:: NECKLA_3 semantic presentation
begin
theorem Th1: :: NECKLA_3:1
for A, B being set holds (id A) | B = (id A) /\ [:B,B:]
proof
let A, B be set ; ::_thesis: (id A) | B = (id A) /\ [:B,B:]
thus (id A) | B c= (id A) /\ [:B,B:] :: according to XBOOLE_0:def_10 ::_thesis: (id A) /\ [:B,B:] c= (id A) | B
proof
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in (id A) | B or a in (id A) /\ [:B,B:] )
assume A1: a in (id A) | B ; ::_thesis: a in (id A) /\ [:B,B:]
(id A) | B is Relation of B,A by RELSET_1:18;
then consider x, y being set 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 a in (id A) /\ [:B,B:] by A2, A4, XBOOLE_0:def_4; ::_thesis: verum
end;
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in (id A) /\ [:B,B:] or a in (id A) | B )
assume A5: a in (id A) /\ [:B,B:] ; ::_thesis: a in (id A) | B
then a in [:B,B:] by XBOOLE_0:def_4;
then A6: ex x1, y1 being set 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 a in (id A) | B by A6, RELAT_1:def_11; ::_thesis: verum
end;
theorem :: NECKLA_3:2
for a, b, c, d being set holds id {a,b,c,d} = {[a,a],[b,b],[c,c],[d,d]}
proof
let a, b, c, d be set ; ::_thesis: id {a,b,c,d} = {[a,a],[b,b],[c,c],[d,d]}
set X = {a,b,c,d};
thus id {a,b,c,d} c= {[a,a],[b,b],[c,c],[d,d]} :: according to XBOOLE_0:def_10 ::_thesis: {[a,a],[b,b],[c,c],[d,d]} c= id {a,b,c,d}
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in id {a,b,c,d} or x in {[a,a],[b,b],[c,c],[d,d]} )
assume A1: x in id {a,b,c,d} ; ::_thesis: x in {[a,a],[b,b],[c,c],[d,d]}
then consider x1, y1 being set such that
A2: x = [x1,y1] and
A3: x1 in {a,b,c,d} and
y1 in {a,b,c,d} by RELSET_1:2;
A4: x1 = y1 by A1, A2, RELAT_1:def_10;
percases ( x1 = a or x1 = b or x1 = c or x1 = d ) by A3, ENUMSET1:def_2;
suppose x1 = a ; ::_thesis: x in {[a,a],[b,b],[c,c],[d,d]}
hence x in {[a,a],[b,b],[c,c],[d,d]} by A2, A4, ENUMSET1:def_2; ::_thesis: verum
end;
suppose x1 = b ; ::_thesis: x in {[a,a],[b,b],[c,c],[d,d]}
hence x in {[a,a],[b,b],[c,c],[d,d]} by A2, A4, ENUMSET1:def_2; ::_thesis: verum
end;
suppose x1 = c ; ::_thesis: x in {[a,a],[b,b],[c,c],[d,d]}
hence x in {[a,a],[b,b],[c,c],[d,d]} by A2, A4, ENUMSET1:def_2; ::_thesis: verum
end;
suppose x1 = d ; ::_thesis: x in {[a,a],[b,b],[c,c],[d,d]}
hence x in {[a,a],[b,b],[c,c],[d,d]} by A2, A4, ENUMSET1:def_2; ::_thesis: verum
end;
end;
end;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {[a,a],[b,b],[c,c],[d,d]} or x in id {a,b,c,d} )
assume A5: x in {[a,a],[b,b],[c,c],[d,d]} ; ::_thesis: x in id {a,b,c,d}
percases ( x = [a,a] or x = [b,b] or x = [c,c] or x = [d,d] ) by A5, ENUMSET1:def_2;
supposeA6: x = [a,a] ; ::_thesis: x in id {a,b,c,d}
a in {a,b,c,d} by ENUMSET1:def_2;
hence x in id {a,b,c,d} by A6, RELAT_1:def_10; ::_thesis: verum
end;
supposeA7: x = [b,b] ; ::_thesis: x in id {a,b,c,d}
b in {a,b,c,d} by ENUMSET1:def_2;
hence x in id {a,b,c,d} by A7, RELAT_1:def_10; ::_thesis: verum
end;
supposeA8: x = [c,c] ; ::_thesis: x in id {a,b,c,d}
c in {a,b,c,d} by ENUMSET1:def_2;
hence x in id {a,b,c,d} by A8, RELAT_1:def_10; ::_thesis: verum
end;
supposeA9: x = [d,d] ; ::_thesis: x in id {a,b,c,d}
d in {a,b,c,d} by ENUMSET1:def_2;
hence x in id {a,b,c,d} by A9, RELAT_1:def_10; ::_thesis: verum
end;
end;
end;
theorem Th3: :: NECKLA_3:3
for a, b, c, d, e, f, g, h being set holds [:{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
let a, b, c, d, e, f, g, h be set ; ::_thesis: [:{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]}
set X1 = {a,b,c,d};
set Y1 = {e,f,g,h};
set X11 = {a,b};
set X12 = {c,d};
set Y11 = {e,f};
set Y12 = {g,h};
A1: ( [:{c,d},{e,f}:] = {[c,e],[c,f],[d,e],[d,f]} & [:{c,d},{g,h}:] = {[c,g],[c,h],[d,g],[d,h]} ) by MCART_1:23;
( {a,b,c,d} = {a,b} \/ {c,d} & {e,f,g,h} = {e,f} \/ {g,h} ) by ENUMSET1:5;
then A2: [:{a,b,c,d},{e,f,g,h}:] = (([:{a,b},{e,f}:] \/ [:{a,b},{g,h}:]) \/ [:{c,d},{e,f}:]) \/ [:{c,d},{g,h}:] by ZFMISC_1:98;
( [:{a,b},{e,f}:] = {[a,e],[a,f],[b,e],[b,f]} & [:{a,b},{g,h}:] = {[a,g],[a,h],[b,g],[b,h]} ) by MCART_1:23;
then [:{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]} 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 [:{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]} ; ::_thesis: verum
end;
registration
let X, Y be trivial set ;
cluster -> trivial for Element of bool [:X,Y:];
correctness
coherence
for b1 being Relation of X,Y holds b1 is trivial ;
proof
let R be Relation of X,Y; ::_thesis: R is trivial
A1: ( X is empty or ex x being set st X = {x} ) by ZFMISC_1:131;
A2: ( Y is empty or ex y being set st Y = {y} ) by ZFMISC_1:131;
assume not R is trivial ; ::_thesis: contradiction
then consider a1, a2 being set such that
A3: a1 in R and
A4: a2 in R and
A5: a1 <> a2 by ZFMISC_1:def_10;
percases ( ( X is empty & Y is empty ) or ( X is empty & ex x being set st Y = {x} ) or ex x being set st
( X = {x} & Y is empty ) or ex x being set st
( X = {x} & ex y being set st Y = {y} ) ) by A1, A2;
suppose ( X is empty & Y is empty ) ; ::_thesis: contradiction
hence contradiction by A3; ::_thesis: verum
end;
suppose ( X is empty & ex x being set st Y = {x} ) ; ::_thesis: contradiction
hence contradiction by A3; ::_thesis: verum
end;
suppose ex x being set st
( X = {x} & Y is empty ) ; ::_thesis: contradiction
hence contradiction by A3; ::_thesis: verum
end;
supposeA6: ex x being set st
( X = {x} & ex y being set st Y = {y} ) ; ::_thesis: contradiction
then consider x being set such that
A7: X = {x} ;
consider x1, x2 being set such that
A8: a1 = [x1,x2] and
A9: x1 in X and
A10: x2 in Y by A3, RELSET_1:2;
A11: x1 = x by A7, A9, TARSKI:def_1;
consider y being set such that
A12: Y = {y} by A6;
consider y1, y2 being set such that
A13: a2 = [y1,y2] and
A14: y1 in X and
A15: y2 in Y by A4, RELSET_1:2;
A16: y1 = x by A7, A14, TARSKI:def_1;
x2 = y by A12, A10, TARSKI:def_1;
hence contradiction by A5, A12, A8, A11, A13, A15, A16, TARSKI:def_1; ::_thesis: verum
end;
end;
end;
end;
theorem Th4: :: NECKLA_3:4
for X being trivial set
for R being Relation of X st not R is empty holds
ex x being set st R = {[x,x]}
proof
let X be trivial set ; ::_thesis: for R being Relation of X st not R is empty holds
ex x being set st R = {[x,x]}
let R be Relation of X; ::_thesis: ( not R is empty implies ex x being set st R = {[x,x]} )
assume not R is empty ; ::_thesis: ex x being set st R = {[x,x]}
then consider x being set such that
A1: x in R by XBOOLE_0:def_1;
consider y, z being set such that
A2: x = [y,z] and
A3: y in X and
A4: z in X by A1, RELSET_1:2;
consider a being set 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]} :: according to XBOOLE_0:def_10 ::_thesis: {[a,a]} c= R
proof
let r be set ; :: according to TARSKI:def_3 ::_thesis: ( not r in R or r in {[a,a]} )
assume r in R ; ::_thesis: r in {[a,a]}
then consider y, z being set 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 r in {[a,a]} by A7, TARSKI:def_1; ::_thesis: verum
end;
let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in {[a,a]} or z in R )
assume z in {[a,a]} ; ::_thesis: z in R
hence z in R by A1, A2, A6, TARSKI:def_1; ::_thesis: verum
end;
hence ex x being set st R = {[x,x]} ; ::_thesis: verum
end;
registration
let X be trivial set ;
cluster -> trivial reflexive symmetric strongly_connected transitive for Element of bool [:X,X:];
correctness
coherence
for b1 being Relation of X holds
( b1 is trivial & b1 is reflexive & b1 is symmetric & b1 is transitive & b1 is strongly_connected );
proof
let R be Relation of X,X; ::_thesis: ( R is trivial & R is reflexive & R is symmetric & R is transitive & R is strongly_connected )
A1: R is_reflexive_in field R
proof
percases ( R is empty or not R is empty ) ;
supposeA2: R is empty ; ::_thesis: R is_reflexive_in field R
let x be set ; :: according to RELAT_2:def_1 ::_thesis: ( not x in field R or [x,x] in R )
assume x in field R ; ::_thesis: [x,x] in R
hence [x,x] in R by A2, RELAT_1:40; ::_thesis: verum
end;
suppose not R is empty ; ::_thesis: R is_reflexive_in field R
then consider z being set such that
A3: R = {[z,z]} by Th4;
let x be set ; :: according to RELAT_2:def_1 ::_thesis: ( not x in field R or [x,x] in R )
assume x in field R ; ::_thesis: [x,x] in 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 [x,x] in R by A3, TARSKI:def_1; ::_thesis: verum
end;
end;
end;
A5: R is_transitive_in field R
proof
percases ( R is empty or not R is empty ) ;
supposeA6: R is empty ; ::_thesis: R is_transitive_in field R
let x, y, z be set ; :: according to RELAT_2:def_8 ::_thesis: ( not x in field R or not y in field R or not z in field R or not [x,y] in R or not [y,z] in R or [x,z] in R )
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 ; ::_thesis: [x,z] in R
thus [x,z] in R by A6, A7; ::_thesis: verum
end;
supposeA8: not R is empty ; ::_thesis: R is_transitive_in field R
let x, y, z be set ; :: according to RELAT_2:def_8 ::_thesis: ( not x in field R or not y in field R or not z in field R or not [x,y] in R or not [y,z] in R or [x,z] in R )
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 ; ::_thesis: [x,z] in R
consider a being set 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 [x,z] in R by A11, A12, TARSKI:def_1; ::_thesis: verum
end;
end;
end;
A13: R is_strongly_connected_in field R
proof
percases ( R is empty or not R is empty ) ;
supposeA14: R is empty ; ::_thesis: R is_strongly_connected_in field R
let x, y be set ; :: according to RELAT_2:def_7 ::_thesis: ( not x in field R or not y in field R or [x,y] in R or [y,x] in R )
assume that
A15: x in field R and
y in field R ; ::_thesis: ( [x,y] in R or [y,x] in R )
thus ( [x,y] in R or [y,x] in R ) by A14, A15, RELAT_1:40; ::_thesis: verum
end;
supposeA16: not R is empty ; ::_thesis: R is_strongly_connected_in field R
let x, y be set ; :: according to RELAT_2:def_7 ::_thesis: ( not x in field R or not y in field R or [x,y] in R or [y,x] in R )
assume that
A17: x in field R and
A18: y in field R ; ::_thesis: ( [x,y] in R or [y,x] in R )
consider a being set 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 ( [x,y] in R or [y,x] in R ) by A19, A21, TARSKI:def_1; ::_thesis: verum
end;
end;
end;
R is_symmetric_in field R
proof
percases ( R is empty or not R is empty ) ;
supposeA22: R is empty ; ::_thesis: R is_symmetric_in field R
let x, y be set ; :: according to RELAT_2:def_3 ::_thesis: ( not x in field R or not y in field R or not [x,y] in R or [y,x] in R )
assume that
x in field R and
y in field R and
A23: [x,y] in R ; ::_thesis: [y,x] in R
thus [y,x] in R by A22, A23; ::_thesis: verum
end;
supposeA24: not R is empty ; ::_thesis: R is_symmetric_in field R
let x, y be set ; :: according to RELAT_2:def_3 ::_thesis: ( not x in field R or not y in field R or not [x,y] in R or [y,x] in R )
assume that
x in field R and
y in field R and
A25: [x,y] in R ; ::_thesis: [y,x] in R
consider a being set 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 [y,x] in R by A26, TARSKI:def_1; ::_thesis: verum
end;
end;
end;
hence ( R is trivial & R is reflexive & R is symmetric & R is transitive & R is strongly_connected ) by A1, A5, A13, RELAT_2:def_9, RELAT_2:def_11, RELAT_2:def_15, RELAT_2:def_16; ::_thesis: verum
end;
end;
theorem Th5: :: NECKLA_3:5
for X being 1 -element set
for R being Relation of X holds R is_symmetric_in X
proof
let X be 1 -element set ; ::_thesis: for R being Relation of X holds R is_symmetric_in X
let R be Relation of X; ::_thesis: R is_symmetric_in X
consider x being set such that
A1: X = {x} by ZFMISC_1:131;
let a, b be set ; :: according to RELAT_2:def_3 ::_thesis: ( not a in X or not b in X or not [a,b] in R or [b,a] in R )
assume that
A2: a in X and
A3: ( b in X & [a,b] in R ) ; ::_thesis: [b,a] in R
a = x by A1, A2, TARSKI:def_1;
hence [b,a] in R by A1, A3, TARSKI:def_1; ::_thesis: verum
end;
registration
cluster non empty finite strict symmetric irreflexive for RelStr ;
correctness
existence
ex b1 being RelStr st
( not b1 is empty & b1 is strict & b1 is finite & b1 is irreflexive & b1 is symmetric );
proof
set X = {0,1};
set r = {[0,1],[1,0]};
( 0 in {0,1} & 1 in {0,1} ) by TARSKI:def_2;
then A1: ( [0,1] in [:{0,1},{0,1}:] & [1,0] in [:{0,1},{0,1}:] ) by ZFMISC_1:def_2;
{[0,1],[1,0]} c= [:{0,1},{0,1}:]
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {[0,1],[1,0]} or x in [:{0,1},{0,1}:] )
assume x in {[0,1],[1,0]} ; ::_thesis: x in [:{0,1},{0,1}:]
hence x in [:{0,1},{0,1}:] by A1, TARSKI:def_2; ::_thesis: verum
end;
then reconsider r = {[0,1],[1,0]} as Relation of {0,1},{0,1} ;
take R = RelStr(# {0,1},r #); ::_thesis: ( not R is empty & R is strict & R is finite & R is irreflexive & R is symmetric )
A2: for x being set st x in the carrier of R holds
not [x,x] in the InternalRel of R
proof
let x be set ; ::_thesis: ( x in the carrier of R implies not [x,x] in the InternalRel of R )
A3: not [0,0] in r
proof
assume [0,0] in r ; ::_thesis: contradiction
then ( [0,0] = [0,1] or [0,0] = [1,0] ) by TARSKI:def_2;
hence contradiction by XTUPLE_0:1; ::_thesis: verum
end;
A4: not [1,1] in r
proof
assume [1,1] in r ; ::_thesis: contradiction
then ( [1,1] = [0,1] or [1,1] = [1,0] ) by TARSKI:def_2;
hence contradiction by XTUPLE_0:1; ::_thesis: verum
end;
assume x in the carrier of R ; ::_thesis: not [x,x] in the InternalRel of R
then ( x = 0 or x = 1 ) by TARSKI:def_2;
hence not [x,x] in the InternalRel of R by A3, A4; ::_thesis: verum
end;
for x, y being set st x in {0,1} & y in {0,1} & [x,y] in r holds
[y,x] in r
proof
let x, y be set ; ::_thesis: ( x in {0,1} & y in {0,1} & [x,y] in r implies [y,x] in r )
assume that
x in {0,1} and
y in {0,1} and
A5: [x,y] in r ; ::_thesis: [y,x] in r
percases ( [x,y] = [0,1] or [x,y] = [1,0] ) by A5, TARSKI:def_2;
suppose [x,y] = [0,1] ; ::_thesis: [y,x] in r
then ( x = 0 & y = 1 ) by XTUPLE_0:1;
hence [y,x] in r by TARSKI:def_2; ::_thesis: verum
end;
suppose [x,y] = [1,0] ; ::_thesis: [y,x] in r
then ( x = 1 & y = 0 ) by XTUPLE_0:1;
hence [y,x] in r by TARSKI:def_2; ::_thesis: verum
end;
end;
end;
then r is_symmetric_in {0,1} by RELAT_2:def_3;
hence ( not R is empty & R is strict & R is finite & R is irreflexive & R is symmetric ) by A2, NECKLACE:def_3, NECKLACE:def_5; ::_thesis: verum
end;
end;
registration
let L be irreflexive RelStr ;
cluster full -> full irreflexive for SubRelStr of L;
correctness
coherence
for b1 being full SubRelStr of L holds b1 is irreflexive ;
proof
let S be full SubRelStr of L; ::_thesis: S is irreflexive
let x be set ; :: according to NECKLACE:def_5 ::_thesis: ( not x in the carrier of S or not [x,x] in the InternalRel of S )
assume A1: x in the carrier of S ; ::_thesis: not [x,x] in the InternalRel 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 not [x,x] in the InternalRel of S by XBOOLE_0:def_4; ::_thesis: verum
end;
end;
registration
let L be symmetric RelStr ;
cluster full -> full symmetric for SubRelStr of L;
correctness
coherence
for b1 being full SubRelStr of L holds b1 is symmetric ;
proof
let S be full SubRelStr of L; ::_thesis: S is symmetric
let x, y be set ; :: according to NECKLACE:def_3,RELAT_2:def_3 ::_thesis: ( not x in the carrier of S or not y in the carrier of S or not [x,y] in the InternalRel of S or [y,x] in the InternalRel of S )
assume that
A1: ( x in the carrier of S & y in the carrier of S ) and
A2: [x,y] in the InternalRel of S ; ::_thesis: [y,x] 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, RELAT_2:def_3;
hence [y,x] in the InternalRel of S by A5, A3, XBOOLE_0:def_4; ::_thesis: verum
end;
end;
theorem Th6: :: NECKLA_3:6
for R being symmetric irreflexive RelStr st card the carrier of R = 2 holds
ex a, b being set st
( the carrier of R = {a,b} & ( the InternalRel of R = {[a,b],[b,a]} or the InternalRel of R = {} ) )
proof
let R be symmetric irreflexive RelStr ; ::_thesis: ( card the carrier of R = 2 implies ex a, b being set st
( the carrier of R = {a,b} & ( the InternalRel of R = {[a,b],[b,a]} or the InternalRel of R = {} ) ) )
set Q = the InternalRel of R;
assume A1: card the carrier of R = 2 ; ::_thesis: ex a, b being set st
( the carrier of R = {a,b} & ( the InternalRel of R = {[a,b],[b,a]} or the InternalRel of R = {} ) )
then reconsider X = the carrier of R as finite set ;
consider a, b being set 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 set ; :: according to TARSKI:def_3 ::_thesis: ( not x in the InternalRel of R or x in {[a,b],[b,a]} )
assume A5: x in the InternalRel of R ; ::_thesis: x in {[a,b],[b,a]}
then consider x1, x2 being set 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;
percases ( x = [a,a] or x = [a,b] or x = [b,a] or x = [b,b] ) by A3, A6, A8, A9, TARSKI:def_2;
