of CmpN4 by
Th11,
ENUMSET1:def 4;
then
[3, 1]
in (the
InternalRel of N4
/\ the
InternalRel of CmpN4) by
A91,
XBOOLE_0:def 4;
then the
InternalRel of N4
meets the
InternalRel of CmpN4;
hence thesis by
Th12;
end;
[2, 1]
in the
InternalRel of N4 by
ENUMSET1:def 4,
NECKLA_2: 2;
then
A92:
[(f
. 2), (f
. 1)]
in the
InternalRel of CmpG by
A58,
A71,
A60;
[
0 , 1]
in the
InternalRel of N4 by
ENUMSET1:def 4,
NECKLA_2: 2;
then
A93:
[(f
.
0 ), (f
. 1)]
in the
InternalRel of CmpG by
A58,
A3,
A71;
A94: for x,y be
Element of CmpN4 holds
[x, y]
in the
InternalRel of CmpN4 iff
[(f
. x), (f
. y)]
in the
InternalRel of G
proof
let x,y be
Element of CmpN4;
A95: the
carrier of N4
= the
carrier of CmpN4 by
NECKLACE:def 8;
thus
[x, y]
in the
InternalRel of CmpN4 implies
[(f
. x), (f
. y)]
in the
InternalRel of G
proof
assume
A96:
[x, y]
in the
InternalRel of CmpN4;
per cases by
A96,
Th11,
ENUMSET1:def 4;
suppose
A97:
[x, y]
=
[
0 , 2];
then x
=
0 by
XTUPLE_0: 1;
hence thesis by
A62,
A97,
XTUPLE_0: 1;
end;
suppose
A98:
[x, y]
=
[2,
0 ];
then x
= 2 by
XTUPLE_0: 1;
hence thesis by
A80,
A98,
XTUPLE_0: 1;
end;
suppose
A99:
[x, y]
=
[
0 , 3];
then x
=
0 by
XTUPLE_0: 1;
hence thesis by
A67,
A99,
XTUPLE_0: 1;
end;
suppose
A100:
[x, y]
=
[3,
0 ];
then x
= 3 by
XTUPLE_0: 1;
hence thesis by
A84,
A100,
XTUPLE_0: 1;
end;
suppose
A101:
[x, y]
=
[1, 3];
then x
= 1 by
XTUPLE_0: 1;
hence thesis by
A72,
A101,
XTUPLE_0: 1;
end;
suppose
A102:
[x, y]
=
[3, 1];
then x
= 3 by
XTUPLE_0: 1;
hence thesis by
A88,
A102,
XTUPLE_0: 1;
end;
end;
assume
A103:
[(f
. x), (f
. y)]
in the
InternalRel of G;
per cases by
A2,
A95,
ENUMSET1:def 2;
suppose
A104: x
=
0 & y
=
0 ;
then (f
.
0 )
in the
carrier of G by
A103,
ZFMISC_1: 87;
hence thesis by
A103,
A104,
NECKLACE:def 5;
end;
suppose x
=
0 & y
= 1;
then
[(f
.
0 ), (f
. 1)]
in (the
InternalRel of G
/\ the
InternalRel of CmpG) by
A93,
A103,
XBOOLE_0:def 4;
then the
InternalRel of G
meets the
InternalRel of CmpG;
hence thesis by
Th12;
end;
suppose x
=
0 & y
= 2;
hence thesis by
Th11,
ENUMSET1:def 4;
end;
suppose x
=
0 & y
= 3;
hence thesis by
Th11,
ENUMSET1:def 4;
end;
suppose x
= 1 & y
=
0 ;
then
[(f
. 1), (f
.
0 )]
in (the
InternalRel of G
/\ the
InternalRel of CmpG) by
A79,
A103,
XBOOLE_0:def 4;
then the
InternalRel of G
meets the
InternalRel of CmpG;
hence thesis by
Th12;
end;
suppose x
= 2 & y
=
0 ;
hence thesis by
Th11,
ENUMSET1:def 4;
end;
suppose x
= 3 & y
=
0 ;
hence thesis by
Th11,
ENUMSET1:def 4;
end;
suppose
A105: x
= 1 & y
= 1;
then (f
. 1)
in the
carrier of G by
A103,
ZFMISC_1: 87;
hence thesis by
A103,
A105,
NECKLACE:def 5;
end;
suppose x
= 1 & y
= 2;
then
[(f
. 1), (f
. 2)]
in (the
InternalRel of G
/\ the
InternalRel of CmpG) by
A78,
A103,
XBOOLE_0:def 4;
then the
InternalRel of G
meets the
InternalRel of CmpG;
hence thesis by
Th12;
end;
suppose x
= 1 & y
= 3;
hence thesis by
Th11,
ENUMSET1:def 4;
end;
suppose x
= 2 & y
= 1;
then
[(f
. 2), (f
. 1)]
in (the
InternalRel of G
/\ the
InternalRel of CmpG) by
A92,
A103,
XBOOLE_0:def 4;
then the
InternalRel of G
meets the
InternalRel of CmpG;
hence thesis by
Th12;
end;
suppose
A106: x
= 2 & y
= 2;
then (f
. 2)
in the
carrier of G by
A103,
ZFMISC_1: 87;
hence thesis by
A103,
A106,
NECKLACE:def 5;
end;
suppose x
= 2 & y
= 3;
then
[(f
. 2), (f
. 3)]
in (the
InternalRel of G
/\ the
InternalRel of CmpG) by
A77,
A103,
XBOOLE_0:def 4;
then the
InternalRel of G
meets the
InternalRel of CmpG;
hence thesis by
Th12;
end;
suppose x
= 3 & y
= 1;
hence thesis by
Th11,
ENUMSET1:def 4;
end;
suppose x
= 3 & y
= 2;
then
[(f
. 3), (f
. 2)]
in (the
InternalRel of G
/\ the
InternalRel of CmpG) by
A76,
A103,
XBOOLE_0:def 4;
then the
InternalRel of G
meets the
InternalRel of CmpG;
hence thesis by
Th12;
end;
suppose
A107: x
= 3 & y
= 3;
then (f
. 3)
in the
carrier of G by
A103,
ZFMISC_1: 87;
hence thesis by
A103,
A107,
NECKLACE:def 5;
end;
end;
reconsider f as
Function of CmpN4, G by
A4,
NECKLACE:def 8;
reconsider h = (f
* g) as
Function of N4, G;
A108: g is
one-to-one
monotone by
A59,
WAYBEL_0:def 38;
for x,y be
Element of N4 holds
[x, y]
in the
InternalRel of N4 iff
[(h
. x), (h
. y)]
in the
InternalRel of G
proof
let x,y be
Element of N4;
thus
[x, y]
in the
InternalRel of N4 implies
[(h
. x), (h
. y)]
in the
InternalRel of G
proof
assume
[x, y]
in the
InternalRel of N4;
then x
<= y by
ORDERS_2:def 5;
then (g
. x)
<= (g
. y) by
A108,
WAYBEL_1:def 2;
then
[(g
. x), (g
. y)]
in the
InternalRel of CmpN4 by
ORDERS_2:def 5;
then
[(f
. (g
. x)), (f
. (g
. y))]
in the
InternalRel of G by
A94;
then
[((f
* g)
. x), (f
. (g
. y))]
in the
InternalRel of G by
FUNCT_2: 15;
hence thesis by
FUNCT_2: 15;
end;
assume
[(h
. x), (h
. y)]
in the
InternalRel of G;
then
[(f
. (g
. x)), (h
. y)]
in the
InternalRel of G by
FUNCT_2: 15;
then
[(f
. (g
. x)), (f
. (g
. y))]
in the
InternalRel of G by
FUNCT_2: 15;
then
[(g
. x), (g
. y)]
in the
InternalRel of CmpN4 by
A94;
then (g
. x)
<= (g
. y) by
ORDERS_2:def 5;
then x
<= y by
A59,
WAYBEL_0: 66;
hence thesis by
ORDERS_2:def 5;
end;
hence thesis by
A57,
A108;
end;
theorem ::
NECKLA_3:25
Th25: for G be non
empty
irreflexive
RelStr holds G is
N-free iff (
ComplRelStr G) is
N-free
proof
let G be non
empty
irreflexive
RelStr;
thus G is
N-free implies (
ComplRelStr G) is
N-free
proof
assume
A1: G is
N-free;
assume not thesis;
then (
ComplRelStr G)
embeds (
Necklace 4) by
NECKLA_2:def 1;
then G
embeds (
Necklace 4) by
Th24;
hence contradiction by
A1,
NECKLA_2:def 1;
end;
assume
A2: (
ComplRelStr G) is
N-free;
assume not thesis;
then G
embeds (
Necklace 4) by
NECKLA_2:def 1;
then (
ComplRelStr G)
embeds (
Necklace 4) by
Th24;
hence contradiction by
A2,
NECKLA_2:def 1;
end;
begin
definition
let R be
RelStr;
mode
path of R is
RedSequence of the
InternalRel of R;
end
definition
let R be
RelStr;
::
NECKLA_3:def1
attr R is
path-connected means for x,y be
set st x
in the
carrier of R & y
in the
carrier of R & x
<> y holds the
InternalRel of R
reduces (x,y) or the
InternalRel of R
reduces (y,x);
end
registration
cluster
empty ->
path-connected for
RelStr;
correctness ;
end
registration
cluster
connected ->
path-connected for non
empty
RelStr;
correctness
proof
let R be non
empty
RelStr;
set cR = the
carrier of R, IR = the
InternalRel of R;
assume
A1: R is
connected;
for x,y be
set st x
in the
carrier of R & y
in the
carrier of R & x
<> y holds the
InternalRel of R
reduces (x,y) or the
InternalRel of R
reduces (y,x)
proof
let x,y be
set such that
A2: x
in cR & y
in cR and x
<> y;
reconsider x, y as
Element of R by
A2;
A3: x
<= y or y
<= x by
A1,
WAYBEL_0:def 29;
per cases by
A3,
ORDERS_2:def 5;
suppose
A4:
[x, y]
in IR;
A5: (
len
<*x, y*>)
= 2 & (
<*x, y*>
. 1)
= x by
FINSEQ_1: 44;
A6: (
<*x, y*>
. 2)
= y by
FINSEQ_1: 44;
<*x, y*> is
RedSequence of IR by
A4,
REWRITE1: 7;
hence thesis by
A5,
A6,
REWRITE1:def 3;
end;
suppose
A7:
[y, x]
in IR;
A8: (
len
<*y, x*>)
= 2 & (
<*y, x*>
. 1)
= y by
FINSEQ_1: 44;
A9: (
<*y, x*>
. 2)
= x by
FINSEQ_1: 44;
<*y, x*> is
RedSequence of IR by
A7,
REWRITE1: 7;
hence thesis by
A8,
A9,
REWRITE1:def 3;
end;
end;
hence thesis;
end;
end
theorem ::
NECKLA_3:26
Th26: for R be non
empty
transitive
reflexive
RelStr, x,y be
Element of R holds the
InternalRel of R
reduces (x,y) implies
[x, y]
in the
InternalRel of R
proof
let R be non
empty
transitive
reflexive
RelStr;
let x,y be
Element of R;
set cR = the
carrier of R, IR = the
InternalRel of R;
assume IR
reduces (x,y);
then
consider p be
RedSequence of IR such that
A1: (p
. 1)
= x and
A2: (p
. (
len p))
= y by
REWRITE1:def 3;
reconsider p as
FinSequence;
defpred
P[
Nat] means $1
in (
dom p) implies
[(p
. 1), (p
. $1)]
in IR;
A3: IR
is_transitive_in cR by
ORDERS_2:def 3;
A4: for k be non
zero
Nat st
P[k] holds
P[(k
+ 1)]
proof
let k be non
zero
Nat such that
A5:
P[k];
assume
A6: (k
+ 1)
in (
dom p);
then k
<= (k
+ 1) & (k
+ 1)
<= (
len p) by
FINSEQ_3: 25,
NAT_1: 11;
then
A7: (
0
+ 1)
<= k & k
<= (
len p) by
NAT_1: 13;
then
A8: (p
. 1)
in cR by
A5,
FINSEQ_3: 25,
ZFMISC_1: 87;
k
in (
dom p) by
A7,
FINSEQ_3: 25;
then
A9:
[(p
. k), (p
. (k
+ 1))]
in IR by
A6,
REWRITE1:def 2;
then (p
. k)
in cR & (p
. (k
+ 1))
in cR by
ZFMISC_1: 87;
hence thesis by
A3,
A5,
A7,
A9,
A8,
FINSEQ_3: 25;
end;
IR
is_reflexive_in cR by
ORDERS_2:def 2;
then
A10:
P[1] by
A1;
A11: for k be non
zero
Nat holds
P[k] from
NAT_1:sch 10(
A10,
A4);
A12: (
len p)
>
0 by
REWRITE1:def 2;
then (
0
+ 1)
<= (
len p) by
NAT_1: 13;
then (
len p)
in (
dom p) by
FINSEQ_3: 25;
hence thesis by
A1,
A2,
A12,
A11;
end;
registration
cluster
path-connected ->
connected for non
empty
transitive
reflexive
RelStr;
correctness
proof
let R be non
empty
transitive
reflexive
RelStr;
set IR = the
InternalRel of R;
assume
A1: R is
path-connected;
for x,y be
Element of R holds x
<= y or y
<= x
proof
let x,y be
Element of R;
per cases ;
suppose x
= y;
hence thesis;
end;
suppose x
<> y;
then IR
reduces (x,y) or IR
reduces (y,x) by
A1;
then
[x, y]
in IR or
[y, x]
in IR by
Th26;
hence thesis by
ORDERS_2:def 5;
end;
end;
hence thesis by
WAYBEL_0:def 29;
end;
end
theorem ::
NECKLA_3:27
Th27: for R be
symmetric
RelStr, x,y be
set holds the
InternalRel of R
reduces (x,y) implies the
InternalRel of R
reduces (y,x)
proof
let R be
symmetric
RelStr;
set IR = the
InternalRel of R;
let x,y be
set;
A1: IR
= (IR
~ ) by
RELAT_2: 13;
assume IR
reduces (x,y);
then
consider p be
RedSequence of IR such that
A2: (p
. 1)
= x and
A3: (p
. (
len p))
= y by
REWRITE1:def 3;
reconsider p as
FinSequence;
A4: ((
Rev p)
. (
len p))
= x by
A2,
FINSEQ_5: 62;
IR
reduces (y,x)
proof
reconsider q = (
Rev p) as
RedSequence of IR by
A1,
REWRITE1: 9;
(q
. 1)
= y & (q
. (
len q))
= x by
A3,
A4,
FINSEQ_5: 62,
FINSEQ_5:def 3;
hence thesis by
REWRITE1:def 3;
end;
hence thesis;
end;
definition
let R be
symmetric
RelStr;
:: original:
path-connected
redefine
::
NECKLA_3:def2
attr R is
path-connected means for x,y be
set st x
in the
carrier of R & y
in the
carrier of R & x
<> y holds the
InternalRel of R
reduces (x,y);
compatibility
proof
set IR = the
InternalRel of R, cR = the
carrier of R;
thus R is
path-connected implies for x,y be
set st x
in the
carrier of R & y
in the
carrier of R & x
<> y holds the
InternalRel of R
reduces (x,y)
proof
assume
A1: R is
path-connected;
let x,y be
set such that
A2: x
in cR & y
in cR & x
<> y;
per cases by
A1,
A2;
suppose IR
reduces (x,y);
hence thesis;
end;
suppose IR
reduces (y,x);
hence thesis by
Th27;
end;
end;
assume for x,y be
set st x
in the
carrier of R & y
in the
carrier of R & x
<> y holds IR
reduces (x,y);
then for x,y be
set st x
in the
carrier of R & y
in the
carrier of R & x
<> y holds the
InternalRel of R
reduces (x,y) or the
InternalRel of R
reduces (y,x);
hence thesis;
end;
end
definition
let R be
RelStr;
let x be
Element of R;
::
NECKLA_3:def3
func
component x ->
Subset of R equals (
Class ((
EqCl the
InternalRel of R),x));
coherence ;
end
registration
let R be non
empty
RelStr;
let x be
Element of R;
cluster (
component x) -> non
empty;
correctness by
EQREL_1: 20;
end
theorem ::
NECKLA_3:28
Th28: for R be
RelStr, x be
Element of R, y be
set st y
in (
component x) holds
[x, y]
in (
EqCl the
InternalRel of R)
proof
let R be
RelStr;
let x be
Element of R;
let y be
set;
set IR = the
InternalRel of R;
assume y
in (
component x);
then
[y, x]
in (
EqCl IR) by
EQREL_1: 19;
hence thesis by
EQREL_1: 6;
end;
theorem ::
NECKLA_3:29
Th29: for R be
RelStr, x be
Element of R, A be
set holds A
= (
component x) iff for y be
object holds y
in A iff
[x, y]
in (
EqCl the
InternalRel of R)
proof
let R be
RelStr;
let x be
Element of R;
let A be
set;
set IR = the
InternalRel of R;
A1: (for y be
object holds y
in A iff
[x, y]
in (
EqCl the
InternalRel of R)) implies A
= (
component x)
proof
assume
A2: for y be
object holds y
in A iff
[x, y]
in (
EqCl the
InternalRel of R);
A3: (
component x)
c= A
proof
let a be
object;
assume a
in (
component x);
then
[a, x]
in (
EqCl IR) by
EQREL_1: 19;
then
[x, a]
in (
EqCl IR) by
EQREL_1: 6;
hence thesis by
A2;
end;
A
c= (
component x)
proof
let a be
object;
assume a
in A;
then
[x, a]
in (
EqCl IR) by
A2;
then
[a, x]
in (
EqCl IR) by
EQREL_1: 6;
hence thesis by
EQREL_1: 19;
end;
hence thesis by
A3;
end;
A
= (
component x) implies for y be
object holds
[x, y]
in (
EqCl IR) implies y
in A
proof
assume
A4: A
= (
component x);
let y be
object;
assume
[x, y]
in (
EqCl IR);
then
[y, x]
in (
EqCl IR) by
EQREL_1: 6;
hence thesis by
A4,
EQREL_1: 19;
end;
hence thesis by
A1,
Th28;
end;
theorem ::
NECKLA_3:30
Th30: for R be non
empty
irreflexive
symmetric
RelStr holds not R is
path-connected implies ex G1,G2 be non
empty
strict
irreflexive
symmetric
RelStr st the
carrier of G1
misses the
carrier of G2 & the RelStr of R
= (
union_of (G1,G2))
proof
let R be non
empty
irreflexive
symmetric
RelStr;
set cR = the
carrier of R, IR = the
InternalRel of R;
assume not R is
path-connected;
then
consider x,y be
set such that
A1: x
in cR & y
in cR and x
<> y and
A2: not IR
reduces (x,y);
reconsider x, y as
Element of R by
A1;
set A1 = (
component x), A2 = (the
carrier of R
\ A1);
reconsider A2 as
Subset of R;
set G1 = (
subrelstr A1), G2 = (
subrelstr A2);
A3: the
carrier of G2
= A2 by
YELLOW_0:def 15;
cR
c= (A1
\/ A2)
proof
let a be
object;
assume
A4: a
in cR;
assume not thesis;
then ( not a
in A1) & not a
in A2 by
XBOOLE_0:def 3;
hence contradiction by
A4,
XBOOLE_0:def 5;
end;
then
A5: cR
= (A1
\/ A2);
A6: the
carrier of G1
= A1 by
YELLOW_0:def 15;
then
A7: the
carrier of G1
misses the
carrier of G2 by
A3,
XBOOLE_1: 79;
A8: the
InternalRel of G1
misses the
InternalRel of G2
proof
set IG1 = the
InternalRel of G1, IG2 = the
InternalRel of G2;
assume not thesis;
then (IG1
/\ IG2)
<>
{} ;
then
consider a be
object such that
A9: a
in (IG1
/\ IG2) by
XBOOLE_0:def 1;
a
in IG1 by
A9,
XBOOLE_0:def 4;
then
consider c1,c2 be
object such that
A10: a
=
[c1, c2] and
A11: c1
in A1 and c2
in A1 by
A6,
RELSET_1: 2;
ex g1,g2 be
object st a
=
[g1, g2] & g1
in A2 & g2
in A2 by
A3,
A9,
RELSET_1: 2;
then c1
in A2 by
A10,
XTUPLE_0: 1;
then c1
in (A1
/\ A2) by
A11,
XBOOLE_0:def 4;
hence contradiction by
A6,
A3,
A7;
end;
A12: the
InternalRel of G2
= (IR
\ the
InternalRel of G1)
proof
set IG1 = the
InternalRel of G1, IG2 = the
InternalRel of G2;
thus IG2
c= (IR
\ IG1)
proof
let a be
object;
assume
A13: a
in IG2;
then
consider g1,g2 be
object such that
A14: a
=
[g1, g2] and
A15: g1
in A2 & g2
in A2 by
A3,
RELSET_1: 2;
reconsider g1, g2 as
Element of G2 by
A15,
YELLOW_0:def 15;
reconsider u1 = g1, u2 = g2 as
Element of R by
A15;
A16: not a
in IG1 by
A13,
XBOOLE_0:def 4,
A8;
g1
<= g2 by
A13,
A14,
ORDERS_2:def 5;
then u1
<= u2 by
YELLOW_0: 59;
then a
in IR by
A14,
ORDERS_2:def 5;
hence thesis by
A16,
XBOOLE_0:def 5;
end;
let a be
object;
assume
A17: a
in (IR
\ IG1);
then
A18: a
in IR by
XBOOLE_0:def 5;
A19: not a
in IG1 by
A17,
XBOOLE_0:def 5;
consider c1,c2 be
object such that
A20: a
=
[c1, c2] and
A21: c1
in cR & c2
in cR by
A17,
RELSET_1: 2;
reconsider c1, c2 as
Element of R by
A21;
A22: c1
<= c2 by
A18,
A20,
ORDERS_2:def 5;
per cases by
A5,
XBOOLE_0:def 3;
suppose
A23: c1
in A1 & c2
in A1;
then
reconsider d2 = c2 as
Element of G1 by
YELLOW_0:def 15;
reconsider d1 = c1 as
Element of G1 by
A23,
YELLOW_0:def 15;
d1
<= d2 by
A6,
A22,
YELLOW_0: 60;
hence thesis by
A19,
A20,
ORDERS_2:def 5;
end;
suppose
A24: c1
in A1 & c2
in A2;
A25:
[:A1, A2:]
misses IR
proof
assume not thesis;
then (
[:A1, A2:]
/\ IR)
<>
{} ;
then
consider b be
object such that
A26: b
in (
[:A1, A2:]
/\ IR) by
XBOOLE_0:def 1;
A27: b
in IR by
A26,
XBOOLE_0:def 4;
b
in
[:A1, A2:] by
A26,
XBOOLE_0:def 4;
then
consider b1,b2 be
object such that
A28: b1
in A1 and
A29: b2
in A2 and
A30: b
=
[b1, b2] by
ZFMISC_1:def 2;
reconsider b2 as
Element of R by
A29;
reconsider b1 as
Element of R by
A28;
IR
c= (
EqCl IR) &
[x, b1]
in (
EqCl IR) by
A28,
Th29,
MSUALG_5:def 1;
then
[x, b2]
in (
EqCl IR) by
A27,
A30,
EQREL_1: 7;
then b2
in A1 by
Th29;
then b2
in (A1
/\ A2) by
A29,
XBOOLE_0:def 4;
hence thesis by
A6,
A3,
A7;
end;
a
in
[:A1, A2:] by
A20,
A24,
ZFMISC_1:def 2;
then a
in (
[:A1, A2:]
/\ IR) by
A18,
XBOOLE_0:def 4;
hence thesis by
A25;
end;
suppose
A31: c1
in A2 & c2
in A1;
A32:
[:A2, A1:]
misses IR
proof
assume not thesis;
then (
[:A2, A1:]
/\ IR)
<>
{} ;
then
consider b be
object such that
A33: b
in (
[:A2, A1:]
/\ IR) by
XBOOLE_0:def 1;
b
in
[:A2, A1:] by
A33,
XBOOLE_0:def 4;
then
consider b1,b2 be
object such that
A34: b1
in A2 and
A35: b2
in A1 and
A36: b
=
[b1, b2] by
ZFMISC_1:def 2;
reconsider b1 as
Element of R by
A34;
reconsider b2 as
Element of R by
A35;
A37:
[x, b2]
in (
EqCl IR) by
A35,
Th29;
A38: IR
c= (
EqCl IR) by
MSUALG_5:def 1;
b
in IR by
A33,
XBOOLE_0:def 4;
then
[b2, b1]
in (
EqCl IR) by
A36,
A38,
EQREL_1: 6;
then
[x, b1]
in (
EqCl IR) by
A37,
EQREL_1: 7;
then b1
in A1 by
Th29;
then b1
in (A1
/\ A2) by
A34,
XBOOLE_0:def 4;
hence thesis by
A6,
A3,
A7;
end;
a
in
[:A2, A1:] by
A20,
A31,
ZFMISC_1:def 2;
then a
in (
[:A2, A1:]
/\ IR) by
A18,
XBOOLE_0:def 4;
hence thesis by
A32;
end;
suppose
A39: c1
in A2 & c2
in A2;
then
reconsider d2 = c2 as
Element of G2 by
YELLOW_0:def 15;
reconsider d1 = c1 as
Element of G2 by
A39,
YELLOW_0:def 15;
d1
<= d2 by
A3,
A22,
A39,
YELLOW_0: 60;
hence thesis by
A20,
ORDERS_2:def 5;
end;
end;
IR
= (the
InternalRel of G1
\/ the
InternalRel of G2)
proof
set IG1 = the
InternalRel of G1, IG2 = the
InternalRel of G2;
thus IR
c= (IG1
\/ IG2)
proof
let a be
object;
assume
A40: a
in IR;
assume not thesis;
then ( not a
in IG1) & not a
in IG2 by
XBOOLE_0:def 3;
hence contradiction by
A12,
A40,
XBOOLE_0:def 5;
end;
let a be
object;
assume
A41: a
in (IG1
\/ IG2);
per cases by
A41,
XBOOLE_0:def 3;
suppose
A42: a
in IG1;
then
consider v,w be
object such that
A43: a
=
[v, w] and
A44: v
in A1 & w
in A1 by
A6,
RELSET_1: 2;
reconsider v, w as
Element of G1 by
A44,
YELLOW_0:def 15;
reconsider u1 = v, u2 = w as
Element of R by
A44;
v
<= w by
A42,
A43,
ORDERS_2:def 5;
then u1
<= u2 by
YELLOW_0: 59;
hence thesis by
A43,
ORDERS_2:def 5;
end;
suppose
A45: a
in IG2;
then
consider v,w be
object such that
A46: a
=
[v, w] and
A47: v
in A2 & w
in A2 by
A3,
RELSET_1: 2;
reconsider v, w as
Element of G2 by
A47,
YELLOW_0:def 15;
reconsider u1 = v, u2 = w as
Element of R by
A47;
v
<= w by
A45,
A46,
ORDERS_2:def 5;
then u1
<= u2 by
YELLOW_0: 59;
hence thesis by
A46,
ORDERS_2:def 5;
end;
end;
then
A48: IR
= the
InternalRel of (
union_of (G1,G2)) by
NECKLA_2:def 2;
IR
= (IR
~ ) by
RELAT_2: 13;
then not (IR
\/ (IR
~ ))
reduces (x,y) by
A2;
then not (x,y)
are_convertible_wrt IR by
REWRITE1:def 4;
then not
[x, y]
in (
EqCl IR) by
MSUALG_6: 41;
then not y
in A1 by
Th29;
then
A49: G2 is non
empty
strict
RelStr by
A3,
XBOOLE_0:def 5;
cR
= the
carrier of (
union_of (G1,G2)) by
A6,
A3,
A5,
NECKLA_2:def 2;
hence thesis by
A6,
A7,
A49,
A48;
end;
theorem ::
NECKLA_3:31
Th31: for R be non
empty
irreflexive
symmetric
RelStr holds not (
ComplRelStr R) is
path-connected implies ex G1,G2 be non
empty
strict
irreflexive
symmetric
RelStr st the
carrier of G1
misses the
carrier of G2 & the RelStr of R
= (
sum_of (G1,G2))
proof
let R be non
empty
irreflexive
symmetric
RelStr;
set cR = the
carrier of R, IR = the
InternalRel of R, CR = (
ComplRelStr R), ICR = the
InternalRel of (
ComplRelStr R), cCR = the
carrier of (
ComplRelStr R);
assume not CR is
path-connected;
then
consider x,y be
set such that
A1: x
in cCR and
A2: y
in cCR and x
<> y and
A3: not ICR
reduces (x,y);
reconsider x, y as
Element of CR by
A1,
A2;
set A1 = (
component x), A2 = (the
carrier of R
\ A1);
reconsider A1 as
Subset of R by
NECKLACE:def 8;
ICR
= (ICR
~ ) by
RELAT_2: 13;
then not (ICR
\/ (ICR
~ ))
reduces (x,y) by
A3;
then not (x,y)
are_convertible_wrt ICR by
REWRITE1:def 4;
then not
[x, y]
in (
EqCl ICR) by
MSUALG_6: 41;
then
A4: not y
in A1 by
Th29;
reconsider A2 as
Subset of R;
set G1 = (
subrelstr A1), G2 = (
subrelstr A2);
A5: the
carrier of G1
= A1 by
YELLOW_0:def 15;
set IG1 = the
InternalRel of G1, IG2 = the
InternalRel of G2, G1G2 =
[:the
carrier of G1, the
carrier of G2:], G2G1 =
[:the
carrier of G2, the
carrier of G1:];
A6: cR
= (A1
\/ A2)
proof
thus cR
c= (A1
\/ A2)
proof
let a be
object;
assume
A7: a
in cR;
assume not thesis;
then ( not a
in A1) & not a
in A2 by
XBOOLE_0:def 3;
hence contradiction by
A7,
XBOOLE_0:def 5;
end;
let a be
object;
assume
A8: a
in (A1
\/ A2);
per cases by
A8,
XBOOLE_0:def 3;
suppose a
in A1;
hence thesis;
end;
suppose a
in A2;
hence thesis;
end;
end;
A9: the
carrier of G2
= A2 by
YELLOW_0:def 15;
then
A10: the
carrier of G1
misses the
carrier of G2 by
A5,
XBOOLE_1: 79;
A11: the
InternalRel of G1
misses the
InternalRel of G2
proof
assume not thesis;
then (IG1
/\ IG2)
<>
{} ;
then
consider a be
object such that
A12: a
in (IG1
/\ IG2) by
XBOOLE_0:def 1;
a
in IG1 by
A12,
XBOOLE_0:def 4;
then
consider c1,c2 be
object such that
A13: a
=
[c1, c2] and
A14: c1
in A1 and c2
in A1 by
A5,
RELSET_1: 2;
ex g1,g2 be
object st a
=
[g1, g2] & g1
in A2 & g2
in A2 by
A9,
A12,
RELSET_1: 2;
then c1
in A2 by
A13,
XTUPLE_0: 1;
then c1
in (A1
/\ A2) by
A14,
XBOOLE_0:def 4;
hence contradiction by
A5,
A9,
A10;
end;
A15: the
InternalRel of G2
= (((IR
\ IG1)
\ G1G2)
\ G2G1)
proof
thus IG2
c= (((IR
\ IG1)
\ G1G2)
\ G2G1)
proof
let a be
object;
assume
A16: a
in IG2;
then
consider g1,g2 be
object such that
A17: a
=
[g1, g2] and
A18: g1
in A2 and
A19: g2
in A2 by
A9,
RELSET_1: 2;
reconsider g1, g2 as
Element of G2 by
A18,
A19,
YELLOW_0:def 15;
reconsider u1 = g1, u2 = g2 as
Element of R by
A18,
A19;
A20: not a
in IG1 by
A16,
XBOOLE_0:def 4,
A11;
A21: not a
in G2G1
proof
assume a
in G2G1;
then g2
in A1 by
A5,
A17,
ZFMISC_1: 87;
then g2
in (A1
/\ A2) by
A19,
XBOOLE_0:def 4;
hence thesis by
A5,
A9,
A10;
end;
A22: not a
in G1G2
proof
assume a
in G1G2;
then g1
in A1 by
A5,
A17,
ZFMISC_1: 87;
then g1
in (A1
/\ A2) by
A18,
XBOOLE_0:def 4;
hence thesis by
A5,
A9,
A10;
end;
g1
<= g2 by
A16,
A17,
ORDERS_2:def 5;
then u1
<= u2 by
YELLOW_0: 59;
then a
in IR by
A17,
ORDERS_2:def 5;
then a
in (IR
\ IG1) by
A20,
XBOOLE_0:def 5;
then a
in ((IR
\ IG1)
\ G1G2) by
A22,
XBOOLE_0:def 5;
hence thesis by
A21,
XBOOLE_0:def 5;
end;
let a be
object;
assume
A23: a
in (((IR
\ IG1)
\ G1G2)
\ G2G1);
then
A24: not a
in G2G1 by
XBOOLE_0:def 5;
A25: a
in ((IR
\ IG1)
\ G1G2) by
A23,
XBOOLE_0:def 5;
then
A26: a
in (IR
\ IG1) by
XBOOLE_0:def 5;
then
A27: not a
in IG1 by
XBOOLE_0:def 5;
A28: not a
in G1G2 by
A25,
XBOOLE_0:def 5;
consider c1,c2 be
object such that
A29: a
=
[c1, c2] and
A30: c1
in cR & c2
in cR by
A23,
RELSET_1: 2;
reconsider c1, c2 as
Element of R by
A30;
a
in IR by
A26,
XBOOLE_0:def 5;
then
A31: c1
<= c2 by
A29,
ORDERS_2:def 5;
per cases by
A6,
XBOOLE_0:def 3;
suppose
A32: c1
in A1 & c2
in A1;
then
reconsider d2 = c2 as
Element of G1 by
YELLOW_0:def 15;
reconsider d1 = c1 as
Element of G1 by
A32,
YELLOW_0:def 15;
d1
<= d2 by
A5,
A31,
YELLOW_0: 60;
hence thesis by
A27,
A29,
ORDERS_2:def 5;
end;
suppose c1
in A1 & c2
in A2;
hence thesis by
A5,
A9,
A28,
A29,
ZFMISC_1: 87;
end;
suppose c1
in A2 & c2
in A1;
hence thesis by
A5,
A9,
A24,
A29,
ZFMISC_1: 87;
end;
suppose
A33: c1
in A2 & c2
in A2;
then
reconsider d1 = c1, d2 = c2 as
Element of G2 by
YELLOW_0:def 15;
d1
<= d2 by
A9,
A31,
A33,
YELLOW_0: 60;
hence thesis by
A29,
ORDERS_2:def 5;
end;
end;
IR
= (((the
InternalRel of G1
\/ the
InternalRel of G2)
\/
[:the
carrier of G1, the
carrier of G2:])
\/
[:the
carrier of G2, the
carrier of G1:])
proof
set G1G2 =
[:the
carrier of G1, the
carrier of G2:], G2G1 =
[:the
carrier of G2, the
carrier of G1:];
thus IR
c= (((IG1
\/ IG2)
\/ G1G2)
\/ G2G1)
proof
let a be
object;
assume
A34: a
in IR;
assume
A35: not thesis;
then
A36: not a
in ((IG1
\/ IG2)
\/ G1G2) by
XBOOLE_0:def 3;
then
A37: not a
in (IG1
\/ IG2) by
XBOOLE_0:def 3;
then not a
in IG2 by
XBOOLE_0:def 3;
then not a
in ((IR
\ IG1)
\ G1G2) or a
in G2G1 by
A15,
XBOOLE_0:def 5;
then
A38: not a
in (IR
\ IG1) or a
in G1G2 or a
in G2G1 by
XBOOLE_0:def 5;
not a
in IG1 by
A37,
XBOOLE_0:def 3;
hence thesis by
A34,
A35,
A36,
A38,
XBOOLE_0:def 3,
XBOOLE_0:def 5;
end;
let a be
object;
assume a
in (((IG1
\/ IG2)
\/ G1G2)
\/ G2G1);
then a
in ((IG1
\/ IG2)
\/ G1G2) or a
in G2G1 by
XBOOLE_0:def 3;
then
A39: a
in (IG1
\/ IG2) or a
in G1G2 or a
in G2G1 by
XBOOLE_0:def 3;
per cases by
A39,
XBOOLE_0:def 3;
suppose
A40: a
in IG1;
then
consider v,w be
object such that
A41: a
=
[v, w] and
A42: v
in A1 & w
in A1 by
A5,
RELSET_1: 2;
reconsider v, w as
Element of G1 by
A42,
YELLOW_0:def 15;
reconsider u1 = v, u2 = w as
Element of R by
A42;
v
<= w by
A40,
A41,
ORDERS_2:def 5;
then u1
<= u2 by
YELLOW_0: 59;
hence thesis by
A41,
ORDERS_2:def 5;
end;
suppose
A43: a
in IG2;
then
consider v,w be
object such that
A44: a
=
[v, w] and
A45: v
in A2 & w
in A2 by
A9,
RELSET_1: 2;
reconsider v, w as
Element of G2 by
A45,
YELLOW_0:def 15;
reconsider u1 = v, u2 = w as
Element of R by
A45;
v
<= w by
A43,
A44,
ORDERS_2:def 5;
then u1
<= u2 by
YELLOW_0: 59;
hence thesis by
A44,
ORDERS_2:def 5;
end;
suppose
A46: a
in G1G2;
assume
A47: not thesis;
consider v,w be
object such that
A48: a
=
[v, w] by
A46,
RELAT_1:def 1;
A49: w
in A2 by
A9,
A46,
A48,
ZFMISC_1: 87;
A50: v
in A1 by
A5,
A46,
A48,
ZFMISC_1: 87;
then
reconsider v, w as
Element of CR by
A49,
NECKLACE:def 8;
v
<> w by
A5,
A9,
A10,
A50,
A49,
XBOOLE_0:def 4;
then
A51: not a
in (
id cR) by
A48,
RELAT_1:def 10;
[v, w]
in
[:cR, cR:] by
A50,
A49,
ZFMISC_1: 87;