supposeA10: x = [a,a] ; ::_thesis: x in {[a,b],[b,a]}
a in the carrier of R by A3, TARSKI:def_2;
hence x in {[a,b],[b,a]} by A5, A10, NECKLACE:def_5; ::_thesis: verum
end;
suppose x = [a,b] ; ::_thesis: x in {[a,b],[b,a]}
hence x in {[a,b],[b,a]} by TARSKI:def_2; ::_thesis: verum
end;
suppose x = [b,a] ; ::_thesis: x in {[a,b],[b,a]}
hence x in {[a,b],[b,a]} by TARSKI:def_2; ::_thesis: verum
end;
supposeA11: x = [b,b] ; ::_thesis: x in {[a,b],[b,a]}
b in the carrier of R by A3, TARSKI:def_2;
hence x in {[a,b],[b,a]} by A5, A11, NECKLACE:def_5; ::_thesis: verum
end;
end;
end;
percases ( the InternalRel of R = {} or the InternalRel of R = {[a,b]} or the InternalRel of R = {[b,a]} or the InternalRel of R = {[a,b],[b,a]} ) by A4, ZFMISC_1:36;
suppose the InternalRel of R = {} ; ::_thesis: ex a, b being set st
( the carrier of R = {a,b} & ( the InternalRel of R = {[a,b],[b,a]} or the InternalRel of R = {} ) )
hence ex a, b being set st
( the carrier of R = {a,b} & ( the InternalRel of R = {[a,b],[b,a]} or the InternalRel of R = {} ) ) by A3; ::_thesis: verum
end;
supposeA12: the InternalRel of R = {[a,b]} ; ::_thesis: ex a, b being set st
( the carrier of R = {a,b} & ( the InternalRel of R = {[a,b],[b,a]} or the InternalRel of R = {} ) )
A13: ( a in X & b in X ) by A3, TARSKI:def_2;
A14: the InternalRel of R is_symmetric_in X by NECKLACE:def_3;
[a,b] in the InternalRel of R by A12, TARSKI:def_1;
then [b,a] in the InternalRel of R by A13, A14, RELAT_2:def_3;
then [b,a] = [a,b] by A12, TARSKI:def_1;
hence ex a, b being set st
( the carrier of R = {a,b} & ( the InternalRel of R = {[a,b],[b,a]} or the InternalRel of R = {} ) ) by A2, XTUPLE_0:1; ::_thesis: verum
end;
supposeA15: the InternalRel of R = {[b,a]} ; ::_thesis: ex a, b being set st
( the carrier of R = {a,b} & ( the InternalRel of R = {[a,b],[b,a]} or the InternalRel of R = {} ) )
A16: ( a in X & b in X ) by A3, TARSKI:def_2;
A17: the InternalRel of R is_symmetric_in X by NECKLACE:def_3;
[b,a] in the InternalRel of R by A15, TARSKI:def_1;
then [a,b] in the InternalRel of R by A16, A17, RELAT_2:def_3;
then [b,a] = [a,b] by A15, TARSKI:def_1;
hence ex a, b being set st
( the carrier of R = {a,b} & ( the InternalRel of R = {[a,b],[b,a]} or the InternalRel of R = {} ) ) by A2, XTUPLE_0:1; ::_thesis: verum
end;
suppose the InternalRel of R = {[a,b],[b,a]} ; ::_thesis: ex a, b being set st
( the carrier of R = {a,b} & ( the InternalRel of R = {[a,b],[b,a]} or the InternalRel of R = {} ) )
hence ex a, b being set st
( the carrier of R = {a,b} & ( the InternalRel of R = {[a,b],[b,a]} or the InternalRel of R = {} ) ) by A3; ::_thesis: verum
end;
end;
end;
begin
registration
let R be non empty RelStr ;
let S be RelStr ;
cluster union_of (R,S) -> non empty ;
correctness
coherence
not union_of (R,S) is empty ;
proof
not the carrier of R \/ the carrier of S is empty ;
hence not union_of (R,S) is empty by NECKLA_2:def_2; ::_thesis: verum
end;
cluster sum_of (R,S) -> non empty ;
correctness
coherence
not sum_of (R,S) is empty ;
proof
not the carrier of R \/ the carrier of S is empty ;
hence not sum_of (R,S) is empty by NECKLA_2:def_3; ::_thesis: verum
end;
end;
registration
let R be RelStr ;
let S be non empty RelStr ;
cluster union_of (R,S) -> non empty ;
correctness
coherence
not union_of (R,S) is empty ;
proof
not the carrier of R \/ the carrier of S is empty ;
hence not union_of (R,S) is empty by NECKLA_2:def_2; ::_thesis: verum
end;
cluster sum_of (R,S) -> non empty ;
correctness
coherence
not sum_of (R,S) is empty ;
proof
not the carrier of R \/ the carrier of S is empty ;
hence not sum_of (R,S) is empty by NECKLA_2:def_3; ::_thesis: verum
end;
end;
registration
let R, S be finite RelStr ;
cluster union_of (R,S) -> finite ;
correctness
coherence
union_of (R,S) is finite ;
proof
the carrier of R \/ the carrier of S is finite ;
hence union_of (R,S) is finite by NECKLA_2:def_2; ::_thesis: verum
end;
cluster sum_of (R,S) -> finite ;
correctness
coherence
sum_of (R,S) is finite ;
proof
the carrier of R \/ the carrier of S is finite ;
hence sum_of (R,S) is finite by NECKLA_2:def_3; ::_thesis: verum
end;
end;
registration
let R, S be symmetric RelStr ;
cluster union_of (R,S) -> symmetric ;
correctness
coherence
union_of (R,S) is symmetric ;
proof
let x, y be set ; :: according to NECKLACE:def_3,RELAT_2:def_3 ::_thesis: ( not x in the carrier of (union_of (R,S)) or not y in the carrier of (union_of (R,S)) or not [x,y] in the InternalRel of (union_of (R,S)) or [y,x] in the InternalRel of (union_of (R,S)) )
set U = union_of (R,S);
set cU = the carrier of (union_of (R,S));
set IU = the InternalRel of (union_of (R,S));
set cR = the carrier of R;
set cS = the carrier of S;
assume that
x in the carrier of (union_of (R,S)) and
y in the carrier of (union_of (R,S)) and
A1: [x,y] in the InternalRel of (union_of (R,S)) ; ::_thesis: [y,x] in the InternalRel of (union_of (R,S))
A2: [x,y] in the InternalRel of R \/ the InternalRel of S by A1, NECKLA_2:def_2;
percases ( [x,y] in the InternalRel of R or [x,y] in the InternalRel of S ) by A2, XBOOLE_0:def_3;
supposeA3: [x,y] in the InternalRel of R ; ::_thesis: [y,x] in the InternalRel of (union_of (R,S))
A4: the InternalRel of R is_symmetric_in the carrier of R by NECKLACE:def_3;
( x in the carrier of R & y in the carrier of R ) by A3, ZFMISC_1:87;
then [y,x] in the InternalRel of R by A3, A4, RELAT_2:def_3;
then [y,x] in the InternalRel of R \/ the InternalRel of S by XBOOLE_0:def_3;
hence [y,x] in the InternalRel of (union_of (R,S)) by NECKLA_2:def_2; ::_thesis: verum
end;
supposeA5: [x,y] in the InternalRel of S ; ::_thesis: [y,x] in the InternalRel of (union_of (R,S))
A6: the InternalRel of S is_symmetric_in the carrier of S by NECKLACE:def_3;
( x in the carrier of S & y in the carrier of S ) by A5, ZFMISC_1:87;
then [y,x] in the InternalRel of S by A5, A6, RELAT_2:def_3;
then [y,x] in the InternalRel of R \/ the InternalRel of S by XBOOLE_0:def_3;
hence [y,x] in the InternalRel of (union_of (R,S)) by NECKLA_2:def_2; ::_thesis: verum
end;
end;
end;
cluster sum_of (R,S) -> symmetric ;
correctness
coherence
sum_of (R,S) is symmetric ;
proof
set SU = sum_of (R,S);
set cSU = the carrier of (sum_of (R,S));
set ISU = the InternalRel of (sum_of (R,S));
set cR = the carrier of R;
set IR = the InternalRel of R;
set cS = the carrier of S;
set IS = the InternalRel of S;
A7: the InternalRel of S is_symmetric_in the carrier of S by NECKLACE:def_3;
A8: the InternalRel of R is_symmetric_in the carrier of R by NECKLACE:def_3;
the InternalRel of (sum_of (R,S)) is_symmetric_in the carrier of (sum_of (R,S))
proof
let x, y be set ; :: according to RELAT_2:def_3 ::_thesis: ( not x in the carrier of (sum_of (R,S)) or not y in the carrier of (sum_of (R,S)) or not [x,y] in the InternalRel of (sum_of (R,S)) or [y,x] in the InternalRel of (sum_of (R,S)) )
assume that
x in the carrier of (sum_of (R,S)) and
y in the carrier of (sum_of (R,S)) and
A9: [x,y] in the InternalRel of (sum_of (R,S)) ; ::_thesis: [y,x] in the InternalRel of (sum_of (R,S))
[x,y] in (( the InternalRel of R \/ the InternalRel of S) \/ [: the carrier of R, the carrier of S:]) \/ [: the carrier of S, the carrier of R:] by A9, NECKLA_2:def_3;
then ( [x,y] in ( the InternalRel of R \/ the InternalRel of S) \/ [: the carrier of R, the carrier of S:] or [x,y] in [: the carrier of S, the carrier of R:] ) by XBOOLE_0:def_3;
then A10: ( [x,y] in the InternalRel of R \/ the InternalRel of S or [x,y] in [: the carrier of R, the carrier of S:] or [x,y] in [: the carrier of S, the carrier of R:] ) by XBOOLE_0:def_3;
percases ( [x,y] in the InternalRel of R or [x,y] in the InternalRel of S or [x,y] in [: the carrier of R, the carrier of S:] or [x,y] in [: the carrier of S, the carrier of R:] ) by A10, XBOOLE_0:def_3;
supposeA11: [x,y] in the InternalRel of R ; ::_thesis: [y,x] in the InternalRel of (sum_of (R,S))
then ( x in the carrier of R & y in the carrier of R ) by ZFMISC_1:87;
then [y,x] in the InternalRel of R by A8, A11, RELAT_2:def_3;
then [y,x] in the InternalRel of R \/ the InternalRel of S by XBOOLE_0:def_3;
then [y,x] in ( the InternalRel of R \/ the InternalRel of S) \/ [: the carrier of R, the carrier of S:] by XBOOLE_0:def_3;
then [y,x] in (( the InternalRel of R \/ the InternalRel of S) \/ [: the carrier of R, the carrier of S:]) \/ [: the carrier of S, the carrier of R:] by XBOOLE_0:def_3;
hence [y,x] in the InternalRel of (sum_of (R,S)) by NECKLA_2:def_3; ::_thesis: verum
end;
supposeA12: [x,y] in the InternalRel of S ; ::_thesis: [y,x] in the InternalRel of (sum_of (R,S))
then ( x in the carrier of S & y in the carrier of S ) by ZFMISC_1:87;
then [y,x] in the InternalRel of S by A7, A12, RELAT_2:def_3;
then [y,x] in the InternalRel of R \/ the InternalRel of S by XBOOLE_0:def_3;
then [y,x] in ( the InternalRel of R \/ the InternalRel of S) \/ [: the carrier of R, the carrier of S:] by XBOOLE_0:def_3;
then [y,x] in (( the InternalRel of R \/ the InternalRel of S) \/ [: the carrier of R, the carrier of S:]) \/ [: the carrier of S, the carrier of R:] by XBOOLE_0:def_3;
hence [y,x] in the InternalRel of (sum_of (R,S)) by NECKLA_2:def_3; ::_thesis: verum
end;
suppose [x,y] in [: the carrier of R, the carrier of S:] ; ::_thesis: [y,x] in the InternalRel of (sum_of (R,S))
then ( x in the carrier of R & y in the carrier of S ) by ZFMISC_1:87;
then [y,x] in [: the carrier of S, the carrier of R:] by ZFMISC_1:87;
then [y,x] in [: the carrier of R, the carrier of S:] \/ [: the carrier of S, the carrier of R:] by XBOOLE_0:def_3;
then [y,x] in the InternalRel of S \/ ([: the carrier of R, the carrier of S:] \/ [: the carrier of S, the carrier of R:]) by XBOOLE_0:def_3;
then [y,x] in ( the InternalRel of S \/ [: the carrier of R, the carrier of S:]) \/ [: the carrier of S, the carrier of R:] by XBOOLE_1:4;
then [y,x] in the InternalRel of R \/ (( the InternalRel of S \/ [: the carrier of R, the carrier of S:]) \/ [: the carrier of S, the carrier of R:]) by XBOOLE_0:def_3;
then [y,x] in the InternalRel of R \/ ( the InternalRel of S \/ ([: the carrier of R, the carrier of S:] \/ [: the carrier of S, the carrier of R:])) by XBOOLE_1:4;
then [y,x] in ( the InternalRel of R \/ the InternalRel of S) \/ ([: the carrier of R, the carrier of S:] \/ [: the carrier of S, the carrier of R:]) by XBOOLE_1:4;
then [y,x] in (( the InternalRel of R \/ the InternalRel of S) \/ [: the carrier of R, the carrier of S:]) \/ [: the carrier of S, the carrier of R:] by XBOOLE_1:4;
hence [y,x] in the InternalRel of (sum_of (R,S)) by NECKLA_2:def_3; ::_thesis: verum
end;
suppose [x,y] in [: the carrier of S, the carrier of R:] ; ::_thesis: [y,x] in the InternalRel of (sum_of (R,S))
then ( x in the carrier of S & y in the carrier of R ) by ZFMISC_1:87;
then [y,x] in [: the carrier of R, the carrier of S:] by ZFMISC_1:87;
then [y,x] in [: the carrier of R, the carrier of S:] \/ [: the carrier of S, the carrier of R:] by XBOOLE_0:def_3;
then [y,x] in the InternalRel of S \/ ([: the carrier of R, the carrier of S:] \/ [: the carrier of S, the carrier of R:]) by XBOOLE_0:def_3;
then [y,x] in ( the InternalRel of S \/ [: the carrier of R, the carrier of S:]) \/ [: the carrier of S, the carrier of R:] by XBOOLE_1:4;
then [y,x] in the InternalRel of R \/ (( the InternalRel of S \/ [: the carrier of R, the carrier of S:]) \/ [: the carrier of S, the carrier of R:]) by XBOOLE_0:def_3;
then [y,x] in the InternalRel of R \/ ( the InternalRel of S \/ ([: the carrier of R, the carrier of S:] \/ [: the carrier of S, the carrier of R:])) by XBOOLE_1:4;
then [y,x] in ( the InternalRel of R \/ the InternalRel of S) \/ ([: the carrier of R, the carrier of S:] \/ [: the carrier of S, the carrier of R:]) by XBOOLE_1:4;
then [y,x] in (( the InternalRel of R \/ the InternalRel of S) \/ [: the carrier of R, the carrier of S:]) \/ [: the carrier of S, the carrier of R:] by XBOOLE_1:4;
hence [y,x] in the InternalRel of (sum_of (R,S)) by NECKLA_2:def_3; ::_thesis: verum
end;
end;
end;
hence sum_of (R,S) is symmetric by NECKLACE:def_3; ::_thesis: verum
end;
end;
registration
let R, S be irreflexive RelStr ;
cluster union_of (R,S) -> irreflexive ;
correctness
coherence
union_of (R,S) is irreflexive ;
proof
set U = union_of (R,S);
set cU = the carrier of (union_of (R,S));
set IU = the InternalRel of (union_of (R,S));
set cR = the carrier of R;
set cS = the carrier of S;
for x being set st x in the carrier of (union_of (R,S)) holds
not [x,x] in the InternalRel of (union_of (R,S))
proof
let x be set ; ::_thesis: ( x in the carrier of (union_of (R,S)) implies not [x,x] in the InternalRel of (union_of (R,S)) )
assume x in the carrier of (union_of (R,S)) ; ::_thesis: not [x,x] in the InternalRel of (union_of (R,S))
assume [x,x] in the InternalRel of (union_of (R,S)) ; ::_thesis: contradiction
then A1: [x,x] in the InternalRel of R \/ the InternalRel of S by NECKLA_2:def_2;
percases ( [x,x] in the InternalRel of R or [x,x] in the InternalRel of S ) by A1, XBOOLE_0:def_3;
supposeA2: [x,x] in the InternalRel of R ; ::_thesis: contradiction
then x in the carrier of R by ZFMISC_1:87;
hence contradiction by A2, NECKLACE:def_5; ::_thesis: verum
end;
supposeA3: [x,x] in the InternalRel of S ; ::_thesis: contradiction
then x in the carrier of S by ZFMISC_1:87;
hence contradiction by A3, NECKLACE:def_5; ::_thesis: verum
end;
end;
end;
hence union_of (R,S) is irreflexive by NECKLACE:def_5; ::_thesis: verum
end;
end;
theorem :: NECKLA_3:7
for R, S being 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 ; ::_thesis: ( the carrier of R misses the carrier of S implies sum_of (R,S) is irreflexive )
assume A1: the carrier of R misses the carrier of S ; ::_thesis: sum_of (R,S) is irreflexive
for x being 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;
set IS = the InternalRel of S;
set RS = [: the carrier of R, the carrier of S:];
set SR = [: the carrier of S, the carrier of R:];
let x be set ; ::_thesis: ( x in the carrier of (sum_of (R,S)) implies not [x,x] in the InternalRel of (sum_of (R,S)) )
assume x in the carrier of (sum_of (R,S)) ; ::_thesis: not [x,x] in the InternalRel of (sum_of (R,S))
assume [x,x] in the InternalRel of (sum_of (R,S)) ; ::_thesis: contradiction
then [x,x] in (( the InternalRel of R \/ the InternalRel of S) \/ [: the carrier of R, the carrier of S:]) \/ [: the carrier of S, the carrier of R:] by NECKLA_2:def_3;
then ( [x,x] in ( the InternalRel of R \/ the InternalRel of S) \/ [: the carrier of R, the carrier of S:] or [x,x] in [: the carrier of S, the carrier of R:] ) by XBOOLE_0:def_3;
then A2: ( [x,x] in the InternalRel of R \/ the InternalRel of S or [x,x] in [: the carrier of R, the carrier of S:] or [x,x] in [: the carrier of S, the carrier of R:] ) by XBOOLE_0:def_3;
percases ( [x,x] in the InternalRel of R or [x,x] in the InternalRel of S or [x,x] in [: the carrier of R, the carrier of S:] or [x,x] in [: the carrier of S, the carrier of R:] ) by A2, XBOOLE_0:def_3;
supposeA3: [x,x] in the InternalRel of R ; ::_thesis: contradiction
then x in the carrier of R by ZFMISC_1:87;
hence contradiction by A3, NECKLACE:def_5; ::_thesis: verum
end;
supposeA4: [x,x] in the InternalRel of S ; ::_thesis: contradiction
then x in the carrier of S by ZFMISC_1:87;
hence contradiction by A4, NECKLACE:def_5; ::_thesis: verum
end;
suppose [x,x] in [: the carrier of R, the carrier of S:] ; ::_thesis: contradiction
then ( x in the carrier of R & x in the carrier of S ) by ZFMISC_1:87;
hence contradiction by A1, XBOOLE_0:3; ::_thesis: verum
end;
suppose [x,x] in [: the carrier of S, the carrier of R:] ; ::_thesis: contradiction
then ( x in the carrier of S & x in the carrier of R ) by ZFMISC_1:87;
hence contradiction by A1, XBOOLE_0:3; ::_thesis: verum
end;
end;
end;
hence sum_of (R,S) is irreflexive by NECKLACE:def_5; ::_thesis: verum
end;
theorem Th8: :: NECKLA_3:8
for R1, R2 being RelStr holds
( union_of (R1,R2) = union_of (R2,R1) & sum_of (R1,R2) = sum_of (R2,R1) )
proof
let R1, R2 be RelStr ; ::_thesis: ( union_of (R1,R2) = union_of (R2,R1) & sum_of (R1,R2) = sum_of (R2,R1) )
set U1 = union_of (R1,R2);
set S1 = sum_of (R1,R2);
A1: the carrier of (sum_of (R1,R2)) = the carrier of R2 \/ the carrier of R1 by NECKLA_2:def_3;
A2: the InternalRel of (sum_of (R1,R2)) = (( 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 (union_of (R1,R2)) = the carrier of R2 \/ the carrier of R1 & the InternalRel of (union_of (R1,R2)) = the InternalRel of R2 \/ the InternalRel of R1 ) by NECKLA_2:def_2;
hence ( union_of (R1,R2) = union_of (R2,R1) & sum_of (R1,R2) = sum_of (R2,R1) ) by A1, A2, NECKLA_2:def_2, NECKLA_2:def_3; ::_thesis: verum