then a
in (
[:cR, cR:]
\ IR) by
A48,
A47,
XBOOLE_0:def 5;
then a
in (IR
` ) by
SUBSET_1:def 4;
then a
in ((IR
` )
\ (
id cR)) by
A51,
XBOOLE_0:def 5;
then
[v, w]
in ICR by
A48,
NECKLACE:def 8;
then (v,w)
are_convertible_wrt ICR by
REWRITE1: 29;
then
A52:
[v, w]
in (
EqCl ICR) by
MSUALG_6: 41;
[x, v]
in (
EqCl ICR) by
A50,
Th29;
then
[x, w]
in (
EqCl ICR) by
A52,
EQREL_1: 7;
then w
in (
component x) by
Th29;
then w
in (A1
/\ A2) by
A49,
XBOOLE_0:def 4;
hence thesis by
A5,
A9,
A10;
end;
suppose
A53: a
in G2G1;
assume
A54: not thesis;
consider v,w be
object such that
A55: a
=
[v, w] by
A53,
RELAT_1:def 1;
A56: w
in A1 by
A5,
A53,
A55,
ZFMISC_1: 87;
A57: v
in A2 by
A9,
A53,
A55,
ZFMISC_1: 87;
then
reconsider v, w as
Element of CR by
A56,
NECKLACE:def 8;
v
<> w by
A5,
A9,
A10,
A57,
A56,
XBOOLE_0:def 4;
then
A58: not a
in (
id cR) by
A55,
RELAT_1:def 10;
[v, w]
in
[:cR, cR:] by
A57,
A56,
ZFMISC_1: 87;
then a
in (
[:cR, cR:]
\ IR) by
A55,
A54,
XBOOLE_0:def 5;
then a
in (IR
` ) by
SUBSET_1:def 4;
then a
in ((IR
` )
\ (
id cR)) by
A58,
XBOOLE_0:def 5;
then
[v, w]
in ICR by
A55,
NECKLACE:def 8;
then (v,w)
are_convertible_wrt ICR by
REWRITE1: 29;
then
[v, w]
in (
EqCl ICR) by
MSUALG_6: 41;
then
A59:
[w, v]
in (
EqCl ICR) by
EQREL_1: 6;
[x, w]
in (
EqCl ICR) by
A56,
Th29;
then
[x, v]
in (
EqCl ICR) by
A59,
EQREL_1: 7;
then v
in (
component x) by
Th29;
then v
in (A1
/\ A2) by
A57,
XBOOLE_0:def 4;
hence thesis by
A5,
A9,
A10;
end;
end;
then
A60: IR
= the
InternalRel of (
sum_of (G1,G2)) by
NECKLA_2:def 3;
y
in cR by
A2,
NECKLACE:def 8;
then
A61: G2 is non
empty
strict
RelStr by
A9,
A4,
XBOOLE_0:def 5;
cR
= the
carrier of (
sum_of (G1,G2)) by
A5,
A9,
A6,
NECKLA_2:def 3;
hence thesis by
A5,
A10,
A61,
A60;
end;
Lm1: for X be non
empty
finite
set, A,B be non
empty
set st X
= (A
\/ B) & A
misses B holds (
card A)
in (
Segm (
card X))
proof
let X be non
empty
finite
set;
let A,B be non
empty
set;
set n = (
card X);
assume that
A1: X
= (A
\/ B) and
A2: A
misses B;
(
card B)
c= n by
A1,
CARD_1: 11,
XBOOLE_1: 7;
then
reconsider B as
finite
set;
(
card A)
c= n by
A1,
CARD_1: 11,
XBOOLE_1: 7;
then
reconsider A as
finite
set;
A3: (
card B)
>= 1 by
NAT_1: 14;
A4: n
= ((
card A)
+ (
card B)) by
A1,
A2,
CARD_2: 40;
(
card A)
< n
proof
assume not thesis;
then ((
card A)
+ (
card B))
>= (n
+ 1) by
A3,
XREAL_1: 7;
hence thesis by
A4,
NAT_1: 13;
end;
hence thesis by
NAT_1: 44;
end;
theorem ::
NECKLA_3:32
for G be
irreflexive
RelStr st G
in
fin_RelStr_sp holds (
ComplRelStr G)
in
fin_RelStr_sp
proof
defpred
P[
Nat] means for G be
irreflexive
RelStr st (
card the
carrier of G)
= $1 & G
in
fin_RelStr_sp holds (
ComplRelStr G)
in
fin_RelStr_sp ;
let G be
irreflexive
RelStr;
A1: for k be
Nat st for n be
Nat st n
< k holds
P[n] holds
P[k]
proof
let k be
Nat such that
A2: for n be
Nat st n
< k holds
P[n];
let G be
irreflexive
RelStr;
assume that
A3: (
card the
carrier of G)
= k and
A4: G
in
fin_RelStr_sp ;
per cases by
A4,
NECKLA_2: 6;
suppose G is
strict1
-element
RelStr;
hence thesis by
A4,
Th15;
end;
suppose ex G1,G2 be
strict
RelStr st the
carrier of G1
misses the
carrier of G2 & G1
in
fin_RelStr_sp & G2
in
fin_RelStr_sp & (G
= (
union_of (G1,G2)) or G
= (
sum_of (G1,G2)));
then
consider G1,G2 be
strict
RelStr such that
A5: the
carrier of G1
misses the
carrier of G2 and
A6: G1
in
fin_RelStr_sp and
A7: G2
in
fin_RelStr_sp and
A8: G
= (
union_of (G1,G2)) or G
= (
sum_of (G1,G2));
A9: G2 is non
empty
finite by
A7,
NECKLA_2: 4;
then
reconsider n2 = (
card the
carrier of G2) as
Nat;
A10: G1 is non
empty
finite by
A6,
NECKLA_2: 4;
then
reconsider n1 = (
card the
carrier of G1) as
Nat;
thus thesis
proof
per cases by
A8;
suppose
A11: G
= (
union_of (G1,G2));
then
reconsider G2 as
irreflexive
RelStr by
Th9;
reconsider G1 as
irreflexive
RelStr by
A11,
Th9;
reconsider cG1 = the
carrier of G1 as non
empty
finite
set by
A10;
reconsider cG2 = the
carrier of G2 as non
empty
finite
set by
A9;
the
carrier of G
= (the
carrier of G1
\/ the
carrier of G2) by
A11,
NECKLA_2:def 2;
then
A12: (
card the
carrier of G)
= ((
card cG1)
+ (
card cG2)) by
A5,
CARD_2: 40;
A13: (
card cG1)
= n1;
n2
< k
proof
assume not thesis;
then (k
+
0 )
< (n1
+ n2) by
A13,
XREAL_1: 8;
hence thesis by
A3,
A12;
end;
then
A14: (
ComplRelStr G2)
in
fin_RelStr_sp by
A2,
A7;
A15: the
carrier of (
ComplRelStr G1)
= the
carrier of G1 & the
carrier of (
ComplRelStr G2)
= the
carrier of G2 by
NECKLACE:def 8;
A16: (
card cG2)
= n2;
n1
< k
proof
assume not thesis;
then (k
+
0 )
< (n2
+ n1) by
A16,
XREAL_1: 8;
hence thesis by
A3,
A12;
end;
then
A17: (
ComplRelStr G1)
in
fin_RelStr_sp by
A2,
A6;
(
ComplRelStr G)
= (
sum_of ((
ComplRelStr G1),(
ComplRelStr G2))) by
A5,
A11,
Th17;
hence thesis by
A5,
A17,
A14,
A15,
NECKLA_2:def 5;
end;
suppose
A18: G
= (
sum_of (G1,G2));
then
reconsider G2 as
irreflexive
RelStr by
Th9;
reconsider G1 as
irreflexive
RelStr by
A18,
Th9;
reconsider cG1 = the
carrier of G1 as non
empty
finite
set by
A10;
reconsider cG2 = the
carrier of G2 as non
empty
finite
set by
A9;
the
carrier of G
= (the
carrier of G1
\/ the
carrier of G2) by
A18,
NECKLA_2:def 3;
then
A19: (
card the
carrier of G)
= ((
card cG1)
+ (
card cG2)) by
A5,
CARD_2: 40;
A20: (
card cG1)
= n1;
n2
< k
proof
assume not thesis;
then (k
+
0 )
< (n1
+ n2) by
A20,
XREAL_1: 8;
hence thesis by
A3,
A19;
end;
then
A21: (
ComplRelStr G2)
in
fin_RelStr_sp by
A2,
A7;
A22: the
carrier of (
ComplRelStr G1)
= the
carrier of G1 & the
carrier of (
ComplRelStr G2)
= the
carrier of G2 by
NECKLACE:def 8;
A23: (
card cG2)
= n2;
n1
< k
proof
assume not thesis;
then (k
+
0 )
< (n2
+ n1) by
A23,
XREAL_1: 8;
hence thesis by
A3,
A19;
end;
then
A24: (
ComplRelStr G1)
in
fin_RelStr_sp by
A2,
A6;
(
ComplRelStr G)
= (
union_of ((
ComplRelStr G1),(
ComplRelStr G2))) by
A5,
A18,
Th18;
hence thesis by
A5,
A24,
A21,
A22,
NECKLA_2:def 5;
end;
end;
end;
end;
A25: for k be
Nat holds
P[k] from
NAT_1:sch 4(
A1);
assume
A26: G
in
fin_RelStr_sp ;
then G is
finite by
NECKLA_2: 4;
then (
card the
carrier of G) is
Nat;
hence thesis by
A26,
A25;
end;
theorem ::
NECKLA_3:33
Th33: for R be
irreflexive
symmetric
RelStr st (
card the
carrier of R)
= 2 & the
carrier of R
in
FinSETS holds the RelStr of R
in
fin_RelStr_sp
proof
let R be
irreflexive
symmetric
RelStr;
assume that
A1: (
card the
carrier of R)
= 2 and
A2: the
carrier of R
in
FinSETS ;
consider a,b be
object such that
A3: the
carrier of R
=
{a, b} and
A4: the
InternalRel of R
=
{
[a, b],
[b, a]} or the
InternalRel of R
=
{} by
A1,
Th6;
set A =
{a}, B =
{b};
A5: A
c= the
carrier of R
proof
let x be
object;
assume x
in A;
then x
= a by
TARSKI:def 1;
hence thesis by
A3,
TARSKI:def 2;
end;
A6: B
c= the
carrier of R
proof
let x be
object;
assume x
in B;
then x
= b by
TARSKI:def 1;
hence thesis by
A3,
TARSKI:def 2;
end;
then
reconsider B as
Subset of R;
reconsider A as
Subset of R by
A5;
set H1 = (
subrelstr A), H2 = (
subrelstr B);
reconsider H2 as non
empty
strict
irreflexive
symmetric
RelStr by
YELLOW_0:def 15;
A7: the
carrier of H2
= B by
YELLOW_0:def 15;
then the
InternalRel of H2
c=
[:
{b},
{b}:];
then the
InternalRel of H2
c=
{
[b, b]} by
ZFMISC_1: 29;
then
A8: the
InternalRel of H2
=
{} or the
InternalRel of H2
=
{
[b, b]} by
ZFMISC_1: 33;
A9: the
InternalRel of H2
=
{}
proof
b
in B by
TARSKI:def 1;
then b
in the
carrier of H2 by
YELLOW_0:def 15;
then
A10: not
[b, b]
in the
InternalRel of H2 by
NECKLACE:def 5;
assume not thesis;
hence thesis by
A8,
A10,
TARSKI:def 1;
end;
the
carrier of H2
c= the
carrier of R by
A6,
YELLOW_0:def 15;
then the
carrier of H2
in
FinSETS by
A2,
CLASSES1: 3,
CLASSES2:def 2;
then
A11: H2
in
fin_RelStr_sp by
A7,
NECKLA_2:def 5;
reconsider H1 as non
empty
strict
irreflexive
symmetric
RelStr by
YELLOW_0:def 15;
A12: the
carrier of H1
= A by
YELLOW_0:def 15;
then
A13: the
carrier of R
= (the
carrier of H1
\/ the
carrier of H2) by
A3,
A7,
ENUMSET1: 1;
a
<> b
proof
assume not thesis;
then the
carrier of R
=
{a} by
A3,
ENUMSET1: 29;
hence thesis by
A1,
CARD_1: 30;
end;
then
A14: A
misses B by
ZFMISC_1: 11;
then
A15: the
carrier of H1
misses the
carrier of H2 by
A7,
YELLOW_0:def 15;
the
InternalRel of H1
c=
[:
{a},
{a}:] by
A12;
then the
InternalRel of H1
c=
{
[a, a]} by
ZFMISC_1: 29;
then
A16: the
InternalRel of H1
=
{} or the
InternalRel of H1
=
{
[a, a]} by
ZFMISC_1: 33;
A17: the
InternalRel of H1
=
{}
proof
a
in A by
TARSKI:def 1;
then a
in the
carrier of H1 by
YELLOW_0:def 15;
then
A18: not
[a, a]
in the
InternalRel of H1 by
NECKLACE:def 5;
assume not thesis;
hence thesis by
A16,
A18,
TARSKI:def 1;
end;
the
carrier of H1
c= the
carrier of R by
A5,
YELLOW_0:def 15;
then the
carrier of H1
in
FinSETS by
A2,
CLASSES1: 3,
CLASSES2:def 2;
then
A19: H1
in
fin_RelStr_sp by
A12,
NECKLA_2:def 5;
per cases by
A4;
suppose
A20: the
InternalRel of R
=
{
[a, b],
[b, a]};
set S = (
sum_of (H1,H2));
the
InternalRel of S
= (((the
InternalRel of H1
\/ the
InternalRel of H2)
\/
[:A, B:])
\/
[:B, A:]) by
A12,
A7,
NECKLA_2:def 3;
then the
InternalRel of S
= (
{
[a, b]}
\/
[:
{b},
{a}:]) by
A17,
A9,
ZFMISC_1: 29;
then the
InternalRel of S
= (
{
[a, b]}
\/
{
[b, a]}) by
ZFMISC_1: 29;
then
A21: the
InternalRel of S
= the
InternalRel of R by
A20,
ENUMSET1: 1;
the
carrier of S
= the
carrier of R by
A13,
NECKLA_2:def 3;
hence thesis by
A12,
A19,
A7,
A11,
A14,
A21,
NECKLA_2:def 5;
end;
suppose
A22: the
InternalRel of R
=
{} ;
set U = (
union_of (H1,H2));
the
InternalRel of U
= (the
InternalRel of H1
\/ the
InternalRel of H2) & the
carrier of U
= the
carrier of R by
A13,
NECKLA_2:def 2;
hence thesis by
A19,
A11,
A15,
A17,
A9,
A22,
NECKLA_2:def 5;
end;
end;
theorem ::
NECKLA_3:34
for R be
RelStr st R
in
fin_RelStr_sp holds R is
symmetric
proof
let X be
RelStr;
assume
A1: X
in
fin_RelStr_sp ;
per cases ;
suppose
A2: X is
trivial;
thus thesis
proof
per cases by
A2,
ZFMISC_1: 131;
suppose
A3: the
carrier of X is
empty;
let a,b be
object;
assume that
A4: a
in the
carrier of X and b
in the
carrier of X and
[a, b]
in the
InternalRel of X;
thus thesis by
A3,
A4;
end;
suppose ex x be
object st the
carrier of X
=
{x};
then
consider x be
object such that
A5: the
carrier of X
=
{x};
A6:
[:the
carrier of X, the
carrier of X:]
=
{
[x, x]} by
A5,
ZFMISC_1: 29;
thus thesis
proof
per cases by
A6,
ZFMISC_1: 33;
suppose
A7: the
InternalRel of X
=
{} ;
let a,b be
object;
assume that a
in the
carrier of X and b
in the
carrier of X and
A8:
[a, b]
in the
InternalRel of X;
thus thesis by
A7,
A8;
end;
suppose
A9: the
InternalRel of X
=
{
[x, x]};
let a,b be
object;
assume that a
in the
carrier of X and b
in the
carrier of X and
A10:
[a, b]
in the
InternalRel of X;
A11:
[a, b]
=
[x, x] by
A9,
A10,
TARSKI:def 1;
then a
= x by
XTUPLE_0: 1;
hence thesis by
A10,
A11,
XTUPLE_0: 1;
end;
end;
end;
end;
end;
suppose
A12: not X is
trivial;
defpred
P[
Nat] means for X be non
empty
RelStr st not X is
trivial & X
in
fin_RelStr_sp holds (
card the
carrier of X)
c= $1 implies X is
symmetric;
A13: ex R be
strict
RelStr st X
= R & the
carrier of R
in
FinSETS by
A1,
NECKLA_2:def 4;
reconsider X as non
empty
RelStr by
A1,
NECKLA_2: 4;
A14: (
card the
carrier of X) is
Nat by
A13;
A15: for k be
Nat st
P[k] holds
P[(k
+ 1)]
proof
let k be
Nat such that
A16:
P[k];
reconsider k1 = k as
Element of
NAT by
ORDINAL1:def 12;
let Y be non
empty
RelStr such that
A17: not Y is
trivial and
A18: Y
in
fin_RelStr_sp ;
consider H1,H2 be
strict
RelStr such that
A19: the
carrier of H1
misses the
carrier of H2 and
A20: H1
in
fin_RelStr_sp and
A21: H2
in
fin_RelStr_sp and
A22: Y
= (
union_of (H1,H2)) or Y
= (
sum_of (H1,H2)) by
A17,
A18,
NECKLA_2: 6;
ex R be
strict
RelStr st Y
= R & the
carrier of R
in
FinSETS by
A18,
NECKLA_2:def 4;
then
reconsider cY = the
carrier of Y as
finite
set;
assume (
card the
carrier of Y)
c= (k
+ 1);
then (
Segm (
card cY))
c= (
Segm (
card (k1
+ 1)));
then (
card cY)
<= (
card (k1
+ 1)) by
NAT_1: 39;
then
A23: (
card cY)
<= (k
+ 1);
set cH1 = the
carrier of H1, cH2 = the
carrier of H2;
A24: (
card cY)
= (
card (cH1
\/ cH2)) by
A22,
NECKLA_2:def 2,
NECKLA_2:def 3;
ex R2 be
strict
RelStr st H2
= R2 & the
carrier of R2
in
FinSETS by
A21,
NECKLA_2:def 4;
then
reconsider cH2 as
finite
set;
ex R1 be
strict
RelStr st H1
= R1 & the
carrier of R1
in
FinSETS by
A20,
NECKLA_2:def 4;
then
reconsider cH1 as
finite
set;
A25: (
card cY)
= ((
card cH1)
+ (
card cH2)) by
A19,
A24,
CARD_2: 40;
H1 is non
empty by
A20,
NECKLA_2: 4;
then
A26: (
card cH1)
>= 1 by
NAT_1: 14;
H2 is non
empty by
A21,
NECKLA_2: 4;
then
A27: (
card cH2)
>= 1 by
NAT_1: 14;
per cases by
A25,
A23,
A26,
A27,
NAT_1: 8,
XXREAL_0: 1;
suppose (
card cY)
<= k;
then (
card cY)
<= (
card k1);
then (
Segm (
card cY))
c= (
Segm (
card k)) by
NAT_1: 39;