end;
theorem Th9: :: NECKLA_3:9
for G being irreflexive RelStr
for G1, G2 being 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 ; ::_thesis: for G1, G2 being RelStr st ( G = union_of (G1,G2) or G = sum_of (G1,G2) ) holds
( G1 is irreflexive & G2 is irreflexive )
let G1, G2 be RelStr ; ::_thesis: ( ( G = union_of (G1,G2) or G = sum_of (G1,G2) ) implies ( G1 is irreflexive & G2 is irreflexive ) )
assume A1: ( G = union_of (G1,G2) or G = sum_of (G1,G2) ) ; ::_thesis: ( G1 is irreflexive & G2 is irreflexive )
percases ( G = union_of (G1,G2) or G = sum_of (G1,G2) ) by A1;
supposeA2: G = union_of (G1,G2) ; ::_thesis: ( G1 is irreflexive & G2 is irreflexive )
assume A3: ( not G1 is irreflexive or not G2 is irreflexive ) ; ::_thesis: contradiction
thus contradiction ::_thesis: verum
proof
percases ( not G1 is irreflexive or not G2 is irreflexive ) by A3;
suppose not G1 is irreflexive ; ::_thesis: contradiction
then consider x being set such that
A4: x in the carrier of G1 and
A5: [x,x] in the InternalRel of G1 by NECKLACE:def_5;
[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 contradiction by A6, NECKLACE:def_5; ::_thesis: verum
end;
suppose not G2 is irreflexive ; ::_thesis: contradiction
then consider x being set such that
A7: x in the carrier of G2 and
A8: [x,x] in the InternalRel of G2 by NECKLACE:def_5;
[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 contradiction by A9, NECKLACE:def_5; ::_thesis: verum
end;
end;
end;
end;
supposeA10: G = sum_of (G1,G2) ; ::_thesis: ( G1 is irreflexive & G2 is irreflexive )
assume A11: ( not G1 is irreflexive or not G2 is irreflexive ) ; ::_thesis: contradiction
thus contradiction ::_thesis: verum
proof
percases ( not G1 is irreflexive or not G2 is irreflexive ) by A11;
suppose not G1 is irreflexive ; ::_thesis: contradiction
then consider x being set such that
A12: x in the carrier of G1 and
A13: [x,x] in the InternalRel of G1 by NECKLACE:def_5;
[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 contradiction by A14, NECKLACE:def_5; ::_thesis: verum
end;
suppose not G2 is irreflexive ; ::_thesis: contradiction
then consider x being set such that
A15: x in the carrier of G2 and
A16: [x,x] in the InternalRel of G2 by NECKLACE:def_5;
[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 contradiction by A17, NECKLACE:def_5; ::_thesis: verum
end;
end;
end;
end;
end;
end;
theorem Th10: :: NECKLA_3:10
for G being non empty RelStr
for H1, H2 being RelStr st the carrier of H1 misses the carrier of H2 & ( RelStr(# the carrier of G, the InternalRel of G #) = union_of (H1,H2) or RelStr(# the carrier of G, the InternalRel 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 ; ::_thesis: for H1, H2 being RelStr st the carrier of H1 misses the carrier of H2 & ( RelStr(# the carrier of G, the InternalRel of G #) = union_of (H1,H2) or RelStr(# the carrier of G, the InternalRel of G #) = sum_of (H1,H2) ) holds
( H1 is full SubRelStr of G & H2 is full SubRelStr of G )
let H1, H2 be RelStr ; ::_thesis: ( the carrier of H1 misses the carrier of H2 & ( RelStr(# the carrier of G, the InternalRel of G #) = union_of (H1,H2) or RelStr(# the carrier of G, the InternalRel of G #) = sum_of (H1,H2) ) implies ( H1 is full SubRelStr of G & H2 is full SubRelStr of G ) )
assume that
A1: the carrier of H1 misses the carrier of H2 and
A2: ( RelStr(# the carrier of G, the InternalRel of G #) = union_of (H1,H2) or RelStr(# the carrier of G, the InternalRel of G #) = sum_of (H1,H2) ) ; ::_thesis: ( H1 is full SubRelStr of G & H2 is full SubRelStr of G )
set cH1 = the carrier of H1;
set cH2 = the carrier of H2;
set IH1 = the InternalRel of H1;
set IH2 = the InternalRel of H2;
set H1H2 = [: the carrier of H1, the carrier of H2:];
set H2H1 = [: the carrier of H2, the carrier of H1:];
percases ( RelStr(# the carrier of G, the InternalRel of G #) = union_of (H1,H2) or RelStr(# the carrier of G, the InternalRel of G #) = sum_of (H1,H2) ) by A2;
supposeA3: RelStr(# the carrier of G, the InternalRel of G #) = union_of (H1,H2) ; ::_thesis: ( H1 is full SubRelStr of G & H2 is full SubRelStr of G )
A4: the InternalRel of H2 = the InternalRel of G |_2 the carrier of H2
proof
thus the InternalRel of H2 c= the InternalRel of G |_2 the carrier of H2 :: according to XBOOLE_0:def_10 ::_thesis: the InternalRel of G |_2 the carrier of H2 c= the InternalRel of H2
proof
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in the InternalRel of H2 or a in the InternalRel of G |_2 the carrier of H2 )
the InternalRel of G = the InternalRel of H1 \/ the InternalRel of H2 by A3, NECKLA_2:def_2;
then A5: the InternalRel of H2 c= the InternalRel of G by XBOOLE_1:7;
assume a in the InternalRel of H2 ; ::_thesis: a in the InternalRel of G |_2 the carrier of H2
hence a in the InternalRel of G |_2 the carrier of H2 by A5, XBOOLE_0:def_4; ::_thesis: verum
end;
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in the InternalRel of G |_2 the carrier of H2 or a in the InternalRel of H2 )
assume A6: a in the InternalRel of G |_2 the carrier of H2 ; ::_thesis: a in the InternalRel of H2
then A7: a in [: the carrier of H2, the carrier of H2:] by XBOOLE_0:def_4;
a in the InternalRel of G by A6, XBOOLE_0:def_4;
then A8: a in the InternalRel of H1 \/ the InternalRel of H2 by A3, NECKLA_2:def_2;
percases ( a in the InternalRel of H1 or a in the InternalRel of H2 ) by A8, XBOOLE_0:def_3;
suppose a in the InternalRel of H1 ; ::_thesis: a in the InternalRel of H2
then consider x, y being set such that
A9: a = [x,y] and
A10: x in the carrier of H1 and
y in the carrier of H1 by RELSET_1:2;
consider x1, y1 being set such that
A11: x1 in the carrier of H2 and
y1 in the carrier of H2 and
A12: a = [x1,y1] by A7, ZFMISC_1:def_2;
x = x1 by A9, A12, XTUPLE_0:1;
then the carrier of H1 /\ the carrier of H2 <> {} by A10, A11, XBOOLE_0:def_4;
hence a in the InternalRel of H2 by A1, XBOOLE_0:def_7; ::_thesis: verum
end;
suppose a in the InternalRel of H2 ; ::_thesis: a in the InternalRel of H2
hence a in the InternalRel of H2 ; ::_thesis: verum
end;
end;
end;
A13: the InternalRel of H1 = the InternalRel of G |_2 the carrier of H1
proof
thus the InternalRel of H1 c= the InternalRel of G |_2 the carrier of H1 :: according to XBOOLE_0:def_10 ::_thesis: the InternalRel of G |_2 the carrier of H1 c= the InternalRel of H1
proof
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in the InternalRel of H1 or a in the InternalRel of G |_2 the carrier of H1 )
the InternalRel of G = the InternalRel of H1 \/ the InternalRel of H2 by A3, NECKLA_2:def_2;
then A14: the InternalRel of H1 c= the InternalRel of G by XBOOLE_1:7;
assume a in the InternalRel of H1 ; ::_thesis: a in the InternalRel of G |_2 the carrier of H1
hence a in the InternalRel of G |_2 the carrier of H1 by A14, XBOOLE_0:def_4; ::_thesis: verum
end;
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in the InternalRel of G |_2 the carrier of H1 or a in the InternalRel of H1 )
assume A15: a in the InternalRel of G |_2 the carrier of H1 ; ::_thesis: a in the InternalRel of H1
then A16: a in [: the carrier of H1, the carrier of H1:] by XBOOLE_0:def_4;
a in the InternalRel of G by A15, XBOOLE_0:def_4;
then A17: a in the InternalRel of H1 \/ the InternalRel of H2 by A3, NECKLA_2:def_2;
percases ( a in the InternalRel of H1 or a in the InternalRel of H2 ) by A17, XBOOLE_0:def_3;
suppose a in the InternalRel of H1 ; ::_thesis: a in the InternalRel of H1
hence a in the InternalRel of H1 ; ::_thesis: verum
end;
suppose a in the InternalRel of H2 ; ::_thesis: a in the InternalRel of H1
then consider x, y being set such that
A18: a = [x,y] and
A19: x in the carrier of H2 and
y in the carrier of H2 by RELSET_1:2;
ex x1, y1 being set st
( x1 in the carrier of H1 & y1 in the carrier of H1 & a = [x1,y1] ) by A16, ZFMISC_1:def_2;
then x in the carrier of H1 by A18, XTUPLE_0:1;
hence a in the InternalRel of H1 by A1, A19, XBOOLE_0:3; ::_thesis: verum
end;
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 = the InternalRel of H1 \/ the InternalRel of H2 by A3, NECKLA_2:def_2;
then ( the InternalRel of H1 c= the InternalRel of G & the InternalRel of H2 c= the InternalRel of G ) by XBOOLE_1:7;
hence ( H1 is full SubRelStr of G & H2 is full SubRelStr of G ) by A20, A13, A4, YELLOW_0:def_13, YELLOW_0:def_14; ::_thesis: verum
end;
supposeA21: RelStr(# the carrier of G, the InternalRel of G #) = sum_of (H1,H2) ; ::_thesis: ( H1 is full SubRelStr of G & H2 is full SubRelStr of G )
A22: the InternalRel of H2 = the InternalRel of G |_2 the carrier of H2
proof
thus the InternalRel of H2 c= the InternalRel of G |_2 the carrier of H2 :: according to XBOOLE_0:def_10 ::_thesis: the InternalRel of G |_2 the carrier of H2 c= the InternalRel of H2
proof
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in the InternalRel of H2 or a in the InternalRel of G |_2 the carrier of H2 )
the InternalRel of G = (( the InternalRel of H1 \/ the InternalRel of H2) \/ [: the carrier of H1, the carrier of H2:]) \/ [: the carrier of H2, the carrier of H1:] by A21, NECKLA_2:def_3;
then the InternalRel of G = the InternalRel of H2 \/ (( the InternalRel of H1 \/ [: the carrier of H1, the carrier of H2:]) \/ [: the carrier of H2, the carrier of H1:]) by XBOOLE_1:113;
then A23: the InternalRel of H2 c= the InternalRel of G by XBOOLE_1:7;
assume a in the InternalRel of H2 ; ::_thesis: a in the InternalRel of G |_2 the carrier of H2
hence a in the InternalRel of G |_2 the carrier of H2 by A23, XBOOLE_0:def_4; ::_thesis: verum
end;
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in the InternalRel of G |_2 the carrier of H2 or a in the InternalRel of H2 )
assume A24: a in the InternalRel of G |_2 the carrier of H2 ; ::_thesis: a in the InternalRel of H2
then A25: a in [: the carrier of H2, the carrier of H2:] by XBOOLE_0:def_4;
a in the InternalRel of G by A24, XBOOLE_0:def_4;
then a in (( the InternalRel of H1 \/ the InternalRel of H2) \/ [: the carrier of H1, the carrier of H2:]) \/ [: the carrier of H2, the carrier of H1:] by A21, NECKLA_2:def_3;
then a in the InternalRel of H1 \/ (( the InternalRel of H2 \/ [: the carrier of H1, the carrier of H2:]) \/ [: the carrier of H2, the carrier of H1:]) by XBOOLE_1:113;
then ( a in the InternalRel of H1 or a in ( the InternalRel of H2 \/ [: the carrier of H1, the carrier of H2:]) \/ [: the carrier of H2, the carrier of H1:] ) by XBOOLE_0:def_3;
then ( a in the InternalRel of H1 or a in the InternalRel of H2 \/ ([: the carrier of H1, the carrier of H2:] \/ [: the carrier of H2, the carrier of H1:]) ) by XBOOLE_1:4;
then A26: ( a in the InternalRel of H1 or a in the InternalRel of H2 or a in [: the carrier of H1, the carrier of H2:] \/ [: the carrier of H2, the carrier of H1:] ) by XBOOLE_0:def_3;
percases ( a in the InternalRel of H1 or a in the InternalRel of H2 or a in [: the carrier of H1, the carrier of H2:] or a in [: the carrier of H2, the carrier of H1:] ) by A26, XBOOLE_0:def_3;
suppose a in the InternalRel of H1 ; ::_thesis: a in the InternalRel of H2
then consider x, y being set such that
A27: a = [x,y] and
A28: x in the carrier of H1 and
y in the carrier of H1 by RELSET_1:2;
consider x1, y1 being set such that
A29: x1 in the carrier of H2 and
y1 in the carrier of H2 and
A30: a = [x1,y1] by A25, ZFMISC_1:def_2;
x = x1 by A27, A30, XTUPLE_0:1;
then the carrier of H1 /\ the carrier of H2 <> {} by A28, A29, XBOOLE_0:def_4;
hence a in the InternalRel of H2 by A1, XBOOLE_0:def_7; ::_thesis: verum
end;
suppose a in the InternalRel of H2 ; ::_thesis: a in the InternalRel of H2
hence a in the InternalRel of H2 ; ::_thesis: verum
end;
suppose a in [: the carrier of H1, the carrier of H2:] ; ::_thesis: a in the InternalRel of H2
then consider x, y being set such that
A31: x in the carrier of H1 and
y in the carrier of H2 and
A32: a = [x,y] by ZFMISC_1:def_2;
consider x1, y1 being set such that
A33: x1 in the carrier of H2 and
y1 in the carrier of H2 and
A34: a = [x1,y1] by A25, ZFMISC_1:def_2;
x = x1 by A32, A34, XTUPLE_0:1;
then the carrier of H1 /\ the carrier of H2 <> {} by A31, A33, XBOOLE_0:def_4;
hence a in the InternalRel of H2 by A1, XBOOLE_0:def_7; ::_thesis: verum
end;
suppose a in [: the carrier of H2, the carrier of H1:] ; ::_thesis: a in the InternalRel of H2
then consider x, y being set such that
x in the carrier of H2 and
A35: y in the carrier of H1 and
A36: a = [x,y] by ZFMISC_1:def_2;
consider x1, y1 being set such that
x1 in the carrier of H2 and
A37: y1 in the carrier of H2 and
A38: a = [x1,y1] by A25, ZFMISC_1:def_2;
y = y1 by A36, A38, XTUPLE_0:1;
then the carrier of H1 /\ the carrier of H2 <> {} by A35, A37, XBOOLE_0:def_4;
hence a in the InternalRel of H2 by A1, XBOOLE_0:def_7; ::_thesis: verum
end;
end;
end;
the InternalRel of H2 c= ( the InternalRel of H1 \/ the InternalRel of H2) \/ [: the carrier of H1, the carrier of H2:] by XBOOLE_1:7, XBOOLE_1:10;
then A39: the InternalRel of H2 c= (( the InternalRel of H1 \/ the InternalRel of H2) \/ [: the carrier of H1, the carrier of H2:]) \/ [: the carrier of H2, the carrier of H1:] by XBOOLE_1:10;
A40: the InternalRel of H1 = the InternalRel of G |_2 the carrier of H1
proof
thus the InternalRel of H1 c= the InternalRel of G |_2 the carrier of H1 :: according to XBOOLE_0:def_10 ::_thesis: the InternalRel of G |_2 the carrier of H1 c= the InternalRel of H1
proof
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in the InternalRel of H1 or a in the InternalRel of G |_2 the carrier of H1 )
the InternalRel of G = (( the InternalRel of H1 \/ the InternalRel of H2) \/ [: the carrier of H1, the carrier of H2:]) \/ [: the carrier of H2, the carrier of H1:] by A21, NECKLA_2:def_3
.= the InternalRel of H1 \/ (( the InternalRel of H2 \/ [: the carrier of H1, the carrier of H2:]) \/ [: the carrier of H2, the carrier of H1:]) by XBOOLE_1:113 ;
then A41: the InternalRel of H1 c= the InternalRel of G by XBOOLE_1:7;
assume a in the InternalRel of H1 ; ::_thesis: a in the InternalRel of G |_2 the carrier of H1
hence a in the InternalRel of G |_2 the carrier of H1 by A41, XBOOLE_0:def_4; ::_thesis: verum
end;
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in the InternalRel of G |_2 the carrier of H1 or a in the InternalRel of H1 )
assume A42: a in the InternalRel of G |_2 the carrier of H1 ; ::_thesis: a in the InternalRel of H1
then A43: a in [: the carrier of H1, the carrier of H1:] by XBOOLE_0:def_4;
a in the InternalRel of G by A42, XBOOLE_0:def_4;
then a in (( the InternalRel of H1 \/ the InternalRel of H2) \/ [: the carrier of H1, the carrier of H2:]) \/ [: the carrier of H2, the carrier of H1:] by A21, NECKLA_2:def_3;
then a in the InternalRel of H1 \/ (( the InternalRel of H2 \/ [: the carrier of H1, the carrier of H2:]) \/ [: the carrier of H2, the carrier of H1:]) by XBOOLE_1:113;
then ( a in the InternalRel of H1 or a in ( the InternalRel of H2 \/ [: the carrier of H1, the carrier of H2:]) \/ [: the carrier of H2, the carrier of H1:] ) by XBOOLE_0:def_3;
then ( a in the InternalRel of H1 or a in the InternalRel of H2 \/ ([: the carrier of H1, the carrier of H2:] \/ [: the carrier of H2, the carrier of H1:]) ) by XBOOLE_1:4;
then A44: ( a in the InternalRel of H1 or a in the InternalRel of H2 or a in [: the carrier of H1, the carrier of H2:] \/ [: the carrier of H2, the carrier of H1:] ) by XBOOLE_0:def_3;
percases ( a in the InternalRel of H1 or a in the InternalRel of H2 or a in [: the carrier of H1, the carrier of H2:] or a in [: the carrier of H2, the carrier of H1:] ) by A44, XBOOLE_0:def_3;
suppose a in the InternalRel of H1 ; ::_thesis: a in the InternalRel of H1
hence a in the InternalRel of H1 ; ::_thesis: verum
end;
suppose a in the InternalRel of H2 ; ::_thesis: a in the InternalRel of H1
then consider x, y being set such that
A45: a = [x,y] and
A46: x in the carrier of H2 and
y in the carrier of H2 by RELSET_1:2;
consider x1, y1 being set such that
A47: x1 in the carrier of H1 and
y1 in the carrier of H1 and
A48: a = [x1,y1] by A43, ZFMISC_1:def_2;
x = x1 by A45, A48, XTUPLE_0:1;
then the carrier of H1 /\ the carrier of H2 <> {} by A46, A47, XBOOLE_0:def_4;
hence a in the InternalRel of H1 by A1, XBOOLE_0:def_7; ::_thesis: verum
end;
suppose a in [: the carrier of H1, the carrier of H2:] ; ::_thesis: a in the InternalRel of H1
then consider x, y being set such that
x in the carrier of H1 and
A49: y in the carrier of H2 and
A50: a = [x,y] by ZFMISC_1:def_2;
consider x1, y1 being set such that
x1 in the carrier of H1 and
A51: y1 in the carrier of H1 and
A52: a = [x1,y1] by A43, ZFMISC_1:def_2;
y = y1 by A50, A52, XTUPLE_0:1;
then the carrier of H1 /\ the carrier of H2 <> {} by A49, A51, XBOOLE_0:def_4;
hence a in the InternalRel of H1 by A1, XBOOLE_0:def_7; ::_thesis: verum
end;
suppose a in [: the carrier of H2, the carrier of H1:] ; ::_thesis: a in the InternalRel of H1
then consider x, y being set such that
A53: x in the carrier of H2 and
y in the carrier of H1 and
A54: a = [x,y] by ZFMISC_1:def_2;
consider x1, y1 being set such that
A55: x1 in the carrier of H1 and
y1 in the carrier of H1 and
A56: a = [x1,y1] by A43, ZFMISC_1:def_2;
x = x1 by A54, A56, XTUPLE_0:1;
then the carrier of H1 /\ the carrier of H2 <> {} by A53, A55, XBOOLE_0:def_4;
hence a in the InternalRel of H1 by A1, XBOOLE_0:def_7; ::_thesis: verum
end;
end;
end;
the InternalRel of H1 c= the InternalRel of H1 \/ ( the InternalRel of H2 \/ [: the carrier of H1, the carrier of H2:]) by XBOOLE_1:7;
then A57: the InternalRel of H1 c= ( the InternalRel of H1 \/ the InternalRel of H2) \/ [: the carrier of H1, the carrier of H2:] 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 = (( the InternalRel of H1 \/ the InternalRel of H2) \/ [: the carrier of H1, the carrier of H2:]) \/ [: the carrier of H2, the carrier of H1:] by A21, NECKLA_2:def_3;
then ( the InternalRel of H1 \/ the InternalRel of H2) \/ [: the carrier of H1, the carrier of H2:] c= the InternalRel of G by XBOOLE_1:7;
then the InternalRel of H1 c= the InternalRel of G by A57, XBOOLE_1:1;
hence ( H1 is full SubRelStr of G & H2 is full SubRelStr of G ) by A59, A58, A39, A40, A22, YELLOW_0:def_13, YELLOW_0:def_14; ::_thesis: verum
end;
end;
end;
begin
theorem Th11: :: NECKLA_3:11
the InternalRel of (ComplRelStr (Necklace 4)) = {[0,2],[2,0],[0,3],[3,0],[1,3],[3,1]}
proof
set N4 = Necklace 4;
set cN4 = the carrier of (Necklace 4);
set CmpN4 = ComplRelStr (Necklace 4);
A1: the carrier of (Necklace 4) = {0,1,2,3} by NECKLACE:1, NECKLACE:20;
thus the InternalRel of (ComplRelStr (Necklace 4)) c= {[0,2],[2,0],[0,3],[3,0],[1,3],[3,1]} :: according to XBOOLE_0:def_10 ::_thesis: {[0,2],[2,0],[0,3],[3,0],[1,3],[3,1]} c= the InternalRel of (ComplRelStr (Necklace 4))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in the InternalRel of (ComplRelStr (Necklace 4)) or x in {[0,2],[2,0],[0,3],[3,0],[1,3],[3,1]} )
assume x in the InternalRel of (ComplRelStr (Necklace 4)) ; ::_thesis: x in {[0,2],[2,0],[0,3],[3,0],[1,3],[3,1]}
then A2: x in ( the InternalRel of (Necklace 4) `) \ (id the carrier of (Necklace 4)) by NECKLACE:def_8;
then A3: not x in id the carrier of (Necklace 4) by XBOOLE_0:def_5;
x in the InternalRel of (Necklace 4) ` by A2, XBOOLE_0:def_5;
then A4: x in [: the carrier of (Necklace 4), the carrier of (Necklace 4):] \ the InternalRel of (Necklace 4) by SUBSET_1:def_4;
consider a1, b1 being set such that
A5: a1 in the carrier of (Necklace 4) and
A6: b1 in the carrier of (Necklace 4) and
A7: x = [a1,b1] by A2, ZFMISC_1:def_2;
percases ( ( a1 = 0 & b1 = 0 ) or ( a1 = 0 & b1 = 1 ) or ( a1 = 0 & b1 = 2 ) or ( a1 = 0 & b1 = 3 ) or ( a1 = 1 & b1 = 0 ) or ( a1 = 2 & b1 = 0 ) or ( a1 = 3 & b1 = 0 ) or ( a1 = 1 & b1 = 1 ) or ( a1 = 1 & b1 = 2 ) or ( a1 = 1 & b1 = 3 ) or ( a1 = 2 & b1 = 2 ) or ( a1 = 2 & b1 = 1 ) or ( a1 = 2 & b1 = 3 ) or ( a1 = 3 & b1 = 3 ) or ( a1 = 3 & b1 = 1 ) or ( a1 = 3 & b1 = 2 ) ) by A1, A5, A6, ENUMSET1:def_2;
suppose ( a1 = 0 & b1 = 0 ) ; ::_thesis: x in {[0,2],[2,0],[0,3],[3,0],[1,3],[3,1]}
hence x in {[0,2],[2,0],[0,3],[3,0],[1,3],[3,1]} by A3, A5, A7, RELAT_1:def_10; ::_thesis: verum
end;
suppose ( a1 = 0 & b1 = 1 ) ; ::_thesis: x in {[0,2],[2,0],[0,3],[3,0],[1,3],[3,1]}
then x in the InternalRel of (Necklace 4) by A7, ENUMSET1:def_4, NECKLA_2:2;
hence x in {[0,2],[2,0],[0,3],[3,0],[1,3],[3,1]} by A4, XBOOLE_0:def_5; ::_thesis: verum
end;
suppose ( a1 = 0 & b1 = 2 ) ; ::_thesis: x in {[0,2],[2,0],[0,3],[3,0],[1,3],[3,1]}
hence x in {[0,2],[2,0],[0,3],[3,0],[1,3],[3,1]} by A7, ENUMSET1:def_4; ::_thesis: verum
end;
suppose ( a1 = 0 & b1 = 3 ) ; ::_thesis: x in {[0,2],[2,0],[0,3],[3,0],[1,3],[3,1]}
hence x in {[0,2],[2,0],[0,3],[3,0],[1,3],[3,1]} by A7, ENUMSET1:def_4; ::_thesis: verum
end;
suppose ( a1 = 1 & b1 = 0 ) ; ::_thesis: x in {[0,2],[2,0],[0,3],[3,0],[1,3],[3,1]}
then x in the InternalRel of (Necklace 4) by A7, ENUMSET1:def_4, NECKLA_2:2;
hence x in {[0,2],[2,0],[0,3],[3,0],[1,3],[3,1]} by A4, XBOOLE_0:def_5; ::_thesis: verum
end;
suppose ( a1 = 2 & b1 = 0 ) ; ::_thesis: x in {[0,2],[2,0],[0,3],[3,0],[1,3],[3,1]}
hence x in {[0,2],[2,0],[0,3],[3,0],[1,3],[3,1]} by A7, ENUMSET1:def_4; ::_thesis: verum
end;
suppose ( a1 = 3 & b1 = 0 ) ; ::_thesis: x in {[0,2],[2,0],[0,3],[3,0],[1,3],[3,1]}
hence x in {[0,2],[2,0],[0,3],[3,0],[1,3],[3,1]} by A7, ENUMSET1:def_4; ::_thesis: verum
end;
suppose ( a1 = 1 & b1 = 1 ) ; ::_thesis: x in {[0,2],[2,0],[0,3],[3,0],[1,3],[3,1]}
hence x in {[0,2],[2,0],[0,3],[3,0],[1,3],[3,1]} by A3, A5, A7, RELAT_1:def_10; ::_thesis: verum
end;
suppose ( a1 = 1 & b1 = 2 ) ; ::_thesis: x in {[0,2],[2,0],[0,3],[3,0],[1,3],[3,1]}
then x in the InternalRel of (Necklace 4) by A7, ENUMSET1:def_4, NECKLA_2:2;
hence x in {[0,2],[2,0],[0,3],[3,0],[1,3],[3,1]} by A4, XBOOLE_0:def_5; ::_thesis: verum
end;
suppose ( a1 = 1 & b1 = 3 ) ; ::_thesis: x in {[0,2],[2,0],[0,3],[3,0],[1,3],[3,1]}
hence x in {[0,2],[2,0],[0,3],[3,0],[1,3],[3,1]} by A7, ENUMSET1:def_4; ::_thesis: verum
end;
suppose ( a1 = 2 & b1 = 2 ) ; ::_thesis: x in {[0,2],[2,0],[0,3],[3,0],[1,3],[3,1]}
hence x in {[0,2],[2,0],[0,3],[3,0],[1,3],[3,1]} by A3, A5, A7, RELAT_1:def_10; ::_thesis: verum
end;
suppose ( a1 = 2 & b1 = 1 ) ; ::_thesis: x in {[0,2],[2,0],[0,3],[3,0],[1,3],[3,1]}
then x in the InternalRel of (Necklace 4) by A7, ENUMSET1:def_4, NECKLA_2:2;
hence x in {[0,2],[2,0],[0,3],[3,0],[1,3],[3,1]} by A4, XBOOLE_0:def_5; ::_thesis: verum
end;
suppose ( a1 = 2 & b1 = 3 ) ; ::_thesis: x in {[0,2],[2,0],[0,3],[3,0],[1,3],[3,1]}
then x in the InternalRel of (Necklace 4) by A7, ENUMSET1:def_4, NECKLA_2:2;
hence x in {[0,2],[2,0],[0,3],[3,0],[1,3],[3,1]} by A4, XBOOLE_0:def_5; ::_thesis: verum
end;
suppose ( a1 = 3 & b1 = 3 ) ; ::_thesis: x in {[0,2],[2,0],[0,3],[3,0],[1,3],[3,1]}
hence x in {[0,2],[2,0],[0,3],[3,0],[1,3],[3,1]} by A3, A5, A7, RELAT_1:def_10; ::_thesis: verum
end;
suppose ( a1 = 3 & b1 = 1 ) ; ::_thesis: x in {[0,2],[2,0],[0,3],[3,0],[1,3],[3,1]}
hence x in {[0,2],[2,0],[0,3],[3,0],[1,3],[3,1]} by A7, ENUMSET1:def_4; ::_thesis: verum
end;
suppose ( a1 = 3 & b1 = 2 ) ; ::_thesis: x in {[0,2],[2,0],[0,3],[3,0],[1,3],[3,1]}
then x in the InternalRel of (Necklace 4) by A7, ENUMSET1:def_4, NECKLA_2:2;
hence x in {[0,2],[2,0],[0,3],[3,0],[1,3],[3,1]} by A4, XBOOLE_0:def_5; ::_thesis: verum
end;
end;
end;
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in {[0,2],[2,0],[0,3],[3,0],[1,3],[3,1]} or a in the InternalRel of (ComplRelStr (Necklace 4)) )
assume A8: a in {[0,2],[2,0],[0,3],[3,0],[1,3],[3,1]} ; ::_thesis: a in the InternalRel of (ComplRelStr (Necklace 4))
percases ( a = [0,2] or a = [2,0] or a = [0,3] or a = [3,0] or a = [1,3] or a = [3,1] ) by A8, ENUMSET1:def_4;
supposeA9: a = [0,2] ; ::_thesis: a in the InternalRel of (ComplRelStr (Necklace 4))
A10: not a in the InternalRel of (Necklace 4)
proof
assume A11: a in the InternalRel of (Necklace 4) ; ::_thesis: contradiction
percases ( a = [0,1] or a = [1,0] or a = [1,2] or a = [2,1] or a = [2,3] or a = [3,2] ) by A11, ENUMSET1:def_4, NECKLA_2:2;
suppose a = [0,1] ; ::_thesis: contradiction
hence contradiction by A9, XTUPLE_0:1; ::_thesis: verum
end;
suppose a = [1,0] ; ::_thesis: contradiction
hence contradiction by A9, XTUPLE_0:1; ::_thesis: verum
end;
suppose a = [1,2] ; ::_thesis: contradiction
hence contradiction by A9, XTUPLE_0:1; ::_thesis: verum
end;
suppose a = [2,1] ; ::_thesis: contradiction
hence contradiction by A9, XTUPLE_0:1; ::_thesis: verum
end;
suppose a = [2,3] ; ::_thesis: contradiction
hence contradiction by A9, XTUPLE_0:1; ::_thesis: verum
end;
suppose a = [3,2] ; ::_thesis: contradiction
hence contradiction by A9, XTUPLE_0:1; ::_thesis: verum
end;
end;
end;
( 0 in the carrier of (Necklace 4) & 2 in the carrier of (Necklace 4) ) by A1, ENUMSET1:def_2;
then a in [: the carrier of (Necklace 4), the carrier of (Necklace 4):] by A9, ZFMISC_1:87;
then a in [: the carrier of (Necklace 4), the carrier of (Necklace 4):] \ the InternalRel of (Necklace 4) by A10, XBOOLE_0:def_5;
then A12: a in the InternalRel of (Necklace 4) ` by SUBSET_1:def_4;
not a in id the carrier of (Necklace 4) by A9, RELAT_1:def_10;
then a in ( the InternalRel of (Necklace 4) `) \ (id the carrier of (Necklace 4)) by A12, XBOOLE_0:def_5;
hence a in the InternalRel of (ComplRelStr (Necklace 4)) by NECKLACE:def_8; ::_thesis: verum
end;
supposeA13: a = [2,0] ; ::_thesis: a in the InternalRel of (ComplRelStr (Necklace 4))
A14: not a in the InternalRel of (Necklace 4)
proof
assume A15: a in the InternalRel of (Necklace 4) ; ::_thesis: contradiction
percases ( a = [0,1] or a = [1,0] or a = [1,2] or a = [2,1] or a = [2,3] or a = [3,2] ) by A15, ENUMSET1:def_4, NECKLA_2:2;
suppose a = [0,1] ; ::_thesis: contradiction
hence contradiction by A13, XTUPLE_0:1; ::_thesis: verum
end;
suppose a = [1,0] ; ::_thesis: contradiction
hence contradiction by A13, XTUPLE_0:1; ::_thesis: verum
end;
suppose a = [1,2] ; ::_thesis: contradiction
hence contradiction by A13, XTUPLE_0:1; ::_thesis: verum
end;
suppose a = [2,1] ; ::_thesis: contradiction
hence contradiction by A13, XTUPLE_0:1; ::_thesis: verum
end;
suppose a = [2,3] ; ::_thesis: contradiction
hence contradiction by A13, XTUPLE_0:1; ::_thesis: verum
end;
suppose a = [3,2] ; ::_thesis: contradiction
hence contradiction by A13, XTUPLE_0:1; ::_thesis: verum
end;
end;
end;
( 0 in the carrier of (Necklace 4) & 2 in the carrier of (Necklace 4) ) by A1, ENUMSET1:def_2;
then a in [: the carrier of (Necklace 4), the carrier of (Necklace 4):] by A13, ZFMISC_1:87;
then a in [: the carrier of (Necklace 4), the carrier of (Necklace 4):] \ the InternalRel of (Necklace 4) by A14, XBOOLE_0:def_5;
then A16: a in the InternalRel of (Necklace 4) ` by SUBSET_1:def_4;
not a in id the carrier of (Necklace 4) by A13, RELAT_1:def_10;
then a in ( the InternalRel of (Necklace 4) `) \ (id the carrier of (Necklace 4)) by A16, XBOOLE_0:def_5;
hence a in the InternalRel of (ComplRelStr (Necklace 4)) by NECKLACE:def_8; ::_thesis: verum
end;
supposeA17: a = [0,3] ; ::_thesis: a in the InternalRel of (ComplRelStr (Necklace 4))
A18: not a in the InternalRel of (Necklace 4)
proof
assume A19: a in the InternalRel of (Necklace 4) ; ::_thesis: contradiction
percases ( a = [0,1] or a = [1,0] or a = [1,2] or a = [2,1] or a = [2,3] or a = [3,2] ) by A19, ENUMSET1:def_4, NECKLA_2:2;
suppose a = [0,1] ; ::_thesis: contradiction
hence contradiction by A17, XTUPLE_0:1; ::_thesis: verum
end;
suppose a = [1,0] ; ::_thesis: contradiction
hence contradiction by A17, XTUPLE_0:1; ::_thesis: verum
end;
suppose a = [1,2] ; ::_thesis: contradiction
hence contradiction by A17, XTUPLE_0:1; ::_thesis: verum
end;
suppose a = [2,1] ; ::_thesis: contradiction
hence contradiction by A17, XTUPLE_0:1; ::_thesis: verum
end;
suppose a = [2,3] ; ::_thesis: contradiction
hence contradiction by A17, XTUPLE_0:1; ::_thesis: verum
end;
suppose a = [3,2] ; ::_thesis: contradiction
hence contradiction by A17, XTUPLE_0:1; ::_thesis: verum
end;
end;
end;
( 0 in the carrier of (Necklace 4) & 3 in the carrier of (Necklace 4) ) by A1, ENUMSET1:def_2;
then a in [: the carrier of (Necklace 4), the carrier of (Necklace 4):] by A17, ZFMISC_1:87;
then a in [: the carrier of (Necklace 4), the carrier of (Necklace 4):] \ the InternalRel of (Necklace 4) by A18, XBOOLE_0:def_5;
then A20: a in the InternalRel of (Necklace 4) ` by SUBSET_1:def_4;
not a in id the carrier of (Necklace 4) by A17, RELAT_1:def_10;
then a in ( the InternalRel of (Necklace 4) `) \ (id the carrier of (Necklace 4)) by A20, XBOOLE_0:def_5;
hence a in the InternalRel of (ComplRelStr (Necklace 4)) by NECKLACE:def_8; ::_thesis: verum
end;
supposeA21: a = [3,0] ; ::_thesis: a in the InternalRel of (ComplRelStr (Necklace 4))
A22: not a in the InternalRel of (Necklace 4)
proof
assume A23: a in the InternalRel of (Necklace 4) ; ::_thesis: contradiction
percases ( a = [0,1] or a = [1,0] or a = [1,2] or a = [2,1] or a = [2,3] or a = [3,2] ) by A23, ENUMSET1:def_4, NECKLA_2:2;
suppose a = [0,1] ; ::_thesis: contradiction
hence contradiction by A21, XTUPLE_0:1; ::_thesis: verum
end;
suppose a = [1,0] ; ::_thesis: contradiction
hence contradiction by A21, XTUPLE_0:1; ::_thesis: verum
end;
suppose a = [1,2] ; ::_thesis: contradiction
hence contradiction by A21, XTUPLE_0:1; ::_thesis: verum
end;
suppose a = [2,1] ; ::_thesis: contradiction
hence contradiction by A21, XTUPLE_0:1; ::_thesis: verum
end;
suppose a = [2,3] ; ::_thesis: contradiction
hence contradiction by A21, XTUPLE_0:1; ::_thesis: verum
end;
suppose a = [3,2] ; ::_thesis: contradiction
hence contradiction by A21, XTUPLE_0:1; ::_thesis: verum
end;
end;
end;
( 0 in the carrier of (Necklace 4) & 3 in the carrier of (Necklace 4) ) by A1, ENUMSET1:def_2;
then a in [: the carrier of (Necklace 4), the carrier of (Necklace 4):] by A21, ZFMISC_1:87;
then a in [: the carrier of (Necklace 4), the carrier of (Necklace 4):] \ the InternalRel of (Necklace 4) by A22, XBOOLE_0:def_5;
then A24: a in the InternalRel of (Necklace 4) ` by SUBSET_1:def_4;
not a in id the carrier of (Necklace 4) by A21, RELAT_1:def_10;
then a in ( the InternalRel of (Necklace 4) `) \ (id the carrier of (Necklace 4)) by A24, XBOOLE_0:def_5;
hence a in the InternalRel of (ComplRelStr (Necklace 4)) by NECKLACE:def_8; ::_thesis: verum
end;
supposeA25: a = [1,3] ; ::_thesis: a in the InternalRel of (ComplRelStr (Necklace 4))
A26: not a in the InternalRel of (Necklace 4)
proof
assume A27: a in the InternalRel of (Necklace 4) ; ::_thesis: contradiction
percases ( a = [0,1] or a = [1,0] or a = [1,2] or a = [2,1] or a = [2,3] or a = [3,2] ) by A27, ENUMSET1:def_4, NECKLA_2:2;
suppose a = [0,1] ; ::_thesis: contradiction
hence contradiction by A25, XTUPLE_0:1; ::_thesis: verum
end;
suppose a = [1,0] ; ::_thesis: contradiction
hence contradiction by A25, XTUPLE_0:1; ::_thesis: verum
end;
suppose a = [1,2] ; ::_thesis: contradiction
hence contradiction by A25, XTUPLE_0:1; ::_thesis: verum
end;
suppose a = [2,1] ; ::_thesis: contradiction
hence contradiction by A25, XTUPLE_0:1; ::_thesis: verum
end;
suppose a = [2,3] ; ::_thesis: contradiction
hence contradiction by A25, XTUPLE_0:1; ::_thesis: verum
end;
suppose a = [3,2] ; ::_thesis: contradiction
hence contradiction by A25, XTUPLE_0:1; ::_thesis: verum
end;
end;
end;
( 1 in the carrier of (Necklace 4) & 3 in the carrier of (Necklace 4) ) by A1, ENUMSET1:def_2;
then a in [: the carrier of (Necklace 4), the carrier of (Necklace 4):] by A25, ZFMISC_1:87;
then a in [: the carrier of (Necklace 4), the carrier of (Necklace 4):] \ the InternalRel of (Necklace 4) by A26, XBOOLE_0:def_5;
then A28: a in the InternalRel of (Necklace 4) ` by SUBSET_1:def_4;
not a in id the carrier of (Necklace 4) by A25, RELAT_1:def_10;
then a in ( the InternalRel of (Necklace 4) `) \ (id the carrier of (Necklace 4)) by A28, XBOOLE_0:def_5;
hence a in the InternalRel of (ComplRelStr (Necklace 4)) by NECKLACE:def_8; ::_thesis: verum
end;
supposeA29: a = [3,1] ; ::_thesis: a in the InternalRel of (ComplRelStr (Necklace 4))
A30: not a in the InternalRel of (Necklace 4)
proof
assume A31: a in the InternalRel of (Necklace 4) ; ::_thesis: contradiction
percases ( a = [0,1] or a = [1,0] or a = [1,2] or a = [2,1] or a = [2,3] or a = [3,2] ) by A31, ENUMSET1:def_4, NECKLA_2:2;
suppose a = [0,1] ; ::_thesis: contradiction
hence contradiction by A29, XTUPLE_0:1; ::_thesis: verum
end;
suppose a = [1,0] ; ::_thesis: contradiction
hence contradiction by A29, XTUPLE_0:1; ::_thesis: verum
end;
suppose a = [1,2] ; ::_thesis: contradiction
hence contradiction by A29, XTUPLE_0:1; ::_thesis: verum
end;
suppose a = [2,1] ; ::_thesis: contradiction
hence contradiction by A29, XTUPLE_0:1; ::_thesis: verum
end;
suppose a = [2,3] ; ::_thesis: contradiction
hence contradiction by A29, XTUPLE_0:1; ::_thesis: verum
end;
suppose a = [3,2] ; ::_thesis: contradiction
hence contradiction by A29, XTUPLE_0:1; ::_thesis: verum
end;
end;
end;
( 1 in the carrier of (Necklace 4) & 3 in the carrier of (Necklace 4) ) by A1, ENUMSET1:def_2;
then a in [: the carrier of (Necklace 4), the carrier of (Necklace 4):] by A29, ZFMISC_1:87;
then a in [: the carrier of (Necklace 4), the carrier of (Necklace 4):] \ the InternalRel of (Necklace 4) by A30, XBOOLE_0:def_5;