then (
card the
carrier of Y)
c= k1;
hence thesis by
A16,
A17,
A18;
end;
suppose
A28: (
card cY)
= (k
+ 1) & k
=
0 ;
set x = the
set;
(
card cY)
= (
card
{x}) by
A28,
CARD_1: 30;
then (cY,
{x})
are_equipotent by
CARD_1: 5;
then ex y be
object st cY
=
{y} by
CARD_1: 28;
hence thesis by
A17;
end;
suppose
A29: ((
card cH1)
+ (
card cH2))
= (k
+ 1) & k
>
0 & (
card cH1)
= 1 & (
card cH2)
= 1;
then ex x be
object st cH1
=
{x} by
CARD_2: 42;
then the
InternalRel of H1
is_symmetric_in cH1 by
Th5;
then
reconsider H1 as
symmetric
RelStr by
NECKLACE:def 3;
ex y be
object st cH2
=
{y} by
A29,
CARD_2: 42;
then the
InternalRel of H2
is_symmetric_in cH2 by
Th5;
then
reconsider H2 as
symmetric
RelStr by
NECKLACE:def 3;
(
union_of (H1,H2)) is
symmetric;
hence thesis by
A22;
end;
suppose
A30: ((
card cH1)
+ (
card cH2))
= (k
+ 1) & k
>
0 & (
card cH1)
= 1 & (
card cH2)
> 1;
then ex x be
object st cH1
=
{x} by
CARD_2: 42;
then the
InternalRel of H1
is_symmetric_in cH1 by
Th5;
then
reconsider H1 as
symmetric
RelStr by
NECKLACE:def 3;
(
card cH2) is non
trivial by
A30,
NAT_2: 28;
then (
card cH2)
>= 2 by
NAT_2: 29;
then H2 is non
empty non
trivial by
NAT_D: 60;
then
reconsider H2 as
symmetric
RelStr by
A16,
A21,
A30;
(
union_of (H1,H2)) is
symmetric;
hence thesis by
A22;
end;
suppose
A31: ((
card cH1)
+ (
card cH2))
= (k
+ 1) & k
>
0 & (
card cH1)
> 1 & (
card cH2)
= 1;
then ex x be
object st cH2
=
{x} by
CARD_2: 42;
then the
InternalRel of H2
is_symmetric_in cH2 by
Th5;
then
reconsider H2 as
symmetric
RelStr by
NECKLACE:def 3;
(
card cH1) is non
trivial by
A31,
NAT_2: 28;
then (
card cH1)
>= 2 by
NAT_2: 29;
then H1 is non
empty non
trivial by
NAT_D: 60;
then
reconsider H1 as
symmetric
RelStr by
A16,
A20,
A31;
(
union_of (H1,H2)) is
symmetric;
hence thesis by
A22;
end;
suppose
A32: ((
card cH1)
+ (
card cH2))
= (k
+ 1) & k
>
0 & (
card cH1)
> 1 & (
card cH2)
> 1;
then (
card cH2) is non
trivial by
NAT_2: 28;
then (
card cH2)
>= 2 by
NAT_2: 29;
then
A33: H2 is non
empty non
trivial by
NAT_D: 60;
(
card cH2)
< (k
+ 1)
proof
assume not thesis;
then ((
card cH1)
+ (
card cH2))
>= ((k
+ 1)
+ 1) by
A26,
XREAL_1: 7;
hence thesis by
A32,
NAT_1: 13;
end;
then (
card cH2)
<= k by
NAT_1: 13;
then (
card cH2)
<= (
card k1);
then (
Segm (
card cH2))
c= (
Segm (
card k)) by
NAT_1: 39;
then (
card cH2)
c= k1;
then
reconsider H2 as
symmetric
RelStr by
A16,
A21,
A33;
(
card cH1) is non
trivial by
A32,
NAT_2: 28;
then (
card cH1)
>= 2 by
NAT_2: 29;
then
A34: H1 is non
empty non
trivial by
NAT_D: 60;
(
card cH1)
< (k
+ 1)
proof
assume not thesis;
then ((
card cH1)
+ (
card cH2))
>= ((k
+ 1)
+ 1) by
A27,
XREAL_1: 7;
hence thesis by
A32,
NAT_1: 13;
end;
then (
card cH1)
<= k by
NAT_1: 13;
then (
card cH1)
<= (
card k1);
then (
Segm (
card cH1))
c= (
Segm (
card k)) by
NAT_1: 39;
then (
card cH1)
c= k1;
then
reconsider H1 as
symmetric
RelStr by
A16,
A20,
A34;
(
union_of (H1,H2)) is
symmetric;
hence thesis by
A22;
end;
end;
A35:
P[
0 ];
for k be
Nat holds
P[k] from
NAT_1:sch 2(
A35,
A15);
hence thesis by
A1,
A12,
A14;
end;
end;
theorem ::
NECKLA_3:35
Th35: for G be
RelStr, H1,H2 be non
empty
RelStr, x be
Element of H1, y be
Element of H2 st G
= (
union_of (H1,H2)) & the
carrier of H1
misses the
carrier of H2 holds not
[x, y]
in the
InternalRel of G
proof
let G be
RelStr;
let H1,H2 be non
empty
RelStr;
let x be
Element of H1;
let y be
Element of H2;
assume that
A1: G
= (
union_of (H1,H2)) and
A2: the
carrier of H1
misses the
carrier of H2;
assume not thesis;
then
A3:
[x, y]
in (the
InternalRel of H1
\/ the
InternalRel of H2) by
A1,
NECKLA_2:def 2;
per cases by
A3,
XBOOLE_0:def 3;
suppose
[x, y]
in the
InternalRel of H1;
then y
in the
carrier of H1 by
ZFMISC_1: 87;
then y
in (the
carrier of H1
/\ the
carrier of H2) by
XBOOLE_0:def 4;
hence thesis by
A2;
end;
suppose
[x, y]
in the
InternalRel of H2;
then x
in the
carrier of H2 by
ZFMISC_1: 87;
then x
in (the
carrier of H1
/\ the
carrier of H2) by
XBOOLE_0:def 4;
hence thesis by
A2;
end;
end;
theorem ::
NECKLA_3:36
for G be
RelStr, H1,H2 be non
empty
RelStr, x be
Element of H1, y be
Element of H2 st G
= (
sum_of (H1,H2)) holds not
[x, y]
in the
InternalRel of (
ComplRelStr G)
proof
let G be
RelStr, H1,H2 be non
empty
RelStr, x be
Element of H1, y be
Element of H2;
set cH1 = the
carrier of H1, cH2 = the
carrier of H2, IH1 = the
InternalRel of H1, IH2 = the
InternalRel of H2;
[x, y]
in (
[:cH1, cH2:]
\/
[:cH2, cH1:]) by
XBOOLE_0:def 3;
then
[x, y]
in (IH2
\/ (
[:cH1, cH2:]
\/
[:cH2, cH1:])) by
XBOOLE_0:def 3;
then
[x, y]
in (IH1
\/ (IH2
\/ (
[:cH1, cH2:]
\/
[:cH2, cH1:]))) by
XBOOLE_0:def 3;
then
[x, y]
in (IH1
\/ ((IH2
\/
[:cH1, cH2:])
\/
[:cH2, cH1:])) by
XBOOLE_1: 4;
then
A1:
[x, y]
in (((IH1
\/ IH2)
\/
[:cH1, cH2:])
\/
[:cH2, cH1:]) by
XBOOLE_1: 113;
assume G
= (
sum_of (H1,H2));
then
A2:
[x, y]
in the
InternalRel of G by
A1,
NECKLA_2:def 3;
not
[x, y]
in the
InternalRel of (
ComplRelStr G)
proof
assume not thesis;
then
[x, y]
in (the
InternalRel of G
/\ the
InternalRel of (
ComplRelStr G)) by
A2,
XBOOLE_0:def 4;
then the
InternalRel of G
meets the
InternalRel of (
ComplRelStr G);
hence contradiction by
Th12;
end;
hence thesis;
end;
theorem ::
NECKLA_3:37
Th37: for G be non
empty
symmetric
RelStr, x be
Element of G, R1,R2 be non
empty
RelStr st the
carrier of R1
misses the
carrier of R2 & (
subrelstr ((
[#] G)
\
{x}))
= (
union_of (R1,R2)) & G is
path-connected holds ex b be
Element of R1 st
[b, x]
in the
InternalRel of G
proof
let G be non
empty
symmetric
RelStr;
let x be
Element of G;
let R1,R2 be non
empty
RelStr;
assume that
A1: the
carrier of R1
misses the
carrier of R2 and
A2: (
subrelstr ((
[#] G)
\
{x}))
= (
union_of (R1,R2)) and
A3: G is
path-connected;
set R = (
subrelstr ((
[#] G)
\
{x})), A = the
carrier of R;
the
carrier of R1
c= (the
carrier of R1
\/ the
carrier of R2) by
XBOOLE_1: 7;
then
A4: the
carrier of R1
c= the
carrier of R by
A2,
NECKLA_2:def 2;
set a = the
Element of R1;
A5: A
= ((
[#] G)
\
{x}) by
YELLOW_0:def 15;
A6: x
<> a
proof
assume not thesis;
then x
in (the
carrier of R1
\/ the
carrier of R2) by
XBOOLE_0:def 3;
then x
in (the
carrier of G
\
{x}) by
A2,
A5,
NECKLA_2:def 2;
then not x
in
{x} by
XBOOLE_0:def 5;
hence thesis by
TARSKI:def 1;
end;
reconsider A as
Subset of G by
YELLOW_0:def 15;
A7: the
carrier of R
= A;
then the
carrier of R1
c= the
carrier of G by
A4,
XBOOLE_1: 1;
then a
in the
carrier of G;
then the
InternalRel of G
reduces (x,a) by
A3,
A6;
then
consider p be
FinSequence such that
A8: (
len p)
>
0 and
A9: (p
. 1)
= x and
A10: (p
. (
len p))
= a and
A11: for i be
Nat st i
in (
dom p) & (i
+ 1)
in (
dom p) holds
[(p
. i), (p
. (i
+ 1))]
in the
InternalRel of G by
REWRITE1: 11;
defpred
P[
Nat] means (p
. $1)
in the
carrier of R1 & $1
in (
dom p) & for k be
Nat st k
> $1 holds k
in (
dom p) implies (p
. k)
in the
carrier of R1;
P[(
len p)] by
A8,
A10,
CARD_1: 27,
FINSEQ_3: 25,
FINSEQ_5: 6;
then
A12: ex k be
Nat st
P[k];
ex n0 be
Nat st
P[n0] & for n be
Nat st
P[n] holds n
>= n0 from
NAT_1:sch 5(
A12);
then
consider n0 be
Element of
NAT such that
A13:
P[n0] and
A14: for n be
Nat st
P[n] holds n
>= n0;
n0
<>
0
proof
assume not thesis;
then
0
in (
Seg (
len p)) by
A13,
FINSEQ_1:def 3;
hence contradiction by
FINSEQ_1: 1;
end;
then
consider k0 be
Nat such that
A15: n0
= (k0
+ 1) by
NAT_1: 6;
A16: n0
<> 1
proof
assume not thesis;
then not x
in
{x} by
A5,
A4,
A9,
A13,
XBOOLE_0:def 5;
hence contradiction by
TARSKI:def 1;
end;
A17: k0
>= 1
proof
assume not thesis;
then k0
=
0 by
NAT_1: 25;
hence contradiction by
A15,
A16;
end;
n0
in (
Seg (
len p)) by
A13,
FINSEQ_1:def 3;
then
A18: n0
<= (
len p) by
FINSEQ_1: 1;
A19: k0
< n0 by
A15,
NAT_1: 13;
A20: for k be
Nat st k
> k0 holds k
in (
dom p) implies (p
. k)
in the
carrier of R1
proof
assume not thesis;
then
consider k be
Nat such that
A21: k
> k0 and
A22: k
in (
dom p) and
A23: not (p
. k)
in the
carrier of R1;
k
> n0
proof
per cases by
XXREAL_0: 1;
suppose k
< n0;
hence thesis by
A15,
A21,
NAT_1: 13;
end;
suppose n0
< k;
hence thesis;
end;
suppose n0
= k;
hence thesis by
A13,
A23;
end;
end;
hence contradiction by
A13,
A22,
A23;
end;
A24: the
carrier of G
= (the
carrier of R
\/
{x})
proof
thus the
carrier of G
c= (the
carrier of R
\/
{x})
proof
let a be
object;
assume
A25: a
in the
carrier of G;
per cases ;
suppose a
= x;
then a
in
{x} by
TARSKI:def 1;
hence thesis by
XBOOLE_0:def 3;
end;
suppose a
<> x;
then not a
in
{x} by
TARSKI:def 1;
then a
in A by
A5,
A25,
XBOOLE_0:def 5;
hence thesis by
XBOOLE_0:def 3;
end;
end;
let a be
object;
assume
A26: a
in (the
carrier of R
\/
{x});
per cases by
A26,
XBOOLE_0:def 3;
suppose a
in the
carrier of R;
hence thesis by
A5;
end;
suppose a
in
{x};
hence thesis;
end;
end;
k0
<= n0 by
A15,
XREAL_1: 29;
then k0
<= (
len p) by
A18,
XXREAL_0: 2;
then
A27: k0
in (
dom p) by
A17,
FINSEQ_3: 25;
then
A28:
[(p
. k0), (p
. (k0
+ 1))]
in the
InternalRel of G by
A11,
A13,
A15;
then
A29: (p
. k0)
in the
carrier of G by
ZFMISC_1: 87;
thus thesis
proof
per cases by
A29,
A24,
XBOOLE_0:def 3;
suppose
A30: (p
. k0)
in the
carrier of R;
set u = (p
. k0), v = (p
. n0);
[u, v]
in
[:the
carrier of R, the
carrier of R:] by
A4,
A13,
A30,
ZFMISC_1: 87;
then
A31:
[u, v]
in (the
InternalRel of G
|_2 the
carrier of R) by
A15,
A28,
XBOOLE_0:def 4;
(p
. k0)
in (the
carrier of R1
\/ the
carrier of R2) by
A2,
A30,
NECKLA_2:def 2;
then (p
. k0)
in the
carrier of R1 or (p
. k0)
in the
carrier of R2 by
XBOOLE_0:def 3;
then
reconsider u as
Element of R2 by
A14,
A27,
A19,
A20;
reconsider v as
Element of R1 by
A13;
not
[u, v]
in the
InternalRel of R
proof
u
in (the
carrier of R1
\/ the
carrier of R2) by
XBOOLE_0:def 3;
then
A32: u
in the
carrier of R by
A2,
NECKLA_2:def 2;
A33: v
in the
carrier of R1 & the
InternalRel of R
is_symmetric_in the
carrier of R by
NECKLACE:def 3;
assume not thesis;
then
[v, u]
in the
InternalRel of R by
A4,
A32,
A33;
hence thesis by
A1,
A2,
Th35;
end;
hence thesis by
A31,
YELLOW_0:def 14;
end;
suppose
A34: (p
. k0)
in
{x};
set b = (p
. n0);
reconsider b as
Element of R1 by
A13;
A35: b
in the
carrier of R & the
InternalRel of G
is_symmetric_in the
carrier of G by
A4,
NECKLACE:def 3;
(p
. k0)
= x by
A34,
TARSKI:def 1;
then
[b, x]
in the
InternalRel of G by
A7,
A15,
A28,
A35;
hence thesis;
end;
end;
end;
theorem ::
NECKLA_3:38
Th38: for G be non
empty
symmetric
irreflexive
RelStr, a,b,c,d be
Element of G, Z be
Subset of G st Z
=
{a, b, c, d} & (a,b,c,d)
are_mutually_distinct &
[a, b]
in the
InternalRel of G &
[b, c]
in the
InternalRel of G &
[c, d]
in the
InternalRel of G & not
[a, c]
in the
InternalRel of G & not
[a, d]
in the
InternalRel of G & not
[b, d]
in the
InternalRel of G holds (
subrelstr Z)
embeds (
Necklace 4)
proof
let G be non
empty
symmetric
irreflexive
symmetric
RelStr;
let a,b,c,d be
Element of G;
let Z be
Subset of G;
assume that
A1: Z
=
{a, b, c, d} and
A2: (a,b,c,d)
are_mutually_distinct and
A3:
[a, b]
in the
InternalRel of G and
A4:
[b, c]
in the
InternalRel of G and
A5:
[c, d]
in the
InternalRel of G and
A6: not
[a, c]
in the
InternalRel of G and
A7: not
[a, d]
in the
InternalRel of G and
A8: not
[b, d]
in the
InternalRel of G;
set g = ((
0 ,1)
--> (a,b)), h = ((2,3)
--> (c,d)), f = (g
+* h);
A9: (
rng h)
=
{c, d} by
FUNCT_4: 64;
A10: a
<> b by
A2,
ZFMISC_1:def 6;
A11: (
rng (
0
.--> a))
misses (
rng (1
.--> b))
proof
assume (
rng (
0
.--> a))
meets (
rng (1
.--> b));
then
consider x be
object such that
A12: x
in (
rng (
0
.--> a)) and
A13: x
in (
rng (1
.--> b)) by
XBOOLE_0: 3;
(
rng (
0
.--> a))
=
{a} by
FUNCOP_1: 8;
then (
rng (1
.--> b))
=
{b} & x
= a by
A12,
FUNCOP_1: 8,
TARSKI:def 1;
hence contradiction by
A10,
A13,
TARSKI:def 1;
end;
set H = (
subrelstr Z), N4 = (
Necklace 4), IH = the
InternalRel of H, cH = the
carrier of H, IG = the
InternalRel of G, X =
{
[a, a],
[a, b],
[b, a],
[b, b],
[a, c],
[a, d],
[b, c],
[b, d]}, Y =
{
[c, a],
[c, b],
[d, a],
[d, b],
[c, c],
[c, d],
[d, c],
[d, d]};
A14: the
carrier of H is non
empty by
A1,
YELLOW_0:def 15;
A15: h
= ((2
.--> c)
+* (3
.--> d)) by
FUNCT_4:def 4;
A16: c
<> d by
A2,
ZFMISC_1:def 6;
(
rng (2
.--> c))
misses (
rng (3
.--> d))
proof
assume (
rng (2
.--> c))
meets (
rng (3
.--> d));
then
consider x be
object such that
A17: x
in (
rng (2
.--> c)) and
A18: x
in (
rng (3
.--> d)) by
XBOOLE_0: 3;
(
rng (2
.--> c))
=
{c} by
FUNCOP_1: 8;
then (
rng (3
.--> d))
=
{d} & x
= c by
A17,
FUNCOP_1: 8,
TARSKI:def 1;
hence contradiction by
A16,
A18,
TARSKI:def 1;
end;
then
A19: h is
one-to-one by
A15,
FUNCT_4: 92;
A20: (
rng g)
=
{a, b} by
FUNCT_4: 64;
A21: (
rng g)
misses (
rng h)
proof
assume not thesis;
then
consider x be
object such that
A22: x
in (
rng g) and
A23: x
in (
rng h) by