then A32: a in the InternalRel of (Necklace 4) ` by SUBSET_1:def_4;
not a in id the carrier of (Necklace 4) by A29, RELAT_1:def_10;
then a in ( the InternalRel of (Necklace 4) `) \ (id the carrier of (Necklace 4)) by A32, XBOOLE_0:def_5;
hence a in the InternalRel of (ComplRelStr (Necklace 4)) by NECKLACE:def_8; ::_thesis: verum
end;
end;
end;
registration
let R be RelStr ;
cluster ComplRelStr R -> irreflexive ;
correctness
coherence
ComplRelStr R is irreflexive ;
proof
set R1 = ComplRelStr R;
for x being set st x in the carrier of (ComplRelStr R) holds
not [x,x] in the InternalRel of (ComplRelStr R)
proof
let x be set ; ::_thesis: ( x in the carrier of (ComplRelStr R) implies not [x,x] in the InternalRel of (ComplRelStr R) )
assume x in the carrier of (ComplRelStr R) ; ::_thesis: not [x,x] in the InternalRel of (ComplRelStr R)
then A1: x in the carrier of R by NECKLACE:def_8;
not [x,x] in the InternalRel of (ComplRelStr R)
proof
assume [x,x] in the InternalRel of (ComplRelStr R) ; ::_thesis: contradiction
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; ::_thesis: verum
end;
hence not [x,x] in the InternalRel of (ComplRelStr R) ; ::_thesis: verum
end;
hence ComplRelStr R is irreflexive by NECKLACE:def_5; ::_thesis: verum
end;
end;
registration
let R be symmetric RelStr ;
cluster ComplRelStr R -> symmetric ;
correctness
coherence
ComplRelStr R is symmetric ;
proof
let x, y be set ; :: according to NECKLACE:def_3,RELAT_2:def_3 ::_thesis: ( not x in the carrier of (ComplRelStr R) or not y in the carrier of (ComplRelStr R) or not [x,y] in the InternalRel of (ComplRelStr R) or [y,x] in the InternalRel of (ComplRelStr R) )
set S = ComplRelStr R;
assume that
A1: ( x in the carrier of (ComplRelStr R) & y in the carrier of (ComplRelStr R) ) and
A2: [x,y] in the InternalRel of (ComplRelStr R) ; ::_thesis: [y,x] in the InternalRel of (ComplRelStr R)
percases ( x = y or x <> y ) ;
suppose x = y ; ::_thesis: [y,x] in the InternalRel of (ComplRelStr R)
hence [y,x] in the InternalRel of (ComplRelStr R) by A2; ::_thesis: verum
end;
supposeA3: x <> y ; ::_thesis: [y,x] in the InternalRel of (ComplRelStr R)
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, RELAT_2:def_3;
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 [y,x] in the InternalRel of (ComplRelStr R) by NECKLACE:def_8; ::_thesis: verum
end;
end;
end;
end;
theorem Th12: :: NECKLA_3:12
for R being RelStr holds the InternalRel of R misses the InternalRel of (ComplRelStr R)
proof
let R be RelStr ; ::_thesis: the InternalRel of R misses the InternalRel of (ComplRelStr R)
assume not the InternalRel of R misses the InternalRel of (ComplRelStr R) ; ::_thesis: contradiction
then the InternalRel of R /\ the InternalRel of (ComplRelStr R) <> {} by XBOOLE_0:def_7;
then consider a being set 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 contradiction by A1, XBOOLE_0:def_4; ::_thesis: verum
end;
theorem Th13: :: NECKLA_3:13
for R being RelStr holds id the carrier of R misses the InternalRel of (ComplRelStr R)
proof
let R be RelStr ; ::_thesis: id the carrier of R misses the InternalRel of (ComplRelStr R)
assume not id the carrier of R misses the InternalRel of (ComplRelStr R) ; ::_thesis: contradiction
then (id the carrier of R) /\ the InternalRel of (ComplRelStr R) <> {} by XBOOLE_0:def_7;
then consider a being set 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; ::_thesis: verum
end;
theorem Th14: :: NECKLA_3:14
for G being 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 ; ::_thesis: [: the carrier of G, the carrier of G:] = ((id the carrier of G) \/ the InternalRel of G) \/ the InternalRel of (ComplRelStr G)
set idcG = id the carrier of G;
set IG = the InternalRel of G;
set ICmpG = the InternalRel of (ComplRelStr G);
set cG = the carrier of G;
thus [: the carrier of G, the carrier of G:] c= ((id the carrier of G) \/ the InternalRel of G) \/ the InternalRel of (ComplRelStr G) :: according to XBOOLE_0:def_10 ::_thesis: ((id the carrier of G) \/ the InternalRel of G) \/ the InternalRel of (ComplRelStr G) c= [: the carrier of G, the carrier of G:]
proof
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in [: the carrier of G, the carrier of G:] or a in ((id the carrier of G) \/ the InternalRel of G) \/ the InternalRel of (ComplRelStr G) )
assume A1: a in [: the carrier of G, the carrier of G:] ; ::_thesis: a in ((id the carrier of G) \/ the InternalRel of G) \/ the InternalRel of (ComplRelStr G)
then consider x, y being set such that
A2: x in the carrier of G and
y in the carrier of G and
A3: a = [x,y] by ZFMISC_1:def_2;
percases ( x = y or x <> y ) ;
supposeA4: x = y ; ::_thesis: a in ((id the carrier of G) \/ the InternalRel of G) \/ the InternalRel of (ComplRelStr G)
[x,x] in id the carrier of G by A2, RELAT_1:def_10;
then a in (id the carrier of G) \/ the InternalRel of G by A3, A4, XBOOLE_0:def_3;
hence a in ((id the carrier of G) \/ the InternalRel of G) \/ the InternalRel of (ComplRelStr G) by XBOOLE_0:def_3; ::_thesis: verum
end;
suppose x <> y ; ::_thesis: a in ((id the carrier of G) \/ the InternalRel of G) \/ the InternalRel of (ComplRelStr G)
then A5: not a in id the carrier of G by A3, RELAT_1:def_10;
thus a in ((id the carrier of G) \/ the InternalRel of G) \/ the InternalRel of (ComplRelStr G) ::_thesis: verum
proof
percases ( a in the InternalRel of G or not a in the InternalRel of G ) ;
suppose a in the InternalRel of G ; ::_thesis: a in ((id the carrier of G) \/ the InternalRel of G) \/ the InternalRel of (ComplRelStr G)
then a in (id the carrier of G) \/ the InternalRel of G by XBOOLE_0:def_3;
hence a in ((id the carrier of G) \/ the InternalRel of G) \/ the InternalRel of (ComplRelStr G) by XBOOLE_0:def_3; ::_thesis: verum
end;
suppose not a in the InternalRel of G ; ::_thesis: a in ((id the carrier of G) \/ the InternalRel of G) \/ the InternalRel of (ComplRelStr G)
then a in [: the carrier of G, the carrier of G:] \ the InternalRel of G by A1, XBOOLE_0:def_5;
then a in the InternalRel of G ` by SUBSET_1:def_4;
then a in ( the InternalRel of G `) \ (id the carrier of G) by A5, XBOOLE_0:def_5;
then a in the InternalRel of (ComplRelStr G) by NECKLACE:def_8;
then a in the InternalRel of G \/ the InternalRel of (ComplRelStr G) by XBOOLE_0:def_3;
then a in (id the carrier of G) \/ ( the InternalRel of G \/ the InternalRel of (ComplRelStr G)) by XBOOLE_0:def_3;
hence a in ((id the carrier of G) \/ the InternalRel of G) \/ the InternalRel of (ComplRelStr G) by XBOOLE_1:4; ::_thesis: verum
end;
end;
end;
end;
end;
end;
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in ((id the carrier of G) \/ the InternalRel of G) \/ the InternalRel of (ComplRelStr G) or a in [: the carrier of G, the carrier of G:] )
assume a in ((id the carrier of G) \/ the InternalRel of G) \/ the InternalRel of (ComplRelStr G) ; ::_thesis: a in [: the carrier of G, the carrier of G:]
then A6: ( a in (id the carrier of G) \/ the InternalRel of G or a in the InternalRel of (ComplRelStr G) ) by XBOOLE_0:def_3;
percases ( a in id the carrier of G or a in the InternalRel of G or a in the InternalRel of (ComplRelStr G) ) by A6, XBOOLE_0:def_3;
suppose a in id the carrier of G ; ::_thesis: a in [: the carrier of G, the carrier of G:]
hence a in [: the carrier of G, the carrier of G:] ; ::_thesis: verum
end;
suppose a in the InternalRel of G ; ::_thesis: a in [: the carrier of G, the carrier of G:]
hence a in [: the carrier of G, the carrier of G:] ; ::_thesis: verum
end;
suppose a in the InternalRel of (ComplRelStr G) ; ::_thesis: a in [: the carrier of G, the carrier of G:]
then a in ( the InternalRel of G `) \ (id the carrier of G) by NECKLACE:def_8;
hence a in [: the carrier of G, the carrier of G:] ; ::_thesis: verum
end;
end;
end;
theorem Th15: :: NECKLA_3:15
for G being strict irreflexive RelStr st G is trivial holds
ComplRelStr G = G
proof
let G be strict irreflexive RelStr ; ::_thesis: ( G is trivial implies ComplRelStr G = G )
set CG = ComplRelStr G;
assume A1: G is trivial ; ::_thesis: ComplRelStr G = G
percases ( the carrier of G is empty or ex x being set st the carrier of G = {x} ) by A1, ZFMISC_1:131;
supposeA2: the carrier of G is empty ; ::_thesis: ComplRelStr G = G
the InternalRel of (ComplRelStr G) = ( the InternalRel of G `) \ (id the carrier of G) by NECKLACE:def_8;
then A3: the InternalRel of (ComplRelStr G) = ({} \ {}) \ (id {}) by A2;
the InternalRel of G = {} by A2;
hence ComplRelStr G = G by A3, NECKLACE:def_8; ::_thesis: verum
end;
suppose ex x being set st the carrier of G = {x} ; ::_thesis: ComplRelStr G = G
then consider x being set such that
A4: the carrier of G = {x} ;
A5: the carrier of (ComplRelStr G) = {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 the InternalRel of G <> {[x,x]} ; ::_thesis: contradiction
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; ::_thesis: verum
end;
the InternalRel of (ComplRelStr G) = ( the InternalRel of G `) \ (id the carrier of G) by NECKLACE:def_8;
then the InternalRel of (ComplRelStr G) = ([:{x},{x}:] \ {}) \ (id {x}) by A4, A6, A7, SUBSET_1:def_4;
then the InternalRel of (ComplRelStr G) = {[x,x]} \ (id {x}) by ZFMISC_1:29;
then the InternalRel of (ComplRelStr G) = {[x,x]} \ {[x,x]} by SYSREL:13;
hence ComplRelStr G = G by A4, A6, A7, A5, XBOOLE_1:37; ::_thesis: verum
end;
end;
end;
theorem Th16: :: NECKLA_3:16
for G being strict irreflexive RelStr holds ComplRelStr (ComplRelStr G) = G
proof
let G be strict irreflexive RelStr ; ::_thesis: ComplRelStr (ComplRelStr G) = G
set CCmpG = ComplRelStr (ComplRelStr G);
set CmpG = ComplRelStr G;
set cG = the carrier of G;
set IG = the InternalRel of G;
set ICmpG = the InternalRel of (ComplRelStr G);
set ICCmpG = the InternalRel of (ComplRelStr (ComplRelStr G));
A1: the carrier of G = the carrier of (ComplRelStr G) by NECKLACE:def_8
.= the carrier of (ComplRelStr (ComplRelStr G)) by NECKLACE:def_8 ;
A2: the carrier of G = the carrier of (ComplRelStr G) by NECKLACE:def_8;
A3: id the carrier of G misses the InternalRel of G
proof
assume not id the carrier of G misses the InternalRel of G ; ::_thesis: contradiction
then (id the carrier of G) /\ the InternalRel of G <> {} by XBOOLE_0:def_7;
then consider a being set such that
A4: a in (id the carrier of G) /\ the InternalRel of G by XBOOLE_0:def_1;
A5: a in the InternalRel of G by A4, XBOOLE_0:def_4;
consider x, y being set such that
A6: a = [x,y] and
A7: x in the carrier of G and
y in the carrier of G by A4, RELSET_1:2;
a in id the carrier of G by A4, XBOOLE_0:def_4;
then x = y by A6, RELAT_1:def_10;
hence contradiction by A5, A6, A7, NECKLACE:def_5; ::_thesis: verum
end;
the InternalRel of (ComplRelStr (ComplRelStr G)) = ( the InternalRel of (ComplRelStr G) `) \ (id the carrier of (ComplRelStr G)) by NECKLACE:def_8
.= ([: the carrier of (ComplRelStr G), the carrier of (ComplRelStr G):] \ the InternalRel of (ComplRelStr G)) \ (id the carrier of (ComplRelStr G)) by SUBSET_1:def_4
.= ([: the carrier of G, the carrier of G:] \ (( the InternalRel of G `) \ (id the carrier of G))) \ (id the carrier of G) by A2, NECKLACE:def_8
.= (([: the carrier of G, the carrier of G:] \ ( the InternalRel of G `)) \/ ([: the carrier of G, the carrier of G:] /\ (id the carrier of G))) \ (id the carrier of G) by XBOOLE_1:52
.= (([: the carrier of G, the carrier of G:] \ ( the InternalRel of G `)) \/ (id the carrier of G)) \ (id the carrier of G) by XBOOLE_1:28
.= ([: the carrier of G, the carrier of G:] \ ( the InternalRel of G `)) \ (id the carrier of G) by XBOOLE_1:40
.= ([: the carrier of G, the carrier of G:] \ ([: the carrier of G, the carrier of G:] \ the InternalRel of G)) \ (id the carrier of G) by SUBSET_1:def_4
.= ([: the carrier of G, the carrier of G:] /\ the InternalRel of G) \ (id the carrier of G) by XBOOLE_1:48
.= the InternalRel of G \ (id the carrier of G) by XBOOLE_1:28
.= the InternalRel of G by A3, XBOOLE_1:83 ;
hence ComplRelStr (ComplRelStr G) = G by A1; ::_thesis: verum
end;
theorem Th17: :: NECKLA_3:17
for G1, G2 being 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 ; ::_thesis: ( the carrier of G1 misses the carrier of G2 implies ComplRelStr (union_of (G1,G2)) = sum_of ((ComplRelStr G1),(ComplRelStr G2)) )
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)));
set R = the InternalRel of (sum_of ((ComplRelStr G1),(ComplRelStr G2)));
set X1 = the InternalRel of (ComplRelStr G1);
set X2 = the InternalRel of (ComplRelStr G2);
set X3 = [: the carrier of (ComplRelStr G1), the carrier of (ComplRelStr G2):];
set X4 = [: the carrier of (ComplRelStr G2), the carrier of (ComplRelStr G1):];
set X5 = [: the carrier of G1, the carrier of G1:];
set X6 = [: the carrier of G2, the carrier of G2:];
set X7 = [: the carrier of G1, the carrier of G2:];
set X8 = [: the carrier of G2, the carrier of G1:];
assume A2: the carrier of G1 misses the carrier of G2 ; ::_thesis: ComplRelStr (union_of (G1,G2)) = sum_of ((ComplRelStr G1),(ComplRelStr G2))
A3: for a, b being set holds
( [a,b] in the InternalRel of (ComplRelStr (union_of (G1,G2))) iff [a,b] in the InternalRel of (sum_of ((ComplRelStr G1),(ComplRelStr G2))) )
proof
let a, b be set ; ::_thesis: ( [a,b] in the InternalRel of (ComplRelStr (union_of (G1,G2))) iff [a,b] in the InternalRel of (sum_of ((ComplRelStr G1),(ComplRelStr G2))) )
set x = [a,b];
thus ( [a,b] in the InternalRel of (ComplRelStr (union_of (G1,G2))) implies [a,b] in the InternalRel of (sum_of ((ComplRelStr G1),(ComplRelStr G2))) ) ::_thesis: ( [a,b] in the InternalRel of (sum_of ((ComplRelStr G1),(ComplRelStr G2))) implies [a,b] in the InternalRel of (ComplRelStr (union_of (G1,G2))) )
proof
assume [a,b] in the InternalRel of (ComplRelStr (union_of (G1,G2))) ; ::_thesis: [a,b] in the InternalRel of (sum_of ((ComplRelStr G1),(ComplRelStr G2)))
then A4: [a,b] in ( the InternalRel of (union_of (G1,G2)) `) \ (id the carrier of (union_of (G1,G2))) by NECKLACE:def_8;
then [a,b] in [: the carrier of (union_of (G1,G2)), the carrier of (union_of (G1,G2)):] ;
then [a,b] in [:( the carrier of G1 \/ the carrier of G2), the carrier of (union_of (G1,G2)):] by NECKLA_2:def_2;
then A5: [a,b] in [:( the carrier of G1 \/ the carrier of G2),( the carrier of G1 \/ the carrier of G2):] by NECKLA_2:def_2;
not [a,b] in id the carrier of (union_of (G1,G2)) by A4, XBOOLE_0:def_5;
then A6: not [a,b] in id ( the carrier of G1 \/ the carrier of G2) by NECKLA_2:def_2;
A7: ( not [a,b] in id the carrier of G1 & not [a,b] in id the carrier of G2 )
proof
assume ( [a,b] in id the carrier of G1 or [a,b] in id the carrier of G2 ) ; ::_thesis: contradiction
then [a,b] in (id the carrier of G1) \/ (id the carrier of G2) by XBOOLE_0:def_3;
hence contradiction by A6, SYSREL:14; ::_thesis: verum
end;
( the carrier of G1 = the carrier of (ComplRelStr G1) & the carrier of G2 = the carrier of (ComplRelStr G2) ) by NECKLACE:def_8;
then [a,b] in (([: the carrier of G1, the carrier of G1:] \/ [: the carrier of (ComplRelStr G1), the carrier of (ComplRelStr G2):]) \/ [: the carrier of (ComplRelStr G2), the carrier of (ComplRelStr G1):]) \/ [: the carrier of G2, the carrier of G2:] by A5, ZFMISC_1:98;
then [a,b] in [: the carrier of G1, the carrier of G1:] \/ (([: the carrier of (ComplRelStr G1), the carrier of (ComplRelStr G2):] \/ [: the carrier of (ComplRelStr G2), the carrier of (ComplRelStr G1):]) \/ [: the carrier of G2, the carrier of G2:]) by XBOOLE_1:113;
then ( [a,b] in [: the carrier of G1, the carrier of G1:] or [a,b] in ([: the carrier of (ComplRelStr G1), the carrier of (ComplRelStr G2):] \/ [: the carrier of (ComplRelStr G2), the carrier of (ComplRelStr G1):]) \/ [: the carrier of G2, the carrier of G2:] ) by XBOOLE_0:def_3;
then ( [a,b] in [: the carrier of G1, the carrier of G1:] or [a,b] in [: the carrier of (ComplRelStr G1), the carrier of (ComplRelStr G2):] \/ ([: the carrier of (ComplRelStr G2), the carrier of (ComplRelStr G1):] \/ [: the carrier of G2, the carrier of G2:]) ) by XBOOLE_1:4;
then A8: ( [a,b] in [: the carrier of G1, the carrier of G1:] or [a,b] in [: the carrier of (ComplRelStr G1), the carrier of (ComplRelStr G2):] or [a,b] in [: the carrier of (ComplRelStr G2), the carrier of (ComplRelStr G1):] \/ [: the carrier of G2, the carrier of G2:] ) by XBOOLE_0:def_3;
[a,b] in the InternalRel of (union_of (G1,G2)) ` by A4, XBOOLE_0:def_5;
then [a,b] 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 [a,b] in the InternalRel of (union_of (G1,G2)) by XBOOLE_0:def_5;
then A9: not [a,b] in the InternalRel of G1 \/ the InternalRel of G2 by NECKLA_2:def_2;