XBOOLE_0: 3;
A24: x
= c or x
= d by
A9,
A23,
TARSKI:def 2;
x
= a or x
= b by
A20,
A22,
TARSKI:def 2;
hence contradiction by
A2,
A24,
ZFMISC_1:def 6;
end;
(
dom f)
= ((
dom g)
\/ (
dom h)) by
FUNCT_4:def 1
.= (
{
0 , 1}
\/ (
dom h)) by
FUNCT_4: 62
.= (
{
0 , 1}
\/
{2, 3}) by
FUNCT_4: 62
.=
{
0 , 1, 2, 3} by
ENUMSET1: 5;
then
A25: (
dom f)
= the
carrier of N4 by
NECKLACE: 1,
NECKLACE: 20;
A26: (
dom g)
misses (
dom h)
proof
assume not thesis;
then
consider x be
object such that
A27: x
in (
dom g) and
A28: x
in (
dom h) by
XBOOLE_0: 3;
x
=
0 or x
= 1 by
A27,
TARSKI:def 2;
hence contradiction by
A28,
TARSKI:def 2;
end;
then (
rng f)
= ((
rng g)
\/ (
rng h)) by
NECKLACE: 6;
then (
rng f)
=
{a, b, c, d} by
A20,
A9,
ENUMSET1: 5;
then
A29: (
rng f)
= the
carrier of H by
A1,
YELLOW_0:def 15;
then
reconsider f as
Function of N4, H by
A25,
FUNCT_2:def 1,
RELSET_1: 4;
g
= ((
0
.--> a)
+* (1
.--> b)) by
FUNCT_4:def 4;
then
A30: g is
one-to-one by
A11,
FUNCT_4: 92;
then
A31: f is
one-to-one by
A19,
A21,
FUNCT_4: 92;
A32: the
InternalRel of H
=
{
[a, b],
[b, a],
[b, c],
[c, b],
[c, d],
[d, c]}
proof
thus the
InternalRel of H
c=
{
[a, b],
[b, a],
[b, c],
[c, b],
[c, d],
[d, c]}
proof
let x be
object;
A33: the
carrier of H
= Z by
YELLOW_0:def 15;
assume
A34: x
in IH;
then
A35: x
in (IG
|_2 cH) by
YELLOW_0:def 14;
then
A36: x
in IG by
XBOOLE_0:def 4;
x
in
[:cH, cH:] by
A34;
then
A37: x
in (X
\/ Y) by
A1,
A33,
Th3;
per cases by
A37,
XBOOLE_0:def 3;
suppose
A38: x
in X;
thus thesis
proof
per cases by
A38,
ENUMSET1:def 6;
suppose
A39: x
=
[a, a];
not
[a, a]
in IG by
NECKLACE:def 5;
hence thesis by
A35,
A39,
XBOOLE_0:def 4;
end;
suppose x
=
[a, b];
hence thesis by
ENUMSET1:def 4;
end;
suppose x
=
[b, a];
hence thesis by
ENUMSET1:def 4;
end;
suppose
A40: x
=
[b, b];
not
[b, b]
in IG by
NECKLACE:def 5;
hence thesis by
A35,
A40,
XBOOLE_0:def 4;
end;
suppose x
=
[a, c];
hence thesis by
A6,
A35,
XBOOLE_0:def 4;
end;
suppose x
=
[a, d];
hence thesis by
A7,
A35,
XBOOLE_0:def 4;
end;
suppose x
=
[b, c];
hence thesis by
ENUMSET1:def 4;
end;
suppose x
=
[b, d];
hence thesis by
A8,
A35,
XBOOLE_0:def 4;
end;
end;
end;
suppose
A41: x
in Y;
A42: IG
is_symmetric_in the
carrier of G by
NECKLACE:def 3;
thus thesis
proof
per cases by
A41,
ENUMSET1:def 6;
suppose x
=
[c, a];
hence thesis by
A6,
A36,
A42;
end;
suppose x
=
[c, b];
hence thesis by
ENUMSET1:def 4;
end;
suppose x
=
[d, a];
hence thesis by
A7,
A36,
A42;
end;
suppose x
=
[d, b];
hence thesis by
A8,
A36,
A42;
end;
suppose
A43: x
=
[c, c];
not
[c, c]
in IG by
NECKLACE:def 5;
hence thesis by
A35,
A43,
XBOOLE_0:def 4;
end;
suppose x
=
[c, d];
hence thesis by
ENUMSET1:def 4;
end;
suppose x
=
[d, c];
hence thesis by
ENUMSET1:def 4;
end;
suppose
A44: x
=
[d, d];
not
[d, d]
in IG by
NECKLACE:def 5;
hence thesis by
A35,
A44,
XBOOLE_0:def 4;
end;
end;
end;
end;
let x be
object;
assume
A45: x
in
{
[a, b],
[b, a],
[b, c],
[c, b],
[c, d],
[d, c]};
per cases by
A45,
ENUMSET1:def 4;
suppose
A46: x
=
[a, b];
b
in Z by
A1,
ENUMSET1:def 2;
then
A47: b
in cH by
YELLOW_0:def 15;
a
in Z by
A1,
ENUMSET1:def 2;
then a
in cH by
YELLOW_0:def 15;
then
[a, b]
in
[:cH, cH:] by
A47,
ZFMISC_1: 87;
then x
in (IG
|_2 cH) by
A3,
A46,
XBOOLE_0:def 4;
hence thesis by
YELLOW_0:def 14;
end;
suppose
A48: x
=
[b, a];
IG
is_symmetric_in the
carrier of G by
NECKLACE:def 3;
then
A49:
[b, a]
in IG by
A3;
a
in Z by
A1,
ENUMSET1:def 2;
then
A50: a
in cH by
YELLOW_0:def 15;
b
in Z by
A1,
ENUMSET1:def 2;
then b
in cH by
YELLOW_0:def 15;
then
[b, a]
in
[:cH, cH:] by
A50,
ZFMISC_1: 87;
then x
in (IG
|_2 cH) by
A48,
A49,
XBOOLE_0:def 4;
hence thesis by
YELLOW_0:def 14;
end;
suppose
A51: x
=
[b, c];
c
in Z by
A1,
ENUMSET1:def 2;
then
A52: c
in cH by
YELLOW_0:def 15;
b
in Z by
A1,
ENUMSET1:def 2;
then b
in cH by
YELLOW_0:def 15;
then
[b, c]
in
[:cH, cH:] by
A52,
ZFMISC_1: 87;
then x
in (IG
|_2 cH) by
A4,
A51,
XBOOLE_0:def 4;
hence thesis by
YELLOW_0:def 14;
end;
suppose
A53: x
=
[c, b];
IG
is_symmetric_in the
carrier of G by
NECKLACE:def 3;
then
A54:
[c, b]
in IG by
A4;
c
in Z by
A1,
ENUMSET1:def 2;
then
A55: c
in cH by
YELLOW_0:def 15;
b
in Z by
A1,
ENUMSET1:def 2;
then b
in cH by
YELLOW_0:def 15;
then
[c, b]
in
[:cH, cH:] by
A55,
ZFMISC_1: 87;
then x
in (IG
|_2 cH) by
A53,
A54,
XBOOLE_0:def 4;
hence thesis by
YELLOW_0:def 14;
end;
suppose
A56: x
=
[c, d];
d
in Z by
A1,
ENUMSET1:def 2;
then
A57: d
in cH by
YELLOW_0:def 15;
c
in Z by
A1,
ENUMSET1:def 2;
then c
in cH by
YELLOW_0:def 15;
then
[c, d]
in
[:cH, cH:] by
A57,
ZFMISC_1: 87;
then x
in (IG
|_2 cH) by
A5,
A56,
XBOOLE_0:def 4;
hence thesis by
YELLOW_0:def 14;
end;
suppose
A58: x
=
[d, c];
IG
is_symmetric_in the
carrier of G by
NECKLACE:def 3;
then
A59:
[d, c]
in IG by
A5;
d
in Z by
A1,
ENUMSET1:def 2;
then
A60: d
in cH by
YELLOW_0:def 15;
c
in Z by
A1,
ENUMSET1:def 2;
then c
in cH by
YELLOW_0:def 15;
then
[d, c]
in
[:cH, cH:] by
A60,
ZFMISC_1: 87;
then x
in (IG
|_2 cH) by
A58,
A59,
XBOOLE_0:def 4;
hence thesis by
YELLOW_0:def 14;
end;
end;
for x,y be
Element of N4 holds
[x, y]
in the
InternalRel of N4 iff
[(f
. x), (f
. y)]
in the
InternalRel of H
proof
let x,y be
Element of N4;
thus
[x, y]
in the
InternalRel of N4 implies
[(f
. x), (f
. y)]
in the
InternalRel of H
proof
assume
A61:
[x, y]
in the
InternalRel of N4;
per cases by
A61,
ENUMSET1:def 4,
NECKLA_2: 2;
suppose
A62:
[x, y]
=
[
0 , 1];
then
A63: y
= 1 by
XTUPLE_0: 1;
then y
in
{
0 , 1} by
TARSKI:def 2;
then y
in (
dom g) by
FUNCT_4: 62;
then
A64: (f
. y)
= (g
. 1) by
A26,
A63,
FUNCT_4: 16
.= b by
FUNCT_4: 63;
A65: x
=
0 by
A62,
XTUPLE_0: 1;
then x
in
{
0 , 1} by
TARSKI:def 2;
then x
in (
dom g) by
FUNCT_4: 62;
then (f
. x)
= (g
.
0 ) by
A26,
A65,
FUNCT_4: 16
.= a by
FUNCT_4: 63;
hence thesis by
A32,
A64,
ENUMSET1:def 4;
end;
suppose
A66:
[x, y]
=
[1,
0 ];
then
A67: y
=
0 by
XTUPLE_0: 1;
then y
in
{
0 , 1} by
TARSKI:def 2;
then y
in (
dom g) by
FUNCT_4: 62;
then
A68: (f
. y)
= (g
.
0 ) by
A26,
A67,
FUNCT_4: 16
.= a by
FUNCT_4: 63;
A69: x
= 1 by
A66,
XTUPLE_0: 1;
then x
in
{
0 , 1} by
TARSKI:def 2;
then x
in (
dom g) by
FUNCT_4: 62;
then (f
. x)
= (g
. 1) by
A26,
A69,
FUNCT_4: 16
.= b by
FUNCT_4: 63;
hence thesis by
A32,
A68,
ENUMSET1:def 4;
end;
suppose
A70:
[x, y]
=
[1, 2];
then
A71: x
= 1 by
XTUPLE_0: 1;
then x
in
{
0 , 1} by
TARSKI:def 2;
then x
in (
dom g) by
FUNCT_4: 62;
then
A72: (f
. x)
= (g
. 1) by
A26,
A71,
FUNCT_4: 16
.= b by
FUNCT_4: 63;
A73: y
= 2 by
A70,
XTUPLE_0: 1;
then y
in
{2, 3} by
TARSKI:def 2;
then
A74: y
in (
dom h) by
FUNCT_4: 62;
(g
+* h)
= (h
+* g) by
A26,
FUNCT_4: 35;
then (f
. y)
= (h
. 2) by
A26,
A73,
A74,
FUNCT_4: 16
.= c by
FUNCT_4: 63;
hence thesis by
A32,
A72,
ENUMSET1:def 4;
end;
suppose
A75:
[x, y]
=
[2, 1];
then
A76: y
= 1 by
XTUPLE_0: 1;
then y
in
{
0 , 1} by
TARSKI:def 2;
then y
in (
dom g) by
FUNCT_4: 62;
then
A77: (f
. y)
= (g
. 1) by
A26,
A76,
FUNCT_4: 16
.= b by
FUNCT_4: 63;
A78: x
= 2 by
A75,
XTUPLE_0: 1;
then x
in
{2, 3} by
TARSKI:def 2;
then
A79: x
in (
dom h) by
FUNCT_4: 62;
(g
+* h)
= (h
+* g) by
A26,
FUNCT_4: 35;
then (f
. x)
= (h
. 2) by
A26,
A78,
A79,
FUNCT_4: 16
.= c by
FUNCT_4: 63;
hence thesis by
A32,
A77,
ENUMSET1:def 4;
end;
suppose
A80:
[x, y]
=
[2, 3];
A81: (g
+* h)
= (h
+* g) by
A26,
FUNCT_4: 35;
A82: y
= 3 by
A80,
XTUPLE_0: 1;
then y
in
{2, 3} by
TARSKI:def 2;
then y
in (
dom h) by
FUNCT_4: 62;
then
A83: (f
. y)
= (h
. 3) by
A26,
A82,
A81,
FUNCT_4: 16
.= d by
FUNCT_4: 63;
A84: x
= 2 by
A80,
XTUPLE_0: 1;
then x
in
{2, 3} by
TARSKI:def 2;
then x
in (
dom h) by
FUNCT_4: 62;
then (f
. x)
= (h
. 2) by
A26,
A84,
A81,
FUNCT_4: 16
.= c by
FUNCT_4: 63;
hence thesis by
A32,
A83,
ENUMSET1:def 4;
end;
suppose
A85:
[x, y]
=
[3, 2];
A86: (g
+* h)
= (h
+* g) by
A26,
FUNCT_4: 35;
A87: y
= 2 by
A85,
XTUPLE_0: 1;
then y
in
{3, 2} by
TARSKI:def 2;
then y
in (
dom h) by
FUNCT_4: 62;
then
A88: (f
. y)
= (h
. 2) by
A26,
A87,
A86,
FUNCT_4: 16
.= c by
FUNCT_4: 63;
A89: x
= 3 by
A85,
XTUPLE_0: 1;
then x
in
{3, 2} by
TARSKI:def 2;
then x
in (
dom h) by
FUNCT_4: 62;
then (f
. x)
= (h
. 3) by
A26,
A89,
A86,
FUNCT_4: 16
.= d by
FUNCT_4: 63;
hence thesis by
A32,
A88,
ENUMSET1:def 4;
end;
end;
thus
[(f
. x), (f
. y)]
in the
InternalRel of H implies
[x, y]
in the
InternalRel of N4
proof
reconsider F = (f
" ) as
Function of the
carrier of H, the
carrier of N4 by
A29,
A31,
FUNCT_2: 25;
A90: (
dom g)
=
{
0 , 1} by
FUNCT_4: 62;
A91: (
rng g)
=
{a, b} by
FUNCT_4: 64;
then
reconsider g as
Function of
{
0 , 1},
{a, b} by
A90,
RELSET_1: 4;
reconsider G = (g
" ) as
Function of
{a, b},
{
0 , 1} by
A20,
A30,
FUNCT_2: 25;
A92: (
dom h)
=
{2, 3} by
FUNCT_4: 62;
A93: (
rng h)
=
{c, d} by
FUNCT_4: 64;
then
reconsider h as
Function of
{2, 3},
{c, d} by
A92,
RELSET_1: 4;
reconsider Hh = (h
" ) as
Function of
{c, d},
{2, 3} by
A9,
A19,
FUNCT_2: 25;
A94: (
dom Hh)
=
{c, d} by
A19,
A93,
FUNCT_1: 33;
A95: Hh
= ((c,d)
--> (2,3)) by
A16,
NECKLACE: 10;
A96: F
= (G
+* Hh) by
A26,
A30,
A19,
A21,
NECKLACE: 7;
A97: G
= ((a,b)
--> (
0 ,1)) by
A10,
NECKLACE: 10;
A98: (
dom G)
=
{a, b} by
A30,
A91,
FUNCT_1: 33;
then (G
+* Hh)
= (Hh
+* G) by
A20,
A9,
A21,
A94,
FUNCT_4: 35;
then
A99: F
= (Hh
+* G) by
A26,
A30,
A19,
A21,
NECKLACE: 7;
assume
A100:
[(f
. x), (f
. y)]
in the
InternalRel of H;
per cases by
A32,
A100,
ENUMSET1:def 4;
suppose
A101:
[(f
. x), (f
. y)]
=
[a, b];
then
A102: (f
. x)
= a by
XTUPLE_0: 1;
then (f
. x)
in
{a, b} by
TARSKI:def 2;
then (F
. (f
. x))
= (G
. a) by
A20,
A9,
A21,
A98,
A94,
A96,
A102,
FUNCT_4: 16
.=
0 by
A10,
A97,
FUNCT_4: 63;
then
A103: x
=
0 by
A14,
A31,
FUNCT_2: 26;
A104: (f
. y)
= b by
A101,
XTUPLE_0: 1;
then (f
. y)
in (
dom G) by
A98,
TARSKI:def 2;
then
A105: (F
. (f
. y))
= (G
. b) by
A20,
A9,
A21,
A98,
A94,
A96,
A104,
FUNCT_4: 16
.= 1 by
A97,
FUNCT_4: 63;
(F
. (f
. y))
= y by
A14,
A31,
FUNCT_2: 26;
hence thesis by
A103,
A105,
ENUMSET1:def 4,
NECKLA_2: 2;
end;
suppose
A106:
[(f
. x), (f
. y)]
=
[b, a];
then
A107: (f
. y)
= a by
XTUPLE_0: 1;
then (f
. y)
in
{a, b} by
TARSKI:def 2;
then (F
. (f
. y))
= (G
. a) by
A20,
A9,
A21,
A98,
A94,
A96,
A107,
FUNCT_4: 16
.=
0 by
A10,
A97,
FUNCT_4: 63;
then
A108: y
=
0 by
A14,
A31,
FUNCT_2: 26;
A109: (f
. x)
= b by
A106,
XTUPLE_0: 1;
then (f
. x)
in (
dom G) by
A98,
TARSKI:def 2;
then
A110: (F
. (f
. x))
= (G
. b) by
A20,
A9,
A21,
A98,
A94,
A96,
A109,
FUNCT_4: 16
.= 1 by
A97,
FUNCT_4: 63;
(F
. (f
. x))
= x by
A14,
A31,
FUNCT_2: 26;
hence thesis by
A108,
A110,
ENUMSET1:def 4,
NECKLA_2: 2;
end;
suppose
A111:
[(f
. x), (f
. y)]
=
[b, c];
then
A112: (f
. x)
= b by
XTUPLE_0: 1;
then (f
. x)
in (
dom G) by
A98,
TARSKI:def 2;
then (F
. (f
. x))
= (G
. b) by
A20,
A9,
A21,
A98,
A94,
A96,
A112,
FUNCT_4: 16
.= 1 by
A97,
FUNCT_4: 63;
then
A113: x
= 1 by
A14,
A31,
FUNCT_2: 26;
A114: (f
. y)
= c by
A111,
XTUPLE_0: 1;
then (f
. y)
in (
dom Hh) by
A94,
TARSKI:def 2;
then
A115: (F
. (f
. y))
= (Hh
. c) by
A20,
A9,
A21,
A98,
A94,
A99,
A114,
FUNCT_4: 16
.= 2 by
A16,
A95,
FUNCT_4: 63;
(F
. (f
. y))
= y by
A14,
A31,
FUNCT_2: 26;
hence thesis by
A113,
A115,
ENUMSET1:def 4,
NECKLA_2: 2;
end;
suppose
A116:
[(f
. x), (f
. y)]
=
[c, b];
then
A117: (f
. y)
= b by
XTUPLE_0: 1;
then (f
. y)
in (
dom G) by
A98,
TARSKI:def 2;
then (F
. (f
. y))
= (G
. b) by
A20,
A9,
A21,
A98,
A94,
A96,
A117,
FUNCT_4: 16
.= 1 by
A97,
FUNCT_4: 63;
then
A118: y
= 1 by
A14,
A31,
FUNCT_2: 26;
A119: (f
. x)
= c by
A116,
XTUPLE_0: 1;
then (f
. x)
in (
dom Hh) by
A94,
TARSKI:def 2;
then
A120: (F
. (f
. x))
= (Hh
. c) by
A20,
A9,
A21,
A98,
A94,
A99,
A119,
FUNCT_4: 16
.= 2 by
A16,
A95,
FUNCT_4: 63;
(F
. (f
. x))
= x by
A14,
A31,
FUNCT_2: 26;
hence thesis by
A118,
A120,
ENUMSET1:def 4,
NECKLA_2: 2;
end;
suppose
A121:
[(f
. x), (f
. y)]
=
[c, d];
then
A122: (f
. x)
= c by
XTUPLE_0: 1;
then (f
. x)
in
{c, d} by
TARSKI:def 2;
then (F
. (f
. x))
= (Hh
. c) by
A20,
A9,
A21,
A98,
A94,
A99,
A122,
FUNCT_4: 16
.= 2 by
A16,
A95,
FUNCT_4: 63;
then
A123: x
= 2 by
A14,
A31,
FUNCT_2: 26;
A124: (f
. y)
= d by
A121,
XTUPLE_0: 1;
then (f
. y)
in (
dom Hh) by
A94,
TARSKI:def 2;
then
A125: (F
. (f
. y))
= (Hh
. d) by
A20,
A9,
A21,
A98,
A94,
A99,
A124,
FUNCT_4: 16
.= 3 by
A95,
FUNCT_4: 63;
(F
. (f
. y))
= y by
A14,
A31,
FUNCT_2: 26;
hence thesis by
A123,
A125,
ENUMSET1:def 4,
NECKLA_2: 2;
end;
suppose
A126:
[(f
. x), (f
. y)]
=
[d, c];
then
A127: (f
. y)
= c by
XTUPLE_0: 1;
then (f
. y)
in
{c, d} by
TARSKI:def 2;
then (F
. (f
. y))
= (Hh
. c) by
A20,
A9,
A21,
A98,
A94,
A99,
A127,
FUNCT_4: 16
.= 2 by
A16,
A95,
FUNCT_4: 63;
then
A128: y
= 2 by
A14,
A31,
FUNCT_2: 26;
A129: (f
. x)
= d by
A126,
XTUPLE_0: 1;
then (f
. x)
in (
dom Hh) by
A94,
TARSKI:def 2;
then
A130: (F
. (f
. x))
= (Hh
. d) by
A20,
A9,
A21,
A98,
A94,
A99,
A129,
FUNCT_4: 16
.= 3 by
A95,
FUNCT_4: 63;
(F
. (f
. x))
= x by
A14,
A31,
FUNCT_2: 26;