then A10: not [a,b] in the InternalRel of G1 by XBOOLE_0:def_3;
A11: not [a,b] in the InternalRel of G2 by A9, XBOOLE_0:def_3;
percases ( [a,b] in [: the carrier of G1, the carrier of G1:] or [a,b] in [: the carrier of (ComplRelStr G1), the carrier of (ComplRelStr G2):] or [a,b] in [: the carrier of (ComplRelStr G2), the carrier of (ComplRelStr G1):] or [a,b] in [: the carrier of G2, the carrier of G2:] ) by A8, XBOOLE_0:def_3;
suppose [a,b] in [: the carrier of G1, the carrier of G1:] ; ::_thesis: [a,b] in the InternalRel of (sum_of ((ComplRelStr G1),(ComplRelStr G2)))
then [a,b] in [: the carrier of G1, the carrier of G1:] \ the InternalRel of G1 by A10, XBOOLE_0:def_5;
then [a,b] in the InternalRel of G1 ` by SUBSET_1:def_4;
then [a,b] in ( the InternalRel of G1 `) \ (id the carrier of G1) by A7, XBOOLE_0:def_5;
then [a,b] in the InternalRel of (ComplRelStr G1) by NECKLACE:def_8;
then [a,b] 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;
then [a,b] 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_1:113;
hence [a,b] in the InternalRel of (sum_of ((ComplRelStr G1),(ComplRelStr G2))) by NECKLA_2:def_3; ::_thesis: verum
end;
suppose [a,b] in [: the carrier of (ComplRelStr G1), the carrier of (ComplRelStr G2):] ; ::_thesis: [a,b] in the InternalRel of (sum_of ((ComplRelStr G1),(ComplRelStr G2)))
then [a,b] in the InternalRel of (ComplRelStr G2) \/ [: the carrier of (ComplRelStr G1), the carrier of (ComplRelStr G2):] by XBOOLE_0:def_3;
then [a,b] in ( 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;
then [a,b] 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;
then [a,b] 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_1:113;
hence [a,b] in the InternalRel of (sum_of ((ComplRelStr G1),(ComplRelStr G2))) by NECKLA_2:def_3; ::_thesis: verum
end;
suppose [a,b] in [: the carrier of (ComplRelStr G2), the carrier of (ComplRelStr G1):] ; ::_thesis: [a,b] in the InternalRel of (sum_of ((ComplRelStr G1),(ComplRelStr G2)))
then [a,b] in [: 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;
then [a,b] in 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;
then [a,b] in ( 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_1:4;
then [a,b] 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;
then [a,b] 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_1:113;
hence [a,b] in the InternalRel of (sum_of ((ComplRelStr G1),(ComplRelStr G2))) by NECKLA_2:def_3; ::_thesis: verum
end;
suppose [a,b] in [: the carrier of G2, the carrier of G2:] ; ::_thesis: [a,b] in the InternalRel of (sum_of ((ComplRelStr G1),(ComplRelStr G2)))
then [a,b] in [: the carrier of G2, the carrier of G2:] \ the InternalRel of G2 by A11, XBOOLE_0:def_5;
then [a,b] in the InternalRel of G2 ` by SUBSET_1:def_4;
then [a,b] in ( the InternalRel of G2 `) \ (id the carrier of G2) by A7, XBOOLE_0:def_5;
then [a,b] in the InternalRel of (ComplRelStr G2) by NECKLACE:def_8;
then [a,b] in the InternalRel of (ComplRelStr G1) \/ the InternalRel of (ComplRelStr G2) by XBOOLE_0:def_3;
then [a,b] 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 [a,b] 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 [a,b] in the InternalRel of (sum_of ((ComplRelStr G1),(ComplRelStr G2))) by NECKLA_2:def_3; ::_thesis: verum
end;
end;
end;
assume [a,b] in the InternalRel of (sum_of ((ComplRelStr G1),(ComplRelStr G2))) ; ::_thesis: [a,b] in the InternalRel of (ComplRelStr (union_of (G1,G2)))
then [a,b] 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 NECKLA_2:def_3;
then ( [a,b] in ( the InternalRel of (ComplRelStr G1) \/ the InternalRel of (ComplRelStr G2)) \/ [: the carrier of (ComplRelStr G1), the carrier of (ComplRelStr G2):] or [a,b] in [: the carrier of (ComplRelStr G2), the carrier of (ComplRelStr G1):] ) by XBOOLE_0:def_3;
then A12: ( [a,b] in the InternalRel of (ComplRelStr G1) \/ the InternalRel of (ComplRelStr G2) or [a,b] in [: the carrier of (ComplRelStr G1), the carrier of (ComplRelStr G2):] or [a,b] in [: the carrier of (ComplRelStr G2), the carrier of (ComplRelStr G1):] ) by XBOOLE_0:def_3;
percases ( [a,b] in the InternalRel of (ComplRelStr G1) or [a,b] in the InternalRel of (ComplRelStr G2) or [a,b] in [: the carrier of (ComplRelStr G1), the carrier of (ComplRelStr G2):] or [a,b] in [: the carrier of (ComplRelStr G2), the carrier of (ComplRelStr G1):] ) by A12, XBOOLE_0:def_3;
suppose [a,b] in the InternalRel of (ComplRelStr G1) ; ::_thesis: [a,b] in the InternalRel of (ComplRelStr (union_of (G1,G2)))
then A13: [a,b] in ( the InternalRel of G1 `) \ (id the carrier of G1) by NECKLACE:def_8;
then [a,b] in the InternalRel of G1 ` by XBOOLE_0:def_5;
then [a,b] in [: the carrier of G1, the carrier of G1:] \ the InternalRel of G1 by SUBSET_1:def_4;
then A14: not [a,b] in the InternalRel of G1 by XBOOLE_0:def_5;
A15: not [a,b] in the InternalRel of (union_of (G1,G2))
proof
assume [a,b] in the InternalRel of (union_of (G1,G2)) ; ::_thesis: contradiction
then [a,b] in the InternalRel of G1 \/ the InternalRel of G2 by NECKLA_2:def_2;
then [a,b] in the InternalRel of G2 by A14, XBOOLE_0:def_3;
then not [: the carrier of G1, the carrier of G1:] /\ [: the carrier of G2, the carrier of G2:] is 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:] by XBOOLE_0:def_7;
hence contradiction by A2, ZFMISC_1:104; ::_thesis: verum
end;
A16: not [a,b] in id the carrier of (union_of (G1,G2))
proof
assume [a,b] in id the carrier of (union_of (G1,G2)) ; ::_thesis: contradiction
then [a,b] in id ( the carrier of G1 \/ the carrier of G2) by NECKLA_2:def_2;
then A17: [a,b] in (id the carrier of G1) \/ (id the carrier of G2) by SYSREL:14;
thus contradiction ::_thesis: verum
proof
percases ( [a,b] in id the carrier of G1 or [a,b] in id the carrier of G2 ) by A17, XBOOLE_0:def_3;
suppose [a,b] in id the carrier of G1 ; ::_thesis: contradiction
hence contradiction by A13, XBOOLE_0:def_5; ::_thesis: verum
end;
suppose [a,b] in id the carrier of G2 ; ::_thesis: contradiction
then not [: the carrier of G1, the carrier of G1:] /\ [: the carrier of G2, the carrier of G2:] is 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:] by XBOOLE_0:def_7;
hence contradiction by A2, ZFMISC_1:104; ::_thesis: verum
end;
end;
end;
end;
[a,b] in [: the carrier of G1, the carrier of G1:] \/ [: the carrier of G1, the carrier of G2:] by A13, XBOOLE_0:def_3;
then [a,b] in ([: the carrier of G1, the carrier of G1:] \/ [: the carrier of G1, the carrier of G2:]) \/ [: the carrier of G2, the carrier of G1:] by XBOOLE_0:def_3;
then [a,b] in (([: the carrier of G1, the carrier of G1:] \/ [: the carrier of G1, the carrier of G2:]) \/ [: the carrier of G2, the carrier of G1:]) \/ [: the carrier of G2, the carrier of G2:] by XBOOLE_0:def_3;
then [a,b] in [:( the carrier of G1 \/ the carrier of G2),( the carrier of G1 \/ the carrier of G2):] by ZFMISC_1:98;
then [a,b] in [:( the carrier of G1 \/ the carrier of G2), the carrier of (union_of (G1,G2)):] by NECKLA_2:def_2;
then [a,b] in [: the carrier of (union_of (G1,G2)), the carrier of (union_of (G1,G2)):] by NECKLA_2:def_2;
then [a,b] 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 [a,b] in the InternalRel of (union_of (G1,G2)) ` by SUBSET_1:def_4;
then [a,b] in ( the InternalRel of (union_of (G1,G2)) `) \ (id the carrier of (union_of (G1,G2))) by A16, XBOOLE_0:def_5;
hence [a,b] in the InternalRel of (ComplRelStr (union_of (G1,G2))) by NECKLACE:def_8; ::_thesis: verum
end;
suppose [a,b] in the InternalRel of (ComplRelStr G2) ; ::_thesis: [a,b] in the InternalRel of (ComplRelStr (union_of (G1,G2)))
then A18: [a,b] in ( the InternalRel of G2 `) \ (id the carrier of G2) by NECKLACE:def_8;
then [a,b] in the InternalRel of G2 ` by XBOOLE_0:def_5;
then [a,b] in [: the carrier of G2, the carrier of G2:] \ the InternalRel of G2 by SUBSET_1:def_4;
then A19: not [a,b] in the InternalRel of G2 by XBOOLE_0:def_5;
A20: not [a,b] in the InternalRel of (union_of (G1,G2))
proof
assume [a,b] in the InternalRel of (union_of (G1,G2)) ; ::_thesis: contradiction
then [a,b] in the InternalRel of G1 \/ the InternalRel of G2 by NECKLA_2:def_2;
then [a,b] in the InternalRel of G1 by A19, XBOOLE_0:def_3;
then not [: the carrier of G1, the carrier of G1:] /\ [: the carrier of G2, the carrier of G2:] is 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:] by XBOOLE_0:def_7;
hence contradiction by A2, ZFMISC_1:104; ::_thesis: verum
end;
A21: not [a,b] in id the carrier of (union_of (G1,G2))
proof
assume [a,b] in id the carrier of (union_of (G1,G2)) ; ::_thesis: contradiction
then [a,b] in id ( the carrier of G1 \/ the carrier of G2) by NECKLA_2:def_2;
then A22: [a,b] in (id the carrier of G1) \/ (id the carrier of G2) by SYSREL:14;
percases ( [a,b] in id the carrier of G2 or [a,b] in id the carrier of G1 ) by A22, XBOOLE_0:def_3;
suppose [a,b] in id the carrier of G2 ; ::_thesis: contradiction
hence contradiction by A18, XBOOLE_0:def_5; ::_thesis: verum
end;
suppose [a,b] in id the carrier of G1 ; ::_thesis: contradiction
then not [: the carrier of G1, the carrier of G1:] /\ [: the carrier of G2, the carrier of G2:] is 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:] by XBOOLE_0:def_7;
hence contradiction by A2, ZFMISC_1:104; ::_thesis: verum
end;
end;
end;
[a,b] in [: the carrier of G2, the carrier of G1:] \/ [: the carrier of G2, the carrier of G2:] by A18, XBOOLE_0:def_3;
then [a,b] in [: the carrier of G1, the carrier of G2:] \/ ([: the carrier of G2, the carrier of G1:] \/ [: the carrier of G2, the carrier of G2:]) by XBOOLE_0:def_3;
then [a,b] in ([: the carrier of G1, the carrier of G2:] \/ [: the carrier of G2, the carrier of G1:]) \/ [: the carrier of G2, the carrier of G2:] by XBOOLE_1:4;
then [a,b] in [: the carrier of G1, the carrier of G1:] \/ (([: the carrier of G1, the carrier of G2:] \/ [: the carrier of G2, the carrier of G1:]) \/ [: the carrier of G2, the carrier of G2:]) by XBOOLE_0:def_3;
then [a,b] in (([: the carrier of G1, the carrier of G1:] \/ [: the carrier of G1, the carrier of G2:]) \/ [: the carrier of G2, the carrier of G1:]) \/ [: the carrier of G2, the carrier of G2:] by XBOOLE_1:113;
then [a,b] in [:( the carrier of G1 \/ the carrier of G2),( the carrier of G1 \/ the carrier of G2):] by ZFMISC_1:98;
then [a,b] in [:( the carrier of G1 \/ the carrier of G2), the carrier of (union_of (G1,G2)):] by NECKLA_2:def_2;
then [a,b] in [: the carrier of (union_of (G1,G2)), the carrier of (union_of (G1,G2)):] by NECKLA_2:def_2;
then [a,b] 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 [a,b] in the InternalRel of (union_of (G1,G2)) ` by SUBSET_1:def_4;
then [a,b] in ( the InternalRel of (union_of (G1,G2)) `) \ (id the carrier of (union_of (G1,G2))) by A21, XBOOLE_0:def_5;
hence [a,b] in the InternalRel of (ComplRelStr (union_of (G1,G2))) by NECKLACE:def_8; ::_thesis: verum
end;
suppose [a,b] in [: the carrier of (ComplRelStr G1), the carrier of (ComplRelStr G2):] ; ::_thesis: [a,b] in the InternalRel of (ComplRelStr (union_of (G1,G2)))
then A23: [a,b] in [: the carrier of G1, the carrier of (ComplRelStr G2):] by NECKLACE:def_8;
then A24: [a,b] in [: the carrier of G1, the carrier of G2:] by NECKLACE:def_8;
A25: not [a,b] in the InternalRel of (union_of (G1,G2))
proof
assume [a,b] in the InternalRel of (union_of (G1,G2)) ; ::_thesis: contradiction
then A26: [a,b] in the InternalRel of G1 \/ the InternalRel of G2 by NECKLA_2:def_2;
percases ( [a,b] in the InternalRel of G1 or [a,b] in the InternalRel of G2 ) by A26, XBOOLE_0:def_3;
supposeA27: [a,b] in the InternalRel of G1 ; ::_thesis: contradiction
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, XBOOLE_0:def_7; ::_thesis: verum
end;
supposeA29: [a,b] in the InternalRel of G2 ; ::_thesis: contradiction
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, XBOOLE_0:def_7; ::_thesis: verum
end;
end;
end;
A31: not [a,b] in id the carrier of (union_of (G1,G2))
proof
assume [a,b] in id the carrier of (union_of (G1,G2)) ; ::_thesis: contradiction
then [a,b] in id ( the carrier of G1 \/ the carrier of G2) by NECKLA_2:def_2;
then A32: [a,b] in (id the carrier of G1) \/ (id the carrier of G2) by SYSREL:14;
percases ( [a,b] in id the carrier of G1 or [a,b] in id the carrier of G2 ) by A32, XBOOLE_0:def_3;
supposeA33: [a,b] in id the carrier of G1 ; ::_thesis: contradiction
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, XBOOLE_0:def_7; ::_thesis: verum
end;
supposeA35: [a,b] in id the carrier of G2 ; ::_thesis: contradiction
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, XBOOLE_0:def_7; ::_thesis: verum
end;
end;
end;
[a,b] in [: the carrier of G1, the carrier of G2:] \/ [: the carrier of G2, the carrier of G1:] by A24, XBOOLE_0:def_3;
then [a,b] in [: the carrier of G1, the carrier of G1:] \/ ([: the carrier of G1, the carrier of G2:] \/ [: the carrier of G2, the carrier of G1:]) by XBOOLE_0:def_3;
then [a,b] in ([: the carrier of G1, the carrier of G1:] \/ [: the carrier of G1, the carrier of G2:]) \/ [: the carrier of G2, the carrier of G1:] by XBOOLE_1:4;
then [a,b] in (([: the carrier of G1, the carrier of G1:] \/ [: the carrier of G1, the carrier of G2:]) \/ [: the carrier of G2, the carrier of G1:]) \/ [: the carrier of G2, the carrier of G2:] by XBOOLE_0:def_3;
then [a,b] in [:( the carrier of G1 \/ the carrier of G2),( the carrier of G1 \/ the carrier of G2):] by ZFMISC_1:98;
then [a,b] in [:( the carrier of G1 \/ the carrier of G2), the carrier of (union_of (G1,G2)):] by NECKLA_2:def_2;
then [a,b] in [: the carrier of (union_of (G1,G2)), the carrier of (union_of (G1,G2)):] by NECKLA_2:def_2;
then [a,b] 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 [a,b] in the InternalRel of (union_of (G1,G2)) ` by SUBSET_1:def_4;
then [a,b] in ( the InternalRel of (union_of (G1,G2)) `) \ (id the carrier of (union_of (G1,G2))) by A31, XBOOLE_0:def_5;
hence [a,b] in the InternalRel of (ComplRelStr (union_of (G1,G2))) by NECKLACE:def_8; ::_thesis: verum
end;
suppose [a,b] in [: the carrier of (ComplRelStr G2), the carrier of (ComplRelStr G1):] ; ::_thesis: [a,b] in the InternalRel of (ComplRelStr (union_of (G1,G2)))
then A37: [a,b] in [: the carrier of G2, the carrier of (ComplRelStr G1):] by NECKLACE:def_8;
then A38: [a,b] in [: the carrier of G2, the carrier of G1:] by NECKLACE:def_8;
A39: not [a,b] in the InternalRel of (union_of (G1,G2))
proof
assume [a,b] in the InternalRel of (union_of (G1,G2)) ; ::_thesis: contradiction
then A40: [a,b] in the InternalRel of G1 \/ the InternalRel of G2 by NECKLA_2:def_2;
percases ( [a,b] in the InternalRel of G1 or [a,b] in the InternalRel of G2 ) by A40, XBOOLE_0:def_3;
supposeA41: [a,b] in the InternalRel of G1 ; ::_thesis: contradiction
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, XBOOLE_0:def_7; ::_thesis: verum
end;
supposeA43: [a,b] in the InternalRel of G2 ; ::_thesis: contradiction
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, XBOOLE_0:def_7; ::_thesis: verum
end;
end;
end;
A45: not [a,b] in id the carrier of (union_of (G1,G2))
proof
assume [a,b] in id the carrier of (union_of (G1,G2)) ; ::_thesis: contradiction
then [a,b] in id ( the carrier of G1 \/ the carrier of G2) by NECKLA_2:def_2;
then A46: [a,b] in (id the carrier of G1) \/ (id the carrier of G2) by SYSREL:14;
percases ( [a,b] in id the carrier of G1 or [a,b] in id the carrier of G2 ) by A46, XBOOLE_0:def_3;
supposeA47: [a,b] in id the carrier of G1 ; ::_thesis: contradiction
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, XBOOLE_0:def_7; ::_thesis: verum
end;
supposeA49: [a,b] in id the carrier of G2 ; ::_thesis: contradiction
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, XBOOLE_0:def_7; ::_thesis: verum
end;
end;
end;
[a,b] in [: the carrier of G1, the carrier of G2:] \/ [: the carrier of G2, the carrier of G1:] by A38, XBOOLE_0:def_3;
then [a,b] in [: the carrier of G1, the carrier of G1:] \/ ([: the carrier of G1, the carrier of G2:] \/ [: the carrier of G2, the carrier of G1:]) by XBOOLE_0:def_3;
then [a,b] in ([: the carrier of G1, the carrier of G1:] \/ [: the carrier of G1, the carrier of G2:]) \/ [: the carrier of G2, the carrier of G1:] by XBOOLE_1:4;
then [a,b] in (([: the carrier of G1, the carrier of G1:] \/ [: the carrier of G1, the carrier of G2:]) \/ [: the carrier of G2, the carrier of G1:]) \/ [: the carrier of G2, the carrier of G2:] by XBOOLE_0:def_3;
then [a,b] in [:( the carrier of G1 \/ the carrier of G2),( the carrier of G1 \/ the carrier of G2):] by ZFMISC_1:98;
then [a,b] in [:( the carrier of G1 \/ the carrier of G2), the carrier of (union_of (G1,G2)):] by NECKLA_2:def_2;
then [a,b] in [: the carrier of (union_of (G1,G2)), the carrier of (union_of (G1,G2)):] by NECKLA_2:def_2;
then [a,b] 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 [a,b] in the InternalRel of (union_of (G1,G2)) ` by SUBSET_1:def_4;
then [a,b] in ( the InternalRel of (union_of (G1,G2)) `) \ (id the carrier of (union_of (G1,G2))) by A45, XBOOLE_0:def_5;
hence [a,b] in the InternalRel of (ComplRelStr (union_of (G1,G2))) by NECKLACE:def_8; ::_thesis: verum
end;
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 ComplRelStr (union_of (G1,G2)) = sum_of ((ComplRelStr G1),(ComplRelStr G2)) by A1, A3, RELAT_1:def_2; ::_thesis: verum
end;
theorem Th18: :: NECKLA_3:18
for G1, G2 being 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 ; ::_thesis: ( the carrier of G1 misses the carrier of G2 implies ComplRelStr (sum_of (G1,G2)) = union_of ((ComplRelStr G1),(ComplRelStr G2)) )
assume A1: the carrier of G1 misses the carrier of G2 ; ::_thesis: ComplRelStr (sum_of (G1,G2)) = union_of ((ComplRelStr G1),(ComplRelStr G2))
set P = the InternalRel of (ComplRelStr (sum_of (G1,G2)));
set R = the InternalRel of (union_of ((ComplRelStr G1),(ComplRelStr G2)));
set X1 = the InternalRel of (ComplRelStr G1);
set X2 = the InternalRel of (ComplRelStr G2);
set X5 = [: the carrier of G1, the carrier of G1:];
set X6 = [: the carrier of G2, the carrier of G2:];
set X7 = [: the carrier of G1, the carrier of G2:];
set 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
.= (([: the carrier of G1, the carrier of G1:] \/ [: the carrier of G1, the carrier of G2:]) \/ [: the carrier of G2, the carrier of G1:]) \/ [: the carrier of G2, the carrier of G2:] by ZFMISC_1:98 ;
A3: for a, b being set holds
( [a,b] in the InternalRel of (ComplRelStr (sum_of (G1,G2))) iff [a,b] in the InternalRel of (union_of ((ComplRelStr G1),(ComplRelStr G2))) )
proof
let a, b be set ; ::_thesis: ( [a,b] in the InternalRel of (ComplRelStr (sum_of (G1,G2))) iff [a,b] in the InternalRel of (union_of ((ComplRelStr G1),(ComplRelStr G2))) )
set x = [a,b];
thus ( [a,b] in the InternalRel of (ComplRelStr (sum_of (G1,G2))) implies [a,b] in the InternalRel of (union_of ((ComplRelStr G1),(ComplRelStr G2))) ) ::_thesis: ( [a,b] in the InternalRel of (union_of ((ComplRelStr G1),(ComplRelStr G2))) implies [a,b] in the InternalRel of (ComplRelStr (sum_of (G1,G2))) )
proof
assume [a,b] in the InternalRel of (ComplRelStr (sum_of (G1,G2))) ; ::_thesis: [a,b] in the InternalRel of (union_of ((ComplRelStr G1),(ComplRelStr G2)))
then A4: [a,b] in ( the InternalRel of (sum_of (G1,G2)) `) \ (id the carrier of (sum_of (G1,G2))) by NECKLACE:def_8;
then ( [a,b] in ([: the carrier of G1, the carrier of G1:] \/ [: the carrier of G1, the carrier of G2:]) \/ [: the carrier of G2, the carrier of G1:] or [a,b] in [: the carrier of G2, the carrier of G2:] ) by A2, XBOOLE_0:def_3;
then A5: ( [a,b] in [: the carrier of G1, the carrier of G1:] \/ [: the carrier of G1, the carrier of G2:] or [a,b] in [: the carrier of G2, the carrier of G1:] or [a,b] in [: the carrier of G2, the carrier of G2:] ) by XBOOLE_0:def_3;
[a,b] in the InternalRel of (sum_of (G1,G2)) ` by A4, XBOOLE_0:def_5;
then [a,b] 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 [a,b] in the InternalRel of (sum_of (G1,G2)) by XBOOLE_0:def_5;
then A6: not [a,b] 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 [a,b] in the InternalRel of G1 & not [a,b] in the InternalRel of G2 & not [a,b] in [: the carrier of G1, the carrier of G2:] & not [a,b] in [: the carrier of G2, the carrier of G1:] )
proof
assume ( [a,b] in the InternalRel of G1 or [a,b] in the InternalRel of G2 or [a,b] in [: the carrier of G1, the carrier of G2:] or [a,b] in [: the carrier of G2, the carrier of G1:] ) ; ::_thesis: contradiction
then ( [a,b] in the InternalRel of G1 \/ the InternalRel of G2 or [a,b] in [: the carrier of G1, the carrier of G2:] or [a,b] in [: the carrier of G2, the carrier of G1:] ) by XBOOLE_0:def_3;
then ( [a,b] in ( the InternalRel of G1 \/ the InternalRel of G2) \/ [: the carrier of G1, the carrier of G2:] or [a,b] in [: the carrier of G2, the carrier of G1:] ) by XBOOLE_0:def_3;
hence contradiction by A6, XBOOLE_0:def_3; ::_thesis: verum
end;
not [a,b] in id the carrier of (sum_of (G1,G2)) by A4, XBOOLE_0:def_5;
then not [a,b] in id ( the carrier of G1 \/ the carrier of G2) by NECKLA_2:def_3;
then A8: not [a,b] in (id the carrier of G1) \/ (id the carrier of G2) by SYSREL:14;
then A9: not [a,b] in id the carrier of G1 by XBOOLE_0:def_3;
A10: not [a,b] in id the carrier of G2 by A8, XBOOLE_0:def_3;
percases ( [a,b] in [: the carrier of G1, the carrier of G1:] or [a,b] in [: the carrier of G1, the carrier of G2:] or [a,b] in [: the carrier of G2, the carrier of G1:] or [a,b] in [: the carrier of G2, the carrier of G2:] ) by A5, XBOOLE_0:def_3;
suppose [a,b] in [: the carrier of G1, the carrier of G1:] ; ::_thesis: [a,b] in the InternalRel of (union_of ((ComplRelStr G1),(ComplRelStr G2)))
then [a,b] in [: the carrier of G1, the carrier of G1:] \ the InternalRel of G1 by A7, XBOOLE_0:def_5;
then [a,b] in the InternalRel of G1 ` by SUBSET_1:def_4;
then [a,b] in ( the InternalRel of G1 `) \ (id the carrier of G1) by A9, XBOOLE_0:def_5;
then [a,b] in the InternalRel of (ComplRelStr G1) by NECKLACE:def_8;
then [a,b] in the InternalRel of (ComplRelStr G1) \/ the InternalRel of (ComplRelStr G2) by XBOOLE_0:def_3;
hence [a,b] in the InternalRel of (union_of ((ComplRelStr G1),(ComplRelStr G2))) by NECKLA_2:def_2; ::_thesis: verum
end;
suppose [a,b] in [: the carrier of G1, the carrier of G2:] ; ::_thesis: [a,b] in the InternalRel of (union_of ((ComplRelStr G1),(ComplRelStr G2)))
hence [a,b] in the InternalRel of (union_of ((ComplRelStr G1),(ComplRelStr G2))) by A7; ::_thesis: verum
end;
suppose [a,b] in [: the carrier of G2, the carrier of G1:] ; ::_thesis: [a,b] in the InternalRel of (union_of ((ComplRelStr G1),(ComplRelStr G2)))
hence [a,b] in the InternalRel of (union_of ((ComplRelStr G1),(ComplRelStr G2))) by A7; ::_thesis: verum
end;
suppose [a,b] in [: the carrier of G2, the carrier of G2:] ; ::_thesis: [a,b] in the InternalRel of (union_of ((ComplRelStr G1),(ComplRelStr G2)))
then [a,b] in [: the carrier of G2, the carrier of G2:] \ the InternalRel of G2 by A7, XBOOLE_0:def_5;
then [a,b] in the InternalRel of G2 ` by SUBSET_1:def_4;
then [a,b] in ( the InternalRel of G2 `) \ (id the carrier of G2) by A10, XBOOLE_0:def_5;
then [a,b] in the InternalRel of (ComplRelStr G2) by NECKLACE:def_8;
then [a,b] in the InternalRel of (ComplRelStr G1) \/ the InternalRel of (ComplRelStr G2) by XBOOLE_0:def_3;
hence [a,b] in the InternalRel of (union_of ((ComplRelStr G1),(ComplRelStr G2))) by NECKLA_2:def_2; ::_thesis: verum
end;
end;
end;
assume [a,b] in the InternalRel of (union_of ((ComplRelStr G1),(ComplRelStr G2))) ; ::_thesis: [a,b] in the InternalRel of (ComplRelStr (sum_of (G1,G2)))
then A11: [a,b] in the InternalRel of (ComplRelStr G1) \/ the InternalRel of (ComplRelStr G2) by NECKLA_2:def_2;
percases ( [a,b] in the InternalRel of (ComplRelStr G1) or [a,b] in the InternalRel of (ComplRelStr G2) ) by A11, XBOOLE_0:def_3;
suppose [a,b] in the InternalRel of (ComplRelStr G1) ; ::_thesis: [a,b] in the InternalRel of (ComplRelStr (sum_of (G1,G2)))
then A12: [a,b] in ( the InternalRel of G1 `) \ (id the carrier of G1) by NECKLACE:def_8;
then A13: not [a,b] in id the carrier of G1 by XBOOLE_0:def_5;
A14: not [a,b] in id the carrier of (sum_of (G1,G2))
proof
assume [a,b] in id the carrier of (sum_of (G1,G2)) ; ::_thesis: contradiction
then [a,b] in id ( the carrier of G1 \/ the carrier of G2) by NECKLA_2:def_3;
then [a,b] in (id the carrier of G1) \/ (id the carrier of G2) by SYSREL:14;
then [a,b] 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 not the carrier of G1 /\ the carrier of G2 is empty by A15, XBOOLE_0:def_4;
hence contradiction by A1, XBOOLE_0:def_7; ::_thesis: verum
end;
[a,b] in the InternalRel of G1 ` by A12, XBOOLE_0:def_5;
then [a,b] in [: the carrier of G1, the carrier of G1:] \ the InternalRel of G1 by SUBSET_1:def_4;
then A16: not [a,b] in the InternalRel of G1 by XBOOLE_0:def_5;
A17: not [a,b] in the InternalRel of (sum_of (G1,G2))
proof
assume [a,b] in the InternalRel of (sum_of (G1,G2)) ; ::_thesis: contradiction
then [a,b] 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;
then ( [a,b] in ( the InternalRel of G1 \/ the InternalRel of G2) \/ [: the carrier of G1, the carrier of G2:] or [a,b] in [: the carrier of G2, the carrier of G1:] ) by XBOOLE_0:def_3;
then A18: ( [a,b] in the InternalRel of G1 \/ the InternalRel of G2 or [a,b] in [: the carrier of G1, the carrier of G2:] or [a,b] in [: the carrier of G2, the carrier of G1:] ) by XBOOLE_0:def_3;
percases ( [a,b] in the InternalRel of G2 or [a,b] in [: the carrier of G1, the carrier of G2:] or [a,b] in [: the carrier of G2, the carrier of G1:] ) by A16, A18, XBOOLE_0:def_3;
supposeA19: [a,b] in the InternalRel of G2 ; ::_thesis: contradiction
A20: a in the carrier of G1 by A12, ZFMISC_1:87;
a in the carrier of G2 by A19, ZFMISC_1:87;
then not the carrier of G1 /\ the carrier of G2 is empty by A20, XBOOLE_0:def_4;
hence contradiction by A1, XBOOLE_0:def_7; ::_thesis: verum
end;
supposeA21: [a,b] in [: the carrier of G1, the carrier of G2:] ; ::_thesis: contradiction
A22: b in the carrier of G1 by A12, ZFMISC_1:87;
b in the carrier of G2 by A21, ZFMISC_1:87;
then not the carrier of G1 /\ the carrier of G2 is empty by A22, XBOOLE_0:def_4;
hence contradiction by A1, XBOOLE_0:def_7; ::_thesis: verum
end;
supposeA23: [a,b] in [: the carrier of G2, the carrier of G1:] ; ::_thesis: contradiction
A24: a in the carrier of G1 by A12, ZFMISC_1:87;
a in the carrier of G2 by A23, ZFMISC_1:87;
then not the carrier of G1 /\ the carrier of G2 is empty by A24, XBOOLE_0:def_4;
hence contradiction by A1, XBOOLE_0:def_7; ::_thesis: verum
end;
end;
end;
[a,b] in [: the carrier of G1, the carrier of G1:] \/ (([: the carrier of G1, the carrier of G2:] \/ [: the carrier of G2, the carrier of G1:]) \/ [: the carrier of G2, the carrier of G2:]) by A12, XBOOLE_0:def_3;
then [a,b] in [: the carrier of (sum_of (G1,G2)), the carrier of (sum_of (G1,G2)):] by A2, XBOOLE_1:113;
then [a,b] 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 [a,b] in the InternalRel of (sum_of (G1,G2)) ` by SUBSET_1:def_4;
then [a,b] in ( the InternalRel of (sum_of (G1,G2)) `) \ (id the carrier of (sum_of (G1,G2))) by A14, XBOOLE_0:def_5;
hence [a,b] in the InternalRel of (ComplRelStr (sum_of (G1,G2))) by NECKLACE:def_8; ::_thesis: verum
end;
suppose [a,b] in the InternalRel of (ComplRelStr G2) ; ::_thesis: [a,b] in the InternalRel of (ComplRelStr (sum_of (G1,G2)))
then A25: [a,b] in ( the InternalRel of G2 `) \ (id the carrier of G2) by NECKLACE:def_8;
then A26: not [a,b] in id the carrier of G2 by XBOOLE_0:def_5;
A27: not [a,b] in id the carrier of (sum_of (G1,G2))
proof
assume [a,b] in id the carrier of (sum_of (G1,G2)) ; ::_thesis: contradiction
then [a,b] in id ( the carrier of G1 \/ the carrier of G2) by NECKLA_2:def_3;
then [a,b] in (id the carrier of G1) \/ (id the carrier of G2) by SYSREL:14;
then [a,b] 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 not the carrier of G1 /\ the carrier of G2 is empty by A28, XBOOLE_0:def_4;
hence contradiction by A1, XBOOLE_0:def_7; ::_thesis: verum
end;
[a,b] in the InternalRel of G2 ` by A25, XBOOLE_0:def_5;
then [a,b] in [: the carrier of G2, the carrier of G2:] \ the InternalRel of G2 by SUBSET_1:def_4;
then A29: not [a,b] in the InternalRel of G2 by XBOOLE_0:def_5;
A30: not [a,b] in the InternalRel of (sum_of (G1,G2))
proof
assume [a,b] in the InternalRel of (sum_of (G1,G2)) ; ::_thesis: contradiction
then [a,b] 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;
then ( [a,b] in ( the InternalRel of G1 \/ the InternalRel of G2) \/ [: the carrier of G1, the carrier of G2:] or [a,b] in [: the carrier of G2, the carrier of G1:] ) by XBOOLE_0:def_3;
then A31: ( [a,b] in the InternalRel of G1 \/ the InternalRel of G2 or [a,b] in [: the carrier of G1, the carrier of G2:] or [a,b] in [: the carrier of G2, the carrier of G1:] ) by XBOOLE_0:def_3;
percases ( [a,b] in the InternalRel of G1 or [a,b] in [: the carrier of G1, the carrier of G2:] or [a,b] in [: the carrier of G2, the carrier of G1:] ) by A29, A31, XBOOLE_0:def_3;
supposeA32: [a,b] in the InternalRel of G1 ; ::_thesis: contradiction
A33: a in the carrier of G2 by A25, ZFMISC_1:87;
a in the carrier of G1 by A32, ZFMISC_1:87;
then not the carrier of G1 /\ the carrier of G2 is empty by A33, XBOOLE_0:def_4;
hence contradiction by A1, XBOOLE_0:def_7; ::_thesis: verum
end;
supposeA34: [a,b] in [: the carrier of G1, the carrier of G2:] ; ::_thesis: contradiction
A35: a in the carrier of G2 by A25, ZFMISC_1:87;
a in the carrier of G1 by A34, ZFMISC_1:87;
then not the carrier of G1 /\ the carrier of G2 is empty by A35, XBOOLE_0:def_4;
hence contradiction by A1, XBOOLE_0:def_7; ::_thesis: verum
end;
supposeA36: [a,b] in [: the carrier of G2, the carrier of G1:] ; ::_thesis: contradiction
A37: b in the carrier of G2 by A25, ZFMISC_1:87;
b in the carrier of G1 by A36, ZFMISC_1:87;
then not the carrier of G1 /\ the carrier of G2 is empty by A37, XBOOLE_0:def_4;
hence contradiction by A1, XBOOLE_0:def_7; ::_thesis: verum
end;
end;
end;
[a,b] in [: the carrier of (sum_of (G1,G2)), the carrier of (sum_of (G1,G2)):] by A2, A25, XBOOLE_0:def_3;
then [a,b] 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 [a,b] in the InternalRel of (sum_of (G1,G2)) ` by SUBSET_1:def_4;
then [a,b] in ( the InternalRel of (sum_of (G1,G2)) `) \ (id the carrier of (sum_of (G1,G2))) by A27, XBOOLE_0:def_5;
hence [a,b] in the InternalRel of (ComplRelStr (sum_of (G1,G2))) by NECKLACE:def_8; ::_thesis: verum
end;
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 ComplRelStr (sum_of (G1,G2)) = union_of ((ComplRelStr G1),(ComplRelStr G2)) by A38, A3, RELAT_1:def_2; ::_thesis: verum
end;
theorem :: NECKLA_3:19
for G being RelStr
for H being 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 ; ::_thesis: for H being full SubRelStr of G holds the InternalRel of (ComplRelStr H) = the InternalRel of (ComplRelStr G) |_2 the carrier of (ComplRelStr H)
let H be full SubRelStr of G; ::_thesis: the InternalRel of (ComplRelStr H) = the InternalRel of (ComplRelStr G) |_2 the carrier of (ComplRelStr H)
set IH = the InternalRel of H;
set ICmpH = the InternalRel of (ComplRelStr H);
set cH = the carrier of H;
set IG = the InternalRel of G;
set cG = the carrier of G;
set ICmpG = the InternalRel of (ComplRelStr G);
A1: the InternalRel of (ComplRelStr H) = ( the InternalRel of H `) \ (id the carrier of H) by NECKLACE:def_8
.= ([: the carrier of H, the carrier of H:] \ the InternalRel of H) \ (id the carrier of H) by SUBSET_1:def_4 ;
A2: the InternalRel of (ComplRelStr G) = ( the InternalRel of G `) \ (id the carrier of G) by NECKLACE:def_8
.= ([: the carrier of G, the carrier of G:] \ the InternalRel of G) \ (id the carrier of G) by SUBSET_1:def_4 ;
A3: the carrier of H c= the carrier of G by YELLOW_0:def_13;
the InternalRel of (ComplRelStr G) |_2 the carrier of (ComplRelStr H) = the InternalRel of (ComplRelStr G) |_2 the carrier of H by NECKLACE:def_8
.= (([: the carrier of G, the carrier of G:] \ the InternalRel of G) /\ [: the carrier of H, the carrier of H:]) \ ((id the carrier of G) /\ [: the carrier of H, the carrier of H:]) by A2, XBOOLE_1:50
.= (([: the carrier of G, the carrier of G:] /\ [: the carrier of H, the carrier of H:]) \ ( the InternalRel of G /\ [: the carrier of H, the carrier of H:])) \ ((id the carrier of G) /\ [: the carrier of H, the carrier of H:]) by XBOOLE_1:50
.= (([: the carrier of G, the carrier of G:] /\ [: the carrier of H, the carrier of H:]) \ ( the InternalRel of G /\ [: the carrier of H, the carrier of H:])) \ ((id the carrier of G) | the carrier of H) by Th1
.= (([: the carrier of G, the carrier of G:] /\ [: the carrier of H, the carrier of H:]) \ ( the InternalRel of G |_2 the carrier of H)) \ (id the carrier of H) by A3, FUNCT_3:1
.= (([: the carrier of G, the carrier of G:] /\ [: the carrier of H, the carrier of H:]) \ the InternalRel of H) \ (id the carrier of H) by YELLOW_0:def_14
.= ([:( the carrier of G /\ the carrier of H),( the carrier of G /\ the carrier of H):] \ the InternalRel of H) \ (id the carrier of H) by ZFMISC_1:100
.= ([: the carrier of H,( the carrier of G /\ the carrier of H):] \ the InternalRel of H) \ (id the carrier of H) by A3, XBOOLE_1:28
.= the InternalRel of (ComplRelStr H) by A1, A3, XBOOLE_1:28 ;