hence thesis by
A128,
A130,
ENUMSET1:def 4,
NECKLA_2: 2;
end;
end;
end;
hence thesis by
A31;
end;
theorem ::
NECKLA_3:39
Th39: for G be non
empty
irreflexive
symmetric
RelStr, x be
Element of G, R1,R2 be non
empty
RelStr st the
carrier of R1
misses the
carrier of R2 & (
subrelstr ((
[#] G)
\
{x}))
= (
union_of (R1,R2)) & G is non
trivial & G is
path-connected & (
ComplRelStr G) is
path-connected holds G
embeds (
Necklace 4)
proof
let G be non
empty
irreflexive
symmetric
RelStr, x be
Element of G, R1,R2 be non
empty
RelStr;
assume that
A1: the
carrier of R1
misses the
carrier of R2 and
A2: (
subrelstr ((
[#] G)
\
{x}))
= (
union_of (R1,R2)) and
A3: G is non
trivial and
A4: G is
path-connected and
A5: (
ComplRelStr G) is
path-connected;
consider a be
Element of R1 such that
A6:
[a, x]
in the
InternalRel of G by
A1,
A2,
A4,
Th37;
set A = (the
carrier of G
\
{x}), X =
{x};
reconsider A as
Subset of G;
set R = (
subrelstr A);
reconsider R as non
empty
irreflexive
symmetric
RelStr by
A3,
YELLOW_0:def 15;
R
= (
subrelstr ((
[#] G)
\
{x})) & R
= (
union_of (R2,R1)) by
A2,
Th8;
then
consider b be
Element of R2 such that
A7:
[b, x]
in the
InternalRel of G by
A1,
A4,
Th37;
reconsider X1 = { y where y be
Element of R1 :
[y, x]
in the
InternalRel of G }, Y1 = { y where y be
Element of R1 : not
[y, x]
in the
InternalRel of G }, X2 = { y where y be
Element of R2 :
[y, x]
in the
InternalRel of G }, Y2 = { y where y be
Element of R2 : not
[y, x]
in the
InternalRel of G } as
set;
reconsider X as
Subset of G;
set H = (
subrelstr X);
A8: X1
misses Y1
proof
assume not thesis;
then
consider a be
object such that
A9: a
in X1 & a
in Y1 by
XBOOLE_0: 3;
(ex y1 be
Element of R1 st y1
= a &
[y1, x]
in the
InternalRel of G) & ex y2 be
Element of R1 st y2
= a & not
[y2, x]
in the
InternalRel of G by
A9;
hence contradiction;
end;
A10: a
in X1 by
A6;
A11: the
carrier of R1
= (X1
\/ Y1)
proof
thus the
carrier of R1
c= (X1
\/ Y1)
proof
let a be
object;
assume
A12: a
in the
carrier of R1;
per cases ;
suppose
[a, x]
in the
InternalRel of G;
then a
in X1 by
A12;
hence thesis by
XBOOLE_0:def 3;
end;
suppose not
[a, x]
in the
InternalRel of G;
then a
in Y1 by
A12;
hence thesis by
XBOOLE_0:def 3;
end;
end;
let a be
object such that
A13: a
in (X1
\/ Y1);
per cases by
A13,
XBOOLE_0:def 3;
suppose a
in X1;
then ex y be
Element of R1 st a
= y &
[y, x]
in the
InternalRel of G;
hence thesis;
end;
suppose a
in Y1;
then ex y be
Element of R1 st a
= y & not
[y, x]
in the
InternalRel of G;
hence thesis;
end;
end;
A14: X2
misses Y2
proof
assume not thesis;
then
consider a be
object such that
A15: a
in X2 & a
in Y2 by
XBOOLE_0: 3;
(ex y1 be
Element of R2 st y1
= a &
[y1, x]
in the
InternalRel of G) & ex y2 be
Element of R2 st y2
= a & not
[y2, x]
in the
InternalRel of G by
A15;
hence contradiction;
end;
A16: the
carrier of H
misses the
carrier of R
proof
assume not thesis;
then (the
carrier of H
/\ the
carrier of R)
<>
{} ;
then (X
/\ the
carrier of R)
<>
{} by
YELLOW_0:def 15;
then (X
/\ A)
<>
{} by
YELLOW_0:def 15;
then
consider a be
object such that
A17: a
in (X
/\ A) by
XBOOLE_0:def 1;
a
in X & a
in A by
A17,
XBOOLE_0:def 4;
hence contradiction by
XBOOLE_0:def 5;
end;
reconsider H as non
empty
irreflexive
symmetric
RelStr by
YELLOW_0:def 15;
A18: b
in X2 by
A7;
A19: the
carrier of G
= (the
carrier of R
\/
{x})
proof
thus the
carrier of G
c= (the
carrier of R
\/
{x})
proof
let a be
object;
assume
A20: a
in the
carrier of G;
per cases ;
suppose a
= x;
then a
in
{x} by
TARSKI:def 1;
hence thesis by
XBOOLE_0:def 3;
end;
suppose a
<> x;
then not a
in
{x} by
TARSKI:def 1;
then a
in (the
carrier of G
\
{x}) by
A20,
XBOOLE_0:def 5;
then a
in the
carrier of R by
YELLOW_0:def 15;
hence thesis by
XBOOLE_0:def 3;
end;
end;
let a be
object;
assume
A21: a
in (the
carrier of R
\/
{x});
per cases by
A21,
XBOOLE_0:def 3;
suppose a
in the
carrier of R;
then a
in (the
carrier of G
\
{x}) by
YELLOW_0:def 15;
hence thesis;
end;
suppose a
in
{x};
hence thesis;
end;
end;
A22: the
carrier of R2
= (X2
\/ Y2)
proof
thus the
carrier of R2
c= (X2
\/ Y2)
proof
let a be
object;
assume
A23: a
in the
carrier of R2;
per cases ;
suppose
[a, x]
in the
InternalRel of G;
then a
in X2 by
A23;
hence thesis by
XBOOLE_0:def 3;
end;
suppose not
[a, x]
in the
InternalRel of G;
then a
in Y2 by
A23;
hence thesis by
XBOOLE_0:def 3;
end;
end;
let a be
object such that
A24: a
in (X2
\/ Y2);
per cases by
A24,
XBOOLE_0:def 3;
suppose a
in X2;
then ex y be
Element of R2 st a
= y &
[y, x]
in the
InternalRel of G;
hence thesis;
end;
suppose a
in Y2;
then ex y be
Element of R2 st a
= y & not
[y, x]
in the
InternalRel of G;
hence thesis;
end;
end;
A25: (Y1
\/ Y2) is non
empty
proof
assume
A26: not thesis;
then
A27: Y2 is
empty;
A28: Y1 is
empty by
A26;
A29: for a be
Element of R holds
[a, x]
in the
InternalRel of G
proof
let a be
Element of R;
A30: the
carrier of R
= (the
carrier of R1
\/ the
carrier of R2) by
A2,
NECKLA_2:def 2;
per cases by
A30,
XBOOLE_0:def 3;
suppose a
in the
carrier of R1;
then ex y be
Element of R1 st a
= y &
[y, x]
in the
InternalRel of G by
A11,
A28;
hence thesis;
end;
suppose a
in the
carrier of R2;
then ex y be
Element of R2 st a
= y &
[y, x]
in the
InternalRel of G by
A22,
A27;
hence thesis;
end;
end;
not (
ComplRelStr G) is
path-connected
proof
A31: a
<> x
proof
assume not thesis;
then x
in (the
carrier of R1
\/ the
carrier of R2) by
XBOOLE_0:def 3;
then
A32: x
in the
carrier of R by
A2,
NECKLA_2:def 2;
x
in
{x} by
TARSKI:def 1;
then x
in the
carrier of H by
YELLOW_0:def 15;
then x
in (the
carrier of R
/\ the
carrier of H) by
A32,
XBOOLE_0:def 4;
hence contradiction by
A16;
end;
a
in (the
carrier of R1
\/ the
carrier of R2) by
XBOOLE_0:def 3;
then
A33: a
in the
carrier of R by
A2,
NECKLA_2:def 2;
the
carrier of R
c= the
carrier of G by
A19,
XBOOLE_1: 7;
then
A34: a is
Element of (
ComplRelStr G) by
A33,
NECKLACE:def 8;
A35: x is
Element of (
ComplRelStr G) by
NECKLACE:def 8;
assume not thesis;
then the
InternalRel of (
ComplRelStr G)
reduces (x,a) by
A31,
A34,
A35;
then
consider p be
FinSequence such that
A36: (
len p)
>
0 and
A37: (p
. 1)
= x and
A38: (p
. (
len p))
= a and
A39: for i be
Nat st i
in (
dom p) & (i
+ 1)
in (
dom p) holds
[(p
. i), (p
. (i
+ 1))]
in the
InternalRel of (
ComplRelStr G) by
REWRITE1: 11;
A40: (
0
+ 1)
<= (
len p) by
A36,
NAT_1: 13;
then (
len p)
> 1 by
A31,
A37,
A38,
XXREAL_0: 1;
then (1
+ 1)
<= (
len p) by
NAT_1: 13;
then
A41: 2
in (
dom p) by
FINSEQ_3: 25;
1
in (
dom p) by
A40,
FINSEQ_3: 25;
then
A42:
[(p
. 1), (p
. (1
+ 1))]
in the
InternalRel of (
ComplRelStr G) by
A39,
A41;
A43: (p
. 2)
<> x
proof
A44:
[x, x]
in (
id the
carrier of G) by
RELAT_1:def 10;
assume not thesis;
then
[x, x]
in (the
InternalRel of (
ComplRelStr G)
/\ (
id the
carrier of G)) by
A37,
A42,
A44,
XBOOLE_0:def 4;
then the
InternalRel of (
ComplRelStr G)
meets (
id the
carrier of G);
hence contradiction by
Th13;
end;
(p
. 2)
in the
carrier of (
ComplRelStr G) by
A42,
ZFMISC_1: 87;
then
A45: (p
. 2)
in the
carrier of G by
NECKLACE:def 8;
(p
. 2)
in the
carrier of R
proof
assume not thesis;
then (p
. 2)
in
{x} by
A19,
A45,
XBOOLE_0:def 3;
hence thesis by
A43,
TARSKI:def 1;
end;
then
A46:
[(p
. 2), x]
in the
InternalRel of G by
A29;
A47: the
InternalRel of (
ComplRelStr G)
is_symmetric_in the
carrier of (
ComplRelStr G) by
NECKLACE:def 3;
(p
. 1)
in the
carrier of (
ComplRelStr G) & (p
. (1
+ 1))
in the
carrier of (
ComplRelStr G) by
A42,
ZFMISC_1: 87;
then
[(p
. (1
+ 1)), (p
. 1)]
in the
InternalRel of (
ComplRelStr G) by
A42,
A47;
then
[(p
. 2), x]
in (the
InternalRel of (
ComplRelStr G)
/\ the
InternalRel of G) by
A37,
A46,
XBOOLE_0:def 4;
then the
InternalRel of (
ComplRelStr G)
meets the
InternalRel of G;
hence thesis by
Th12;
end;
hence thesis by
A5;
end;
thus thesis
proof
per cases by
A25;
suppose
A48: Y1 is non
empty;
ex b be
Element of Y1, c be
Element of X1 st
[b, c]
in the
InternalRel of G
proof
set b = the
Element of Y1;
a
in (the
carrier of R1
\/ the
carrier of R2) by
XBOOLE_0:def 3;
then
A49: a
in the
carrier of R by
A2,
NECKLA_2:def 2;
b
in Y1 by
A48;
then ex y be
Element of R1 st y
= b & not
[y, x]
in the
InternalRel of G;
then b
in (the
carrier of R1
\/ the
carrier of R2) by
XBOOLE_0:def 3;
then
A50: b
in the
carrier of R by
A2,
NECKLA_2:def 2;
A51: the
carrier of R
c= the
carrier of G by
A19,
XBOOLE_1: 7;
then
reconsider a as
Element of G by
A49;
reconsider b as
Element of G by
A51,
A50;
a
<> b
proof
assume
A52: not thesis;
a
in X1 by
A6;
then a
in (X1
/\ Y1) by
A48,
A52,
XBOOLE_0:def 4;
hence contradiction by
A8;
end;
then the
InternalRel of G
reduces (a,b) by
A4;
then
consider p be
FinSequence such that
A53: (
len p)
>
0 and
A54: (p
. 1)
= a and
A55: (p
. (
len p))
= b and
A56: for i be
Nat st i
in (
dom p) & (i
+ 1)
in (
dom p) holds
[(p
. i), (p
. (i
+ 1))]
in the
InternalRel of G by
REWRITE1: 11;
defpred
P[
Nat] means (p
. $1)
in Y1 & $1
in (
dom p) & for k be
Nat st k
> $1 holds k
in (
dom p) implies (p
. k)
in Y1;
for k be
Nat st k
> (
len p) holds k
in (
dom p) implies (p
. k)
in Y1
proof
let k be
Nat such that
A57: k
> (
len p);
assume k
in (
dom p);
then k
in (
Seg (
len p)) by
FINSEQ_1:def 3;
hence thesis by
A57,
FINSEQ_1: 1;
end;
then
P[(
len p)] by
A48,
A53,
A55,
CARD_1: 27,
FINSEQ_5: 6;
then
A58: ex k be
Nat st
P[k];
ex n0 be
Nat st
P[n0] & for n be
Nat st
P[n] holds n
>= n0 from
NAT_1:sch 5(
A58);
then
consider n0 be
Nat such that
A59:
P[n0] and
A60: for n be
Nat st
P[n] holds n
>= n0;
n0
<>
0
proof
assume not thesis;
then
0
in (
Seg (
len p)) by
A59,
FINSEQ_1:def 3;
hence contradiction by
FINSEQ_1: 1;
end;
then
consider k0 be
Nat such that
A61: n0
= (k0
+ 1) by
NAT_1: 6;
A62: n0
<> 1
proof
assume
A63: not thesis;
a
in X1 by
A6;
then (X1
/\ Y1) is non
empty by
A54,
A59,
A63,
XBOOLE_0:def 4;
hence contradiction by
A8;
end;
A64: k0
>= 1
proof
assume not thesis;
then k0
=
0 by
NAT_1: 25;
hence contradiction by
A61,
A62;
end;
n0
in (
Seg (
len p)) by
A59,
FINSEQ_1:def 3;
then k0
<= (k0
+ 1) & n0
<= (
len p) by
FINSEQ_1: 1,
XREAL_1: 29;
then
A65: k0
<= (
len p) by
A61,
XXREAL_0: 2;
then
A66: k0
in (
dom p) by
A64,
FINSEQ_3: 25;
then
A67:
[(p
. k0), (p
. (k0
+ 1))]
in the
InternalRel of G by
A56,
A59,
A61;
then
A68: the
InternalRel of G
is_symmetric_in the
carrier of G & (p
. k0)
in the
carrier of G by
NECKLACE:def 3,
ZFMISC_1: 87;
(p
. n0)
in the
carrier of G by
A61,
A67,
ZFMISC_1: 87;
then
A69:
[(p
. n0), (p
. k0)]
in the
InternalRel of G by
A61,
A67,
A68;
A70: for k be
Nat st k
> k0 holds k
in (
dom p) implies (p
. k)
in Y1
proof
assume not thesis;
then
consider k be
Nat such that
A71: k
> k0 and
A72: k
in (
dom p) and
A73: not (p
. k)
in Y1;
k
> n0
proof
per cases by
XXREAL_0: 1;
suppose k
< n0;
hence thesis by
A61,
A71,
NAT_1: 13;
end;
suppose n0
< k;
hence thesis;
end;
suppose n0
= k;
hence thesis by
A59,
A73;
end;
end;
hence contradiction by
A59,
A72,
A73;
end;
k0
< n0 by
A61,
NAT_1: 13;
then
A74: not
P[k0] by
A60;
(p
. k0)
in the
carrier of G by
A67,
ZFMISC_1: 87;
then (p
. k0)
in the
carrier of R or (p
. k0)
in
{x} by
A19,
XBOOLE_0:def 3;
then
A75: (p
. k0)
in (the
carrier of R1
\/ the
carrier of R2) or (p
. k0)
in
{x} by
A2,
NECKLA_2:def 2;
thus thesis
proof
per cases by
A61,
A67,
A75,
XBOOLE_0:def 3,
ZFMISC_1: 87;
suppose
A76: (p
. k0)
in the
carrier of R1 & (p
. n0)
in the
carrier of G;
then
reconsider m = (p
. k0) as
Element of X1 by
A11,
A64,
A65,
A74,
A70,
FINSEQ_3: 25,
XBOOLE_0:def 3;
m
in (the
carrier of R1
\/ the
carrier of R2) by
A76,
XBOOLE_0:def 3;
then
A77: m
in the
carrier of R by
A2,
NECKLA_2:def 2;
reconsider l = (p
. n0) as
Element of Y1 by
A59;
A78: the
carrier of R
c= the
carrier of G by
A19,
XBOOLE_1: 7;
l
in the
carrier of R1 by
A11,
A59,
XBOOLE_0:def 3;
then l
in (the
carrier of R1
\/ the
carrier of R2) by
XBOOLE_0:def 3;
then
A79: l
in the
carrier of R by
A2,
NECKLA_2:def 2;
[m, l]
in the
InternalRel of G & the
InternalRel of G
is_symmetric_in the
carrier of G by
A56,
A59,
A61,
A66,
NECKLACE:def 3;
then
[l, m]
in the
InternalRel of G by
A79,
A77,
A78;
hence thesis;
end;
suppose (p
. k0)
in the
carrier of R2 & (p
. n0)
in the
carrier of G;
then
reconsider m = (p
. k0) as
Element of R2;
reconsider l = (p
. n0) as
Element of R1 by
A11,
A59,
XBOOLE_0:def 3;
m
in (the
carrier of R1
\/ the
carrier of R2) by
XBOOLE_0:def 3;
then
A80: m
in the
carrier of R by
A2,
NECKLA_2:def 2;
l
in (the
carrier of R1
\/ the
carrier of R2) by
XBOOLE_0:def 3;
then l
in the
carrier of R by
A2,
NECKLA_2:def 2;
then
[l, m]
in
[:the
carrier of R, the
carrier of R:] by
A80,
ZFMISC_1: 87;
then
[l, m]
in (the
InternalRel of G
|_2 the
carrier of R) by
A69,
XBOOLE_0:def 4;
then
[l, m]
in the
InternalRel of R by
YELLOW_0:def 14;
hence thesis by
A1,
A2,
Th35;
end;
suppose
A81: (p
. k0)
in
{x} & (p
. n0)
in the
carrier of G;
ex y1 be
Element of R1 st (p
. n0)
= y1 & not
[y1, x]
in the
InternalRel of G by
A59;
hence thesis by
A69,
A81,
TARSKI:def 1;
end;
end;
end;
then
consider u be
Element of Y1, v be
Element of X1 such that
A82:
[u, v]
in the
InternalRel of G;
set w = the
Element of X2;
w
in X2 by
A18;
then
A83: ex y be
Element of R2 st y
= w &
[y, x]
in the
InternalRel of G;
set Z =
{u, v, x, w};
Z
c= the
carrier of G
proof
w
in X2 by
A18;
then ex y2 be
Element of R2 st y2
= w &
[y2, x]
in the
InternalRel of G;
then w
in (the
carrier of R1
\/ the
carrier of R2) by
XBOOLE_0:def 3;
then
A84: w
in the
carrier of R by
A2,
NECKLA_2:def 2;
v
in X1 by
A10;
then ex y1 be
Element of R1 st y1
= v &
[y1, x]
in the
InternalRel of G;
then v
in (the
carrier of R1
\/ the
carrier of R2) by
XBOOLE_0:def 3;
then
A85: v
in the
carrier of R by
A2,
NECKLA_2:def 2;
u
in the
carrier of R1 by
A11,
A48,
XBOOLE_0:def 3;
then u
in (the
carrier of R1
\/ the
carrier of R2) by
XBOOLE_0:def 3;
then
A86: u
in the
carrier of R by
A2,
NECKLA_2:def 2;
let q be
object;
assume q
in Z;
then
A87: q
= u or q
= v or q
= x or q
= w by
ENUMSET1:def 2;
the
carrier of R
c= the
carrier of G by
A19,
XBOOLE_1: 7;
hence thesis by
A87,
A86,
A85,
A84;
end;
then
reconsider Z as
Subset of G;
reconsider H = (
subrelstr Z) as non
empty
full
SubRelStr of G by
YELLOW_0:def 15;
A88: w
in X2 by
A18;
reconsider w as
Element of G by
A83,
ZFMISC_1: 87;
A89: v
in X1 by
A10;
A90:
[x, w]
in the
InternalRel of G
proof
(ex y1 be
Element of R2 st w
= y1 &
[y1, x]
in the
InternalRel of G) & the
InternalRel of G
is_symmetric_in the
carrier of G by
A88,
NECKLACE:def 3;
hence thesis;
end;
A91: u
in Y1 by
A48;
reconsider u, v as
Element of G by
A82,
ZFMISC_1: 87;
A92:
[v, x]
in the
InternalRel of G
proof
ex y1 be
Element of R1 st v
= y1 &
[y1, x]
in the
InternalRel of G by
A89;
hence thesis;
end;
A93: w
<> u
proof
assume
A94: not thesis;
(ex y1 be
Element of R2 st w
= y1 &
[y1, x]
in the
InternalRel of G) & ex y2 be
Element of R1 st u
= y2 & not
[y2, x]
in the
InternalRel of G by
A91,
A88;
hence contradiction by
A94;
end;
A95: not
[u, x]
in the
InternalRel of G
proof
ex y1 be
Element of R1 st u
= y1 & not
[y1, x]
in the
InternalRel of G by
A91;
hence thesis;
end;
A96: not
[v, w]
in the
InternalRel of G
proof
A97: ex y2 be
Element of R2 st w
= y2 &
[y2, x]
in the
InternalRel of G by
A88;
then w
in (the
carrier of R1
\/ the
carrier of R2) by
XBOOLE_0:def 3;
then
reconsider w as
Element of R by
A2,
NECKLA_2:def 2;
A98: ex y1 be
Element of R1 st v
= y1 &
[y1, x]
in the
InternalRel of G by
A89;
then v
in (the
carrier of R1
\/ the
carrier of R2) by
XBOOLE_0:def 3;
then
reconsider v as
Element of R by
A2,
NECKLA_2:def 2;
assume not thesis;
then
[v, w]
in (the
InternalRel of G
|_2 the
carrier of R) by
XBOOLE_0:def 4;
then
[v, w]
in the
InternalRel of R by
YELLOW_0:def 14;
then
A99:
[v, w]
in (the
InternalRel of R1
\/ the
InternalRel of R2) by
A2,
NECKLA_2:def 2;
per cases by
A99,
XBOOLE_0:def 3;
suppose
[v, w]
in the
InternalRel of R1;
then w
in the
carrier of R1 by
ZFMISC_1: 87;
then w
in (the
carrier of R1
/\ the
carrier of R2) by
A97,
XBOOLE_0:def 4;
hence contradiction by
A1;
end;
suppose
[v, w]
in the
InternalRel of R2;
then v
in the
carrier of R2 by
ZFMISC_1: 87;
then v
in (the
carrier of R1
/\ the
carrier of R2) by
A98,
XBOOLE_0:def 4;
hence contradiction by
A1;
end;
end;
A100: w
<> x
proof
assume
A101: not thesis;
ex y1 be
Element of R2 st w
= y1 &
[y1, x]
in the
InternalRel of G by
A88;
then x
in (the
carrier of R1
\/ the
carrier of R2) by
A101,
XBOOLE_0:def 3;
then x
in the
carrier of R by
A2,
NECKLA_2:def 2;
then x
in (the
carrier of G
\
{x}) by
YELLOW_0:def 15;
then not x
in
{x} by
XBOOLE_0:def 5;
hence contradiction by
TARSKI:def 1;
end;
A102: not
[u, w]
in the
InternalRel of G
proof
A103: ex y2 be
Element of R2 st w
= y2 &
[y2, x]
in the
InternalRel of G by
A88;
then w
in (the
carrier of R1
\/ the
carrier of R2) by
XBOOLE_0:def 3;
then
reconsider w as
Element of R by
A2,
NECKLA_2:def 2;
A104: ex y1 be
Element of R1 st u
= y1 & not
[y1, x]
in the
InternalRel of G by
A91;
then u
in (the
carrier of R1
\/ the
carrier of R2) by
XBOOLE_0:def 3;
then
reconsider u as
Element of R by
A2,
NECKLA_2:def 2;
assume not thesis;
then
[u, w]
in (the
InternalRel of G
|_2 the
carrier of R) by
XBOOLE_0:def 4;
then
[u, w]
in the
InternalRel of R by
YELLOW_0:def 14;
then
A105:
[u, w]
in (the
InternalRel of R1
\/ the
InternalRel of R2) by
A2,
NECKLA_2:def 2;
per cases by
A105,
XBOOLE_0:def 3;
suppose
[u, w]
in the
InternalRel of R1;
then w
in the
carrier of R1 by
ZFMISC_1: 87;
then w
in (the
carrier of R1
/\ the
carrier of R2) by
A103,
XBOOLE_0:def 4;
hence contradiction by
A1;
end;
suppose
[u, w]
in the
InternalRel of R2;
then u
in the
carrier of R2 by
ZFMISC_1: 87;
then u
in (the
carrier of R1
/\ the
carrier of R2) by
A104,
XBOOLE_0:def 4;
hence contradiction by
A1;
end;
end;
A106: x
<> u
proof
assume
A107: not thesis;
ex y1 be
Element of R1 st u
= y1 & not
[y1, x]
in the
InternalRel of G by
A91;
then x
in (the
carrier of R1
\/ the
carrier of R2) by
A107,
XBOOLE_0:def 3;
then x
in the
carrier of R by
A2,
NECKLA_2:def 2;
then x
in (the
carrier of G
\
{x}) by
YELLOW_0:def 15;
then not x
in
{x} by
XBOOLE_0:def 5;
hence contradiction by
TARSKI:def 1;
end;
A108: w
<> v
proof
consider y1 be
Element of R2 such that
A109: w
= y1 and
[y1, x]
in the
InternalRel of G by
A88;
assume
A110: not thesis;
ex y2 be
Element of R1 st v
= y2 &
[y2, x]
in the
InternalRel of G by
A89;
then y1
in (the
carrier of R1
/\ the
carrier of R2) by
A110,
A109,
XBOOLE_0:def 4;
hence contradiction by
A1;
end;
A111: v
<> x
proof
assume
A112: not thesis;
ex y1 be
Element of R1 st v
= y1 &
[y1, x]
in the
InternalRel of G by
A89;
then x
in (the
carrier of R1
\/ the
carrier of R2) by
A112,
XBOOLE_0:def 3;
then x
in the
carrier of R by
A2,
NECKLA_2:def 2;
then x
in (the
carrier of G
\
{x}) by
YELLOW_0:def 15;
then not x
in
{x} by
XBOOLE_0:def 5;
hence contradiction by
TARSKI:def 1;
end;
u
<> v
proof
assume
A113: not thesis;
(ex y1 be
Element of R1 st u
= y1 & not
[y1, x]
in the
InternalRel of G) & ex y2 be
Element of R1 st v
= y2 &
[y2, x]
in the
InternalRel of G by
A91,
A89;
hence contradiction by
A113;
end;
then (u,v,x,w)
are_mutually_distinct by
A111,
A106,
A93,
A108,
A100,
ZFMISC_1:def 6;
then
A114: (
subrelstr Z)
embeds (
Necklace 4) by
A82,
A92,
A90,
A95,
A102,
A96,
Th38;
G
embeds (
Necklace 4)
proof
assume not thesis;
then G is
N-free by
NECKLA_2:def 1;
then H is
N-free by
Th23;
hence thesis by
A114,
NECKLA_2:def 1;
end;
hence thesis;
end;
suppose
A115: Y2 is non
empty;
ex c be
Element of Y2, d be
Element of X2 st
[c, d]
in the
InternalRel of G
proof
set c = the
Element of Y2;
b
in (the
carrier of R1
\/ the
carrier of R2) by
XBOOLE_0:def 3;
then
A116: b
in the
carrier of R by
A2,
NECKLA_2:def 2;
c
in Y2 by
A115;
then ex y be
Element of R2 st y
= c & not
[y, x]
in the
InternalRel of G;
then c
in (the
carrier of R1
\/ the
carrier of R2) by
XBOOLE_0:def 3;
then
A117: c
in the
carrier of R by
A2,
NECKLA_2:def 2;
A118: the
carrier of R
c= the
carrier of G by
A19,
XBOOLE_1: 7;
then
reconsider b as
Element of G by
A116;
reconsider c as
Element of G by
A118,
A117;
b
<> c
proof
assume not thesis;
then c
in X2 by
A7;
then c
in (X2
/\ Y2) by
A115,
XBOOLE_0:def 4;
hence contradiction by
A14;
end;
then the
InternalRel of G
reduces (b,c) by
A4;
then
consider p be
FinSequence such that
A119: (
len p)
>
0 and
A120: (p
. 1)
= b and
A121: (p
. (
len p))
= c and
A122: for i be
Nat st i
in (
dom p) & (i
+ 1)
in (
dom p) holds
[(p
. i), (p
. (i
+ 1))]
in the
InternalRel of G by
REWRITE1: 11;
defpred
P[
Nat] means (p
. $1)
in Y2 & $1
in (
dom p) & for k be
Nat st k
> $1 holds k
in (
dom p) implies (p
. k)
in Y2;
for k be
Nat st k
> (
len p) holds k
in (
dom p) implies (p
. k)
in Y2
proof
let k be
Nat such that
A123: k
> (
len p);
assume k
in (
dom p);
then k
in (
Seg (
len p)) by
FINSEQ_1:def 3;
hence thesis by
A123,
FINSEQ_1: 1;
end;
then
P[(
len p)] by
A115,
A119,
A121,
CARD_1: 27,
FINSEQ_5: 6;
then
A124: ex k be
Nat st
P[k];
ex n0 be
Nat st
P[n0] & for n be
Nat st
P[n] holds n
>= n0 from
NAT_1:sch 5(
A124);
then
consider n0 be
Nat such that
A125:
P[n0] and
A126: for n be
Nat st
P[n] holds n
>= n0;
n0
<>
0
proof
assume not thesis;
then
0
in (
Seg (
len p)) by
A125,
FINSEQ_1:def 3;
hence contradiction by
FINSEQ_1: 1;
end;
then
consider k0 be
Nat such that
A127: n0
= (k0
+ 1) by
NAT_1: 6;
A128: n0
<> 1
proof
assume
A129: not thesis;
b
in X2 by
A7;
then (X2
/\ Y2) is non
empty by
A120,
A125,
A129,
XBOOLE_0:def 4;
hence contradiction by
A14;
end;
A130: k0
>= 1
proof
assume not thesis;
then k0
=
0 by
NAT_1: 25;
hence contradiction by
A127,
A128;
end;
n0
in (
Seg (
len p)) by
A125,
FINSEQ_1:def 3;
then k0
<= (k0
+ 1) & n0
<= (
len p) by
FINSEQ_1: 1,
XREAL_1: 29;
then k0
<= (
len p) by
A127,
XXREAL_0: 2;
then
A131: k0
in (
Seg (
len p)) by
A130,
FINSEQ_1: 1;
then
A132: k0
in (
dom p) by
FINSEQ_1:def 3;
then
A133:
[(p
. k0), (p
. (k0
+ 1))]
in the
InternalRel of G by
A122,
A125,
A127;
then
A134: the
InternalRel of G
is_symmetric_in the
carrier of G & (p
. k0)
in the
carrier of G by
NECKLACE:def 3,
ZFMISC_1: 87;
(p
. n0)
in the
carrier of G by
A127,
A133,
ZFMISC_1: 87;
then
A135:
[(p
. n0), (p
. k0)]
in the
InternalRel of G by
A127,
A133,
A134;
A136: for k be
Nat st k
> k0 holds k
in (
dom p) implies (p
. k)
in Y2
proof
assume not thesis;
then
consider k be
Nat such that
A137: k
> k0 and
A138: k
in (
dom p) and
A139: not (p
. k)
in Y2;
k
> n0
proof
per cases by
XXREAL_0: 1;
suppose k
< n0;
hence thesis by
A127,
A137,
NAT_1: 13;
end;
suppose n0
< k;
hence thesis;
end;
suppose n0
= k;
hence thesis by
A125,
A139;
end;
end;
hence contradiction by
A125,
A138,
A139;
end;
k0
< n0 by
A127,
NAT_1: 13;
then
A140: not
P[k0] by
A126;
(p
. k0)
in the
carrier of G by
A133,
ZFMISC_1: 87;
then (p
. k0)
in the
carrier of R or (p
. k0)
in
{x} by
A19,
XBOOLE_0:def 3;
then
A141: (p
. k0)
in (the
carrier of R1
\/ the
carrier of R2) or (p
. k0)
in
{x} by
A2,
NECKLA_2:def 2;
thus thesis
proof
per cases by
A127,
A133,
A141,
XBOOLE_0:def 3,
ZFMISC_1: 87;
suppose (p
. k0)
in the
carrier of R2 & (p
. n0)
in the
carrier of G;
then
reconsider m = (p
. k0) as
Element of X2 by
A22,
A131,
A140,
A136,
FINSEQ_1:def 3,
XBOOLE_0:def 3;
reconsider l = (p
. n0) as
Element of Y2 by
A125;
[m, l]
in the
InternalRel of G by
A122,
A125,
A127,
A132;
hence thesis by
A135;
end;
suppose (p
. k0)
in the
carrier of R1 & (p
. n0)
in the
carrier of G;
then
reconsider m = (p
. k0) as
Element of R1;
reconsider l = (p
. n0) as
Element of R2 by
A22,
A125,
XBOOLE_0:def 3;
A142: the
InternalRel of R
is_symmetric_in the
carrier of R by
NECKLACE:def 3;
m
in (the
carrier of R1
\/ the
carrier of R2) by
XBOOLE_0:def 3;
then
A143: m
in the
carrier of R by
A2,
NECKLA_2:def 2;
l
in (the
carrier of R1
\/ the
carrier of R2) by
XBOOLE_0:def 3;
then
A144: l
in the
carrier of R by
A2,
NECKLA_2:def 2;
then
[l, m]
in
[:the
carrier of R, the
carrier of R:] by
A143,
ZFMISC_1: 87;
then
[l, m]
in (the
InternalRel of G
|_2 the
carrier of R) by
A135,
XBOOLE_0:def 4;
then
[l, m]
in the
InternalRel of R by
YELLOW_0:def 14;
then
[m, l]
in the
InternalRel of R by
A144,
A143,
A142;
hence thesis by
A1,
A2,
Th35;
end;
suppose
A145: (p
. k0)
in
{x} & (p
. n0)
in the
carrier of G;
ex y1 be
Element of R2 st (p
. n0)
= y1 & not
[y1, x]
in the
InternalRel of G by
A125;
hence thesis by
A135,
A145,
TARSKI:def 1;
end;
end;
end;
then
consider u be
Element of Y2, v be
Element of X2 such that
A146:
[u, v]
in the
InternalRel of G;
set w = the
Element of X1;
w
in X1 by
A10;
then
A147: ex y be
Element of R1 st y
= w &
[y, x]
in the
InternalRel of G;
set Z =
{u, v, x, w};
Z
c= the
carrier of G
proof
w
in X1 by
A10;
then ex y2 be
Element of R1 st y2
= w &
[y2, x]
in the
InternalRel of G;
then w
in (the
carrier of R1
\/ the
carrier of R2) by
XBOOLE_0:def 3;
then
A148: w
in the
carrier of R by
A2,
NECKLA_2:def 2;
v
in X2 by
A18;
then ex y1 be
Element of R2 st y1
= v &
[y1, x]
in the
InternalRel of G;
then v
in (the
carrier of R1
\/ the
carrier of R2) by
XBOOLE_0:def 3;
then
A149: v
in the
carrier of R by
A2,
NECKLA_2:def 2;
u
in the
carrier of R2 by
A22,
A115,
XBOOLE_0:def 3;
then u
in (the
carrier of R1
\/ the
carrier of R2) by
XBOOLE_0:def 3;
then
A150: u
in the
carrier of R by
A2,
NECKLA_2:def 2;
let q be
object;
assume q
in Z;
then
A151: q
= u or q
= v or q
= x or q
= w by
ENUMSET1:def 2;
the
carrier of R
c= the
carrier of G by
A19,
XBOOLE_1: 7;
hence thesis by
A151,
A150,
A149,
A148;
end;
then
reconsider Z as
Subset of G;