hence the InternalRel of (ComplRelStr H) = the InternalRel of (ComplRelStr G) |_2 the carrier of (ComplRelStr H) ; ::_thesis: verum
end;
theorem Th20: :: NECKLA_3:20
for G being non empty irreflexive RelStr
for x being Element of G
for x9 being Element of (ComplRelStr G) st x = x9 holds
ComplRelStr (subrelstr (([#] G) \ {x})) = subrelstr (([#] (ComplRelStr G)) \ {x9})
proof
let G be non empty irreflexive RelStr ; ::_thesis: for x being Element of G
for x9 being Element of (ComplRelStr G) st x = x9 holds
ComplRelStr (subrelstr (([#] G) \ {x})) = subrelstr (([#] (ComplRelStr G)) \ {x9})
let x be Element of G; ::_thesis: for x9 being Element of (ComplRelStr G) st x = x9 holds
ComplRelStr (subrelstr (([#] G) \ {x})) = subrelstr (([#] (ComplRelStr G)) \ {x9})
let x9 be Element of (ComplRelStr G); ::_thesis: ( x = x9 implies ComplRelStr (subrelstr (([#] G) \ {x})) = subrelstr (([#] (ComplRelStr G)) \ {x9}) )
assume A1: x = x9 ; ::_thesis: ComplRelStr (subrelstr (([#] G) \ {x})) = subrelstr (([#] (ComplRelStr G)) \ {x9})
set R = subrelstr (([#] G) \ {x});
set cR = the carrier of (subrelstr (([#] G) \ {x}));
set cG = the carrier of G;
A2: [#] (ComplRelStr G) = the carrier of G by NECKLACE:def_8;
A3: [:( the carrier of G \ {x}),( the carrier of G \ {x}):] = [: the carrier of (subrelstr (([#] G) \ {x})),(([#] G) \ {x}):] by YELLOW_0:def_15
.= [: the carrier of (subrelstr (([#] G) \ {x})), the carrier of (subrelstr (([#] G) \ {x})):] by YELLOW_0:def_15 ;
A4: the carrier of (subrelstr (([#] G) \ {x})) c= the carrier of G 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 ( the carrier of G \ {x}) by A1, A2, YELLOW_0:def_15
.= (( the InternalRel of G `) \ (id the carrier of G)) /\ [:( the carrier of G \ {x}),( the carrier of G \ {x}):] by NECKLACE:def_8
.= ([: the carrier of (subrelstr (([#] G) \ {x})), the carrier of (subrelstr (([#] G) \ {x})):] /\ ( the InternalRel of G `)) \ (id the carrier of G) by A3, XBOOLE_1:49
.= ([: the carrier of (subrelstr (([#] G) \ {x})), the carrier of (subrelstr (([#] G) \ {x})):] /\ ([: the carrier of G, the carrier of G:] \ the InternalRel of G)) \ (id the carrier of G) by SUBSET_1:def_4
.= (([: the carrier of (subrelstr (([#] G) \ {x})), the carrier of (subrelstr (([#] G) \ {x})):] /\ [: the carrier of G, the carrier of G:]) \ the InternalRel of G) \ (id the carrier of G) by XBOOLE_1:49
.= ([: the carrier of (subrelstr (([#] G) \ {x})), the carrier of (subrelstr (([#] G) \ {x})):] \ the InternalRel of G) \ (id the carrier of G) by A4, XBOOLE_1:28, ZFMISC_1:96 ;
A6: the InternalRel of (ComplRelStr (subrelstr (([#] G) \ {x}))) = ( the InternalRel of (subrelstr (([#] G) \ {x})) `) \ (id the carrier of (subrelstr (([#] G) \ {x}))) by NECKLACE:def_8
.= ([: the carrier of (subrelstr (([#] G) \ {x})), the carrier of (subrelstr (([#] G) \ {x})):] \ the InternalRel of (subrelstr (([#] G) \ {x}))) \ (id the carrier of (subrelstr (([#] G) \ {x}))) by SUBSET_1:def_4
.= ([: the carrier of (subrelstr (([#] G) \ {x})), the carrier of (subrelstr (([#] G) \ {x})):] \ ( the InternalRel of G |_2 the carrier of (subrelstr (([#] G) \ {x})))) \ (id the carrier of (subrelstr (([#] G) \ {x}))) by YELLOW_0:def_14
.= (([: the carrier of (subrelstr (([#] G) \ {x})), the carrier of (subrelstr (([#] G) \ {x})):] \ the InternalRel of G) \/ ([: the carrier of (subrelstr (([#] G) \ {x})), the carrier of (subrelstr (([#] G) \ {x})):] \ [: the carrier of (subrelstr (([#] G) \ {x})), the carrier of (subrelstr (([#] G) \ {x})):])) \ (id the carrier of (subrelstr (([#] G) \ {x}))) by XBOOLE_1:54
.= (([: the carrier of (subrelstr (([#] G) \ {x})), the carrier of (subrelstr (([#] G) \ {x})):] \ the InternalRel of G) \/ {}) \ (id the carrier of (subrelstr (([#] G) \ {x}))) by XBOOLE_1:37
.= ([: the carrier of (subrelstr (([#] G) \ {x})), the carrier of (subrelstr (([#] G) \ {x})):] \ the InternalRel of G) \ (id the carrier of (subrelstr (([#] G) \ {x}))) ;
A7: [: the carrier of (subrelstr (([#] G) \ {x})), the carrier of (subrelstr (([#] G) \ {x})):] = [:([#] G),(([#] G) \ {x}):] \ [:{x},(([#] G) \ {x}):] by A3, ZFMISC_1:102
.= ([:([#] G),([#] G):] \ [:([#] G),{x}:]) \ [:{x},(([#] G) \ {x}):] by ZFMISC_1:102
.= ([: the carrier of G, the carrier of G:] \ [: the carrier of G,{x}:]) \ ([:{x}, the carrier of G:] \ [:{x},{x}:]) by ZFMISC_1:102
.= (([: the carrier of G, the carrier of G:] \ [: the carrier of G,{x}:]) \ [:{x}, the carrier of G:]) \/ (([: the carrier of G, the carrier of G:] \ [: the carrier of G,{x}:]) /\ [:{x},{x}:]) by XBOOLE_1:52
.= ([: the carrier of G, the carrier of G:] \ ([: the carrier of G,{x}:] \/ [:{x}, the carrier of G:])) \/ (([: the carrier of G, the carrier of G:] \ [: the carrier of G,{x}:]) /\ [:{x},{x}:]) by XBOOLE_1:41 ;
A8: the InternalRel of (subrelstr (([#] (ComplRelStr G)) \ {x9})) = the InternalRel of (ComplRelStr (subrelstr (([#] G) \ {x})))
proof
thus the InternalRel of (subrelstr (([#] (ComplRelStr G)) \ {x9})) c= the InternalRel of (ComplRelStr (subrelstr (([#] G) \ {x}))) :: according to XBOOLE_0:def_10 ::_thesis: the InternalRel of (ComplRelStr (subrelstr (([#] G) \ {x}))) c= the InternalRel of (subrelstr (([#] (ComplRelStr G)) \ {x9}))
proof
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in the InternalRel of (subrelstr (([#] (ComplRelStr G)) \ {x9})) or a in the InternalRel of (ComplRelStr (subrelstr (([#] G) \ {x}))) )
assume A9: a in the InternalRel of (subrelstr (([#] (ComplRelStr G)) \ {x9})) ; ::_thesis: a in the InternalRel of (ComplRelStr (subrelstr (([#] G) \ {x})))
then A10: not a in id the carrier of G by A5, XBOOLE_0:def_5;
A11: not a in id the carrier of (subrelstr (([#] G) \ {x}))
proof
assume A12: a in id the carrier of (subrelstr (([#] G) \ {x})) ; ::_thesis: contradiction
then consider x2, y2 being set such that
A13: a = [x2,y2] and
A14: x2 in the carrier of (subrelstr (([#] G) \ {x})) and
y2 in the carrier of (subrelstr (([#] G) \ {x})) by RELSET_1:2;
A15: x2 in the carrier of G \ {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; ::_thesis: verum
end;
a in [: the carrier of (subrelstr (([#] G) \ {x})), the carrier of (subrelstr (([#] G) \ {x})):] \ the InternalRel of G by A5, A9, XBOOLE_0:def_5;
hence a in the InternalRel of (ComplRelStr (subrelstr (([#] G) \ {x}))) by A6, A11, XBOOLE_0:def_5; ::_thesis: verum
end;
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in the InternalRel of (ComplRelStr (subrelstr (([#] G) \ {x}))) or a in the InternalRel of (subrelstr (([#] (ComplRelStr G)) \ {x9})) )
assume A16: a in the InternalRel of (ComplRelStr (subrelstr (([#] G) \ {x}))) ; ::_thesis: a in the InternalRel of (subrelstr (([#] (ComplRelStr G)) \ {x9}))
then not a in id the carrier of (subrelstr (([#] G) \ {x})) by A6, XBOOLE_0:def_5;
then not a in id ( the carrier of G \ {x}) by YELLOW_0:def_15;
then A17: not a in (id the carrier of G) \ (id {x}) by SYSREL:14;
percases ( not a in id the carrier of G or a in id {x} ) by A17, XBOOLE_0:def_5;
supposeA18: not a in id the carrier of G ; ::_thesis: a in the InternalRel of (subrelstr (([#] (ComplRelStr G)) \ {x9}))
a in [: the carrier of (subrelstr (([#] G) \ {x})), the carrier of (subrelstr (([#] G) \ {x})):] \ the InternalRel of G by A6, A16, XBOOLE_0:def_5;
hence a in the InternalRel of (subrelstr (([#] (ComplRelStr G)) \ {x9})) by A5, A18, XBOOLE_0:def_5; ::_thesis: verum
end;
suppose a in id {x} ; ::_thesis: a in the InternalRel of (subrelstr (([#] (ComplRelStr G)) \ {x9}))
then A19: a in {[x,x]} by SYSREL:13;
thus a in the InternalRel of (subrelstr (([#] (ComplRelStr G)) \ {x9})) ::_thesis: verum
proof
percases ( a in [: the carrier of G, the carrier of G:] \ ([: the carrier of G,{x}:] \/ [:{x}, the carrier of G:]) or a in ([: the carrier of G, the carrier of G:] \ [: the carrier of G,{x}:]) /\ [:{x},{x}:] ) by A7, A6, A16, XBOOLE_0:def_3;
supposeA20: a in [: the carrier of G, the carrier of G:] \ ([: the carrier of G,{x}:] \/ [:{x}, the carrier of G:]) ; ::_thesis: a in the InternalRel of (subrelstr (([#] (ComplRelStr G)) \ {x9}))
x in {x} by TARSKI:def_1;
then A21: [x,x] in [:{x}, the carrier of G:] by ZFMISC_1:87;
not a in [: the carrier of G,{x}:] \/ [:{x}, the carrier of G:] by A20, XBOOLE_0:def_5;
then not a in [:{x}, the carrier of G:] by XBOOLE_0:def_3;
hence a in the InternalRel of (subrelstr (([#] (ComplRelStr G)) \ {x9})) by A19, A21, TARSKI:def_1; ::_thesis: verum
end;
supposeA22: a in ([: the carrier of G, the carrier of G:] \ [: the carrier of G,{x}:]) /\ [:{x},{x}:] ; ::_thesis: a in the InternalRel of (subrelstr (([#] (ComplRelStr G)) \ {x9}))
x in {x} by TARSKI:def_1;
then A23: [x,x] in [: the carrier of G,{x}:] by ZFMISC_1:87;
a in [: the carrier of G, the carrier of G:] \ [: the carrier of G,{x}:] by A22, XBOOLE_0:def_4;
then not a in [: the carrier of G,{x}:] by XBOOLE_0:def_5;
hence a in the InternalRel of (subrelstr (([#] (ComplRelStr G)) \ {x9})) by A19, A23, TARSKI:def_1; ::_thesis: verum
end;
end;
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 ComplRelStr (subrelstr (([#] G) \ {x})) = subrelstr (([#] (ComplRelStr G)) \ {x9}) by A8; ::_thesis: verum
end;
begin
registration
cluster non empty trivial strict -> non empty N-free for RelStr ;
correctness
coherence
for b1 being non empty RelStr st b1 is trivial & b1 is strict holds
b1 is N-free ;
proof
set Y = Necklace 4;
let R be non empty RelStr ; ::_thesis: ( R is trivial & R is strict implies R is N-free )
assume ( R is trivial & R is strict ) ; ::_thesis: R is N-free
then consider y being set such that
A1: the carrier of R = {y} by GROUP_6:def_2;
assume not R is N-free ; ::_thesis: contradiction
then R embeds Necklace 4 by NECKLA_2:def_1;
then consider f being Function of (Necklace 4),R such that
A2: f is V21() and
for x, y being Element of (Necklace 4) holds
( [x,y] in the InternalRel of (Necklace 4) iff [(f . x),(f . y)] in the InternalRel of R ) by NECKLACE:def_1;
A3: dom f = the carrier of (Necklace 4) 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; ::_thesis: verum
end;
end;
theorem :: NECKLA_3:21
for R being reflexive antisymmetric RelStr
for S being RelStr holds
( ex f being Function of R,S st
for x, y being 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 ; ::_thesis: for S being RelStr holds
( ex f being Function of R,S st
for x, y being 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 )
let S be RelStr ; ::_thesis: ( ex f being Function of R,S st
for x, y being 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 )
A1: now__::_thesis:_(_ex_f_being_Function_of_R,S_st_
for_x,_y_being_Element_of_R_holds_
(_[x,y]_in_the_InternalRel_of_R_iff_[(f_._x),(f_._y)]_in_the_InternalRel_of_S_)_implies_S_embeds_R_)
assume ex f being Function of R,S st
for x, y being Element of R holds
( [x,y] in the InternalRel of R iff [(f . x),(f . y)] in the InternalRel of S ) ; ::_thesis: S embeds R
then consider f being Function of R,S such that
A2: for x, y being Element of R holds
( [x,y] in the InternalRel of R iff [(f . x),(f . y)] in the InternalRel of S ) ;
for x1, x2 being set st x1 in dom f & x2 in dom f & f . x1 = f . x2 holds
x1 = x2
proof
let x1, x2 be set ; ::_thesis: ( x1 in dom f & x2 in dom f & f . x1 = f . x2 implies x1 = x2 )
assume that
A3: x1 in dom f and
A4: x2 in dom f and
A5: f . x1 = f . x2 ; ::_thesis: x1 = x2
reconsider x1 = x1, x2 = 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, RELAT_2:def_1;
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, RELAT_2:def_1;
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 x1 = x2 by A7, ORDERS_2:2; ::_thesis: verum
end;
then f is one-to-one by FUNCT_1:def_4;
hence S embeds R by A2, NECKLACE:def_1; ::_thesis: verum
end;
now__::_thesis:_(_S_embeds_R_implies_ex_f_being_Function_of_R,S_st_
(_ex_f_being_Function_of_R,S_st_
for_x,_y_being_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_)_)
assume A8: S embeds R ; ::_thesis: ex f being Function of R,S st
( ex f being Function of R,S st
for x, y being 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 )
then consider f being Function of R,S such that
f is one-to-one and
A9: for x, y being Element of R holds
( [x,y] in the InternalRel of R iff [(f . x),(f . y)] in the InternalRel of S ) by NECKLACE:def_1;
take f = f; ::_thesis: ( ex f being Function of R,S st
for x, y being 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 )
thus ( ex f being Function of R,S st
for x, y being 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 ) by A8, A9; ::_thesis: verum
end;
hence ( ex f being Function of R,S st
for x, y being 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 ) by A1; ::_thesis: verum
end;
theorem Th22: :: NECKLA_3:22
for G being non empty RelStr
for H being non empty full SubRelStr of G holds G embeds H
proof
let G be non empty RelStr ; ::_thesis: for H being non empty full SubRelStr of G holds G embeds H
let H be non empty full SubRelStr of G; ::_thesis: G embeds H
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 by FUNCT_1:17;
A2: the carrier of H c= the carrier of G by YELLOW_0:def_13;
for x being set st x in the carrier of H holds
f . x in the carrier of G
proof
let x be set ; ::_thesis: ( x in the carrier of H implies f . x in the carrier of G )
assume x in the carrier of H ; ::_thesis: f . x in the carrier of G
then f . x in the carrier of H by FUNCT_1:17;
hence f . x in the carrier of G by A2; ::_thesis: verum
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 = f as Function of H,G ;
for x, y being 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;
set IG = the InternalRel of G;
set cH = the carrier of H;
let x, y be Element of H; ::_thesis: ( [x,y] in the InternalRel of H iff [(f . x),(f . y)] in the InternalRel of G )
A3: ( f . x = x & f . y = y ) by FUNCT_1:17;
thus ( [x,y] in the InternalRel of H implies [(f . x),(f . y)] in the InternalRel of G ) ::_thesis: ( [(f . x),(f . y)] in the InternalRel of G implies [x,y] in the InternalRel of H )
proof
assume [x,y] in the InternalRel of H ; ::_thesis: [(f . x),(f . y)] in the InternalRel of G
then [x,y] in the InternalRel of G |_2 the carrier of H by YELLOW_0:def_14;
hence [(f . x),(f . y)] in the InternalRel of G by A3, XBOOLE_0:def_4; ::_thesis: verum
end;
assume [(f . x),(f . y)] in the InternalRel of G ; ::_thesis: [x,y] in the InternalRel of H
then [x,y] in the InternalRel of G |_2 the carrier of H by A3, XBOOLE_0:def_4;
hence [x,y] in the InternalRel of H by YELLOW_0:def_14; ::_thesis: verum
end;
hence G embeds H by NECKLACE:def_1; ::_thesis: verum
end;
theorem Th23: :: NECKLA_3:23
for G being non empty RelStr
for H being non empty full SubRelStr of G st G is N-free holds
H is N-free
proof
let G be non empty RelStr ; ::_thesis: for H being non empty full SubRelStr of G st G is N-free holds
H is N-free
let H be non empty full SubRelStr of G; ::_thesis: ( G is N-free implies H is N-free )
assume A1: G is N-free ; ::_thesis: H is N-free
A2: G embeds H by Th22;
assume not H is N-free ; ::_thesis: contradiction
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; ::_thesis: verum
end;
theorem Th24: :: NECKLA_3:24
for G being non empty irreflexive RelStr holds
( G embeds Necklace 4 iff ComplRelStr G embeds Necklace 4 )
proof
let G be non empty irreflexive RelStr ; ::_thesis: ( G embeds Necklace 4 iff ComplRelStr G embeds Necklace 4 )
set N4 = Necklace 4;
set CmpN4 = ComplRelStr (Necklace 4);
set CmpG = ComplRelStr G;
A1: the carrier of (ComplRelStr G) = 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 (Necklace 4) by ENUMSET1:def_2;
A4: the carrier of (ComplRelStr (Necklace 4)) = the carrier of (Necklace 4) by NECKLACE:def_8;
thus ( G embeds Necklace 4 implies ComplRelStr G embeds Necklace 4 ) ::_thesis: ( ComplRelStr G embeds Necklace 4 implies G embeds Necklace 4 )
proof
ComplRelStr (Necklace 4), Necklace 4 are_isomorphic by NECKLACE:29, WAYBEL_1:6;
then consider g being Function of (ComplRelStr (Necklace 4)),(Necklace 4) such that
A5: g is isomorphic by WAYBEL_1:def_8;
assume G embeds Necklace 4 ; ::_thesis: ComplRelStr G embeds Necklace 4
then consider f being Function of (Necklace 4),G such that
A6: f is V21() and
A7: for x, y being Element of (Necklace 4) holds
( [x,y] in the InternalRel of (Necklace 4) iff [(f . x),(f . y)] in the InternalRel of G ) by NECKLACE:def_1;
reconsider h = f * g as Function of (ComplRelStr (Necklace 4)),G ;
A8: ( g is V21() & g is monotone ) by A5, WAYBEL_0:def_38;
A9: for x, y being Element of (ComplRelStr (Necklace 4)) holds
( [x,y] in the InternalRel of (ComplRelStr (Necklace 4)) iff [(h . x),(h . y)] in the InternalRel of G )
proof
let x, y be Element of (ComplRelStr (Necklace 4)); ::