reconsider H = (
subrelstr Z) as non
empty
full
SubRelStr of G by
YELLOW_0:def 15;
A152: w
in X1 by
A10;
reconsider w as
Element of G by
A147,
ZFMISC_1: 87;
A153: v
in X2 by
A18;
A154:
[x, w]
in the
InternalRel of G
proof
(ex y1 be
Element of R1 st w
= y1 &
[y1, x]
in the
InternalRel of G) & the
InternalRel of G
is_symmetric_in the
carrier of G by
A152,
NECKLACE:def 3;
hence thesis;
end;
A155: u
in Y2 by
A115;
reconsider u, v as
Element of G by
A146,
ZFMISC_1: 87;
A156:
[v, x]
in the
InternalRel of G
proof
ex y1 be
Element of R2 st v
= y1 &
[y1, x]
in the
InternalRel of G by
A153;
hence thesis;
end;
A157: w
<> u
proof
assume
A158: not thesis;
(ex y1 be
Element of R1 st w
= y1 &
[y1, x]
in the
InternalRel of G) & ex y2 be
Element of R2 st u
= y2 & not
[y2, x]
in the
InternalRel of G by
A155,
A152;
hence contradiction by
A158;
end;
A159: not
[u, x]
in the
InternalRel of G
proof
ex y1 be
Element of R2 st u
= y1 & not
[y1, x]
in the
InternalRel of G by
A155;
hence thesis;
end;
A160: not
[v, w]
in the
InternalRel of G
proof
A161: ex y2 be
Element of R1 st w
= y2 &
[y2, x]
in the
InternalRel of G by
A152;
then w
in (the
carrier of R1
\/ the
carrier of R2) by
XBOOLE_0:def 3;
then
reconsider w as
Element of R by
A2,
NECKLA_2:def 2;
A162: ex y1 be
Element of R2 st v
= y1 &
[y1, x]
in the
InternalRel of G by
A153;
then v
in (the
carrier of R1
\/ the
carrier of R2) by
XBOOLE_0:def 3;
then
reconsider v as
Element of R by
A2,
NECKLA_2:def 2;
assume not thesis;
then
[v, w]
in (the
InternalRel of G
|_2 the
carrier of R) by
XBOOLE_0:def 4;
then
[v, w]
in the
InternalRel of R by
YELLOW_0:def 14;
then
A163:
[v, w]
in (the
InternalRel of R1
\/ the
InternalRel of R2) by
A2,
NECKLA_2:def 2;
per cases by
A163,
XBOOLE_0:def 3;
suppose
[v, w]
in the
InternalRel of R1;
then v
in the
carrier of R1 by
ZFMISC_1: 87;
then v
in (the
carrier of R1
/\ the
carrier of R2) by
A162,
XBOOLE_0:def 4;
hence contradiction by
A1;
end;
suppose
[v, w]
in the
InternalRel of R2;
then w
in the
carrier of R2 by
ZFMISC_1: 87;
then w
in (the
carrier of R1
/\ the
carrier of R2) by
A161,
XBOOLE_0:def 4;
hence contradiction by
A1;
end;
end;
A164: w
<> x
proof
assume
A165: not thesis;
ex y1 be
Element of R1 st w
= y1 &
[y1, x]
in the
InternalRel of G by
A152;
then x
in (the
carrier of R1
\/ the
carrier of R2) by
A165,
XBOOLE_0:def 3;
then x
in the
carrier of R by
A2,
NECKLA_2:def 2;
then x
in (the
carrier of G
\
{x}) by
YELLOW_0:def 15;
then not x
in
{x} by
XBOOLE_0:def 5;
hence contradiction by
TARSKI:def 1;
end;
A166: not
[u, w]
in the
InternalRel of G
proof
A167: ex y2 be
Element of R1 st w
= y2 &
[y2, x]
in the
InternalRel of G by
A152;
then w
in (the
carrier of R1
\/ the
carrier of R2) by
XBOOLE_0:def 3;
then
reconsider w as
Element of R by
A2,
NECKLA_2:def 2;
A168: ex y1 be
Element of R2 st u
= y1 & not
[y1, x]
in the
InternalRel of G by
A155;
then u
in (the
carrier of R1
\/ the
carrier of R2) by
XBOOLE_0:def 3;
then
reconsider u as
Element of R by
A2,
NECKLA_2:def 2;
assume not thesis;
then
[u, w]
in (the
InternalRel of G
|_2 the
carrier of R) by
XBOOLE_0:def 4;
then
[u, w]
in the
InternalRel of R by
YELLOW_0:def 14;
then
A169:
[u, w]
in (the
InternalRel of R1
\/ the
InternalRel of R2) by
A2,
NECKLA_2:def 2;
per cases by
A169,
XBOOLE_0:def 3;
suppose
[u, w]
in the
InternalRel of R1;
then u
in the
carrier of R1 by
ZFMISC_1: 87;
then u
in (the
carrier of R1
/\ the
carrier of R2) by
A168,
XBOOLE_0:def 4;
hence contradiction by
A1;
end;
suppose
[u, w]
in the
InternalRel of R2;
then w
in the
carrier of R2 by
ZFMISC_1: 87;
then w
in (the
carrier of R1
/\ the
carrier of R2) by
A167,
XBOOLE_0:def 4;
hence contradiction by
A1;
end;
end;
A170: x
<> u
proof
assume
A171: not thesis;
ex y1 be
Element of R2 st u
= y1 & not
[y1, x]
in the
InternalRel of G by
A155;
then x
in (the
carrier of R1
\/ the
carrier of R2) by
A171,
XBOOLE_0:def 3;
then x
in the
carrier of R by
A2,
NECKLA_2:def 2;
then x
in (the
carrier of G
\
{x}) by
YELLOW_0:def 15;
then not x
in
{x} by
XBOOLE_0:def 5;
hence contradiction by
TARSKI:def 1;
end;
A172: w
<> v
proof
consider y1 be
Element of R1 such that
A173: w
= y1 and
[y1, x]
in the
InternalRel of G by
A152;
assume
A174: not thesis;
ex y2 be
Element of R2 st v
= y2 &
[y2, x]
in the
InternalRel of G by
A153;
then y1
in (the
carrier of R1
/\ the
carrier of R2) by
A174,
A173,
XBOOLE_0:def 4;
hence contradiction by
A1;
end;
A175: v
<> x
proof
assume
A176: not thesis;
ex y1 be
Element of R2 st v
= y1 &
[y1, x]
in the
InternalRel of G by
A153;
then x
in (the
carrier of R1
\/ the
carrier of R2) by
A176,
XBOOLE_0:def 3;
then x
in the
carrier of R by
A2,
NECKLA_2:def 2;
then x
in (the
carrier of G
\
{x}) by
YELLOW_0:def 15;
then not x
in
{x} by
XBOOLE_0:def 5;
hence contradiction by
TARSKI:def 1;
end;
u
<> v
proof
assume
A177: not thesis;
(ex y1 be
Element of R2 st u
= y1 & not
[y1, x]
in the
InternalRel of G) & ex y2 be
Element of R2 st v
= y2 &
[y2, x]
in the
InternalRel of G by
A155,
A153;
hence contradiction by
A177;
end;
then (u,v,x,w)
are_mutually_distinct by
A175,
A170,
A157,
A172,
A164,
ZFMISC_1:def 6;
then
A178: (
subrelstr Z)
embeds (
Necklace 4) by
A146,
A156,
A154,
A159,
A166,
A160,
Th38;
G
embeds (
Necklace 4)
proof
assume not thesis;
then G is
N-free by
NECKLA_2:def 1;
then H is
N-free by
Th23;
hence thesis by
A178,
NECKLA_2:def 1;
end;
hence thesis;
end;
end;
end;
theorem ::
NECKLA_3:40
for G be non
empty
strict
finite
irreflexive
symmetric
RelStr st G is
N-free & the
carrier of G
in
FinSETS holds the RelStr of G
in
fin_RelStr_sp
proof
let R be non
empty
strict
finite
irreflexive
symmetric
RelStr;
defpred
P[
Nat] means for G be non
empty
strict
finite
irreflexive
symmetric
RelStr st G is
N-free & (
card the
carrier of G)
= $1 & the
carrier of G
in
FinSETS holds the RelStr of G
in
fin_RelStr_sp ;
A1: for n be
Nat st for k be
Nat st k
< n holds
P[k] holds
P[n]
proof
let n be
Nat such that
A2: for k be
Nat st k
< n holds
P[k];
let G be non
empty
strict
finite
irreflexive
symmetric
RelStr;
set CG = (
ComplRelStr G);
assume that
A3: G is
N-free and
A4: (
card the
carrier of G)
= n and
A5: the
carrier of G
in
FinSETS ;
per cases ;
suppose G is
trivial;
then the
carrier of G is 1
-element;
then
reconsider G as 1
-element
RelStr by
STRUCT_0:def 19;
the RelStr of G
in
fin_RelStr_sp by
A5,
NECKLA_2:def 5;
hence thesis;
end;
suppose not G is
path-connected & G is non
trivial;
then
consider G1,G2 be non
empty
strict
irreflexive
symmetric
RelStr such that
A6: the
carrier of G1
misses the
carrier of G2 and
A7: the RelStr of G
= (
union_of (G1,G2)) by
Th30;
set cG1 = the
carrier of G1, cG2 = the
carrier of G2, R = the RelStr of G, cR = the
carrier of R;
reconsider cR as
finite
set;
A8: cR
= (cG1
\/ cG2) by
A7,
NECKLA_2:def 2;
then
A9: (
card cG1)
in (
Segm (
card cR)) by
A6,
Lm1;
then
reconsider G1 as non
empty
strict
finite
irreflexive
symmetric
RelStr;
reconsider cR as
finite
set;
A10: (
card cG2)
in (
Segm (
card cR)) by
A6,
A8,
Lm1;
then
reconsider G2 as non
empty
strict
finite
irreflexive
symmetric
RelStr;
reconsider cG2 as
finite
set by
A10;
A11: (
card cG2)
< (
card cR) by
A10,
NAT_1: 44;
G2 is
full
SubRelStr of G by
A6,
A7,
Th10;
then
A12: G2 is
N-free by
A3,
Th23;
the
carrier of G2
in
FinSETS by
A5,
A8,
CLASSES1: 3,
CLASSES2:def 2,
XBOOLE_1: 7;
then
A13: G2
in
fin_RelStr_sp by
A2,
A4,
A11,
A12;
G1 is
full
SubRelStr of G by
A6,
A7,
Th10;
then
A14: G1 is
N-free by
A3,
Th23;
reconsider cG1 as
finite
set by
A9;
A15: (
card cG1)
< (
card cR) by
A9,
NAT_1: 44;
the
carrier of G1
in
FinSETS by
A5,
A8,
CLASSES1: 3,
CLASSES2:def 2,
XBOOLE_1: 7;
then G1
in
fin_RelStr_sp by
A2,
A4,
A15,
A14;
hence thesis by
A6,
A7,
A13,
NECKLA_2:def 5;
end;
suppose not CG is
path-connected & G is non
trivial;
then
consider G1,G2 be non
empty
strict
irreflexive
symmetric
RelStr such that
A16: the
carrier of G1
misses the
carrier of G2 and
A17: the RelStr of G
= (
sum_of (G1,G2)) by
Th31;
set cG1 = the
carrier of G1, cG2 = the
carrier of G2, R = the RelStr of G, cR = the
carrier of R;
reconsider cR as
finite
set;
A18: cR
= (cG1
\/ cG2) by
A17,
NECKLA_2:def 3;
then
A19: (
card cG1)
in (
Segm (
card cR)) by
A16,
Lm1;
then
reconsider G1 as non
empty
strict
finite
irreflexive
symmetric
RelStr;
A20: (
card cG2)
in (
Segm (
card cR)) by
A16,
A18,
Lm1;
then
reconsider G2 as non
empty
strict
finite
irreflexive
symmetric
RelStr;
reconsider cG2 as
finite
set by
A20;
A21: (
card cG2)
< (
card cR) by
A20,
NAT_1: 44;
G2 is
full
SubRelStr of G by
A16,
A17,
Th10;
then
A22: G2 is
N-free by
A3,
Th23;
the
carrier of G2
in
FinSETS by
A5,
A18,
CLASSES1: 3,
CLASSES2:def 2,
XBOOLE_1: 7;
then
A23: G2
in
fin_RelStr_sp by
A2,
A4,
A21,
A22;
G1 is
full
SubRelStr of G by
A16,
A17,
Th10;
then
A24: G1 is
N-free by
A3,
Th23;
reconsider cG1 as
finite
set by
A19;
A25: (
card cG1)
< (
card cR) by
A19,
NAT_1: 44;
the
carrier of G1
in
FinSETS by
A5,
A18,
CLASSES1: 3,
CLASSES2:def 2,
XBOOLE_1: 7;
then G1
in
fin_RelStr_sp by
A2,
A4,
A25,
A24;
hence thesis by
A16,
A17,
A23,
NECKLA_2:def 5;
end;
suppose
A26: G is non
trivial & G is
path-connected & CG is
path-connected;
consider x be
object such that
A27: x
in the
carrier of G by
XBOOLE_0:def 1;
reconsider x as
Element of G by
A27;
set A = (the
carrier of G
\
{x});
A28: A
c= the
carrier of G;
reconsider A as
Subset of G;
set R = (
subrelstr A);
reconsider R as non
empty
finite
irreflexive
symmetric
RelStr by
A26,
YELLOW_0:def 15;
A29: the
carrier of R
c= the
carrier of G by
A28,
YELLOW_0:def 15;
(
card A)
= ((
card the
carrier of G)
- (
card
{x})) by
CARD_2: 44;
then
A30: (
card A)
= (n
- 1) by
A4,
CARD_2: 42;
(n
- 1)
< ((n
- 1)
+ 1) by
XREAL_1: 29;
then
A31: (
card the
carrier of R)
< n by
A30,
YELLOW_0:def 15;
R is
N-free by
A3,
Th23;
then
A32: R
in
fin_RelStr_sp by
A2,
A5,
A31,
A29,
CLASSES1: 3,
CLASSES2:def 2;
thus thesis
proof
per cases by
A32,
NECKLA_2: 6;
suppose
A33: R is
trivial
RelStr;
the
carrier of R is non
empty;
then
A34: A is non
empty by
YELLOW_0:def 15;
A is
trivial by
A33,
YELLOW_0:def 15;
then
consider a be
object such that
A35: A
=
{a} by
A34,
ZFMISC_1: 131;
A36: (the
carrier of G
\/
{x})
= the
carrier of G
proof
thus (the
carrier of G
\/
{x})
c= the
carrier of G
proof
let c be
object;
assume c
in (the
carrier of G
\/
{x});
then c
in the
carrier of G or c
in
{x} by
XBOOLE_0:def 3;
hence thesis;
end;
let c be
object;
assume c
in the
carrier of G;
hence thesis by
XBOOLE_0:def 3;
end;
(
{a}
\/
{x})
= (the
carrier of G
\/
{x}) by
A35,
XBOOLE_1: 39;
then the
carrier of G
=
{a, x} & a
<> x by
A26,
A36,
ENUMSET1: 1;
then (
card the
carrier of G)
= 2 by
CARD_2: 57;
hence thesis by
A5,
Th33;
end;
suppose ex R1,R2 be
strict
RelStr st the
carrier of R1
misses the
carrier of R2 & R1
in
fin_RelStr_sp & R2
in
fin_RelStr_sp & (R
= (
union_of (R1,R2)) or R
= (
sum_of (R1,R2)));
then
consider R1,R2 be
strict
RelStr such that
A37: the
carrier of R1
misses the
carrier of R2 and
A38: R1
in
fin_RelStr_sp and
A39: R2
in
fin_RelStr_sp and
A40: R
= (
union_of (R1,R2)) or R
= (
sum_of (R1,R2));
thus thesis
proof
per cases by
A40;
suppose
A41: R
= (
union_of (R1,R2));
R2 is
SubRelStr of R by
A37,
A40,
Th10;
then
reconsider R2 as non
empty
SubRelStr of G by
A39,
NECKLA_2: 4,
YELLOW_6: 7;
R1 is
SubRelStr of R by
A37,
A40,
Th10;
then
reconsider R1 as non
empty
SubRelStr of G by
A38,
NECKLA_2: 4,
YELLOW_6: 7;
(
subrelstr ((
[#] G)
\
{x}))
= (
union_of (R1,R2)) by
A41;
then G
embeds (
Necklace 4) by
A26,
A37,
Th39;
hence thesis by
A3,
NECKLA_2:def 1;
end;
suppose
A42: R
= (
sum_of (R1,R2));
(
ComplRelStr R2) is non
empty
proof
assume not thesis;
then R2 is
empty;
hence contradiction by
A39,
NECKLA_2: 4;
end;
then
reconsider R22 = (
ComplRelStr R2) as non
empty
RelStr;
(
ComplRelStr R1) is non
empty
proof
assume not thesis;
then R1 is
empty;
hence contradiction by
A38,
NECKLA_2: 4;
end;
then
reconsider R11 = (
ComplRelStr R1) as non
empty
RelStr;
reconsider G9 = (
ComplRelStr G) as non
empty
irreflexive
symmetric
RelStr;
reconsider x9 = x as
Element of G9 by
NECKLACE:def 8;
A43: the
carrier of R11
= the
carrier of R1 & the
carrier of R22
= the
carrier of R2 by
NECKLACE:def 8;
A44: (
ComplRelStr R)
= (
ComplRelStr (
subrelstr ((
[#] G)
\
{x})))
.= (
subrelstr ((
[#] G9)
\
{x9})) by
Th20;
A45: G9 is
N-free by
A3,
Th25;
A46: (
ComplRelStr G9) is
path-connected & G9 is non
trivial by
A26,
Th16,
NECKLACE:def 8;
(
ComplRelStr R)
= (
union_of ((
ComplRelStr R1),(
ComplRelStr R2))) by
A37,
A42,
Th18;
then G9
embeds (
Necklace 4) by
A26,
A37,
A43,
A46,
A44,
Th39;
hence thesis by
A45,
NECKLA_2:def 1;
end;
end;
end;
end;
end;
end;
A47: for k be
Nat holds
P[k] from
NAT_1:sch 4(
A1);
(
card the
carrier of R) is
Nat;
hence thesis by
A47;
end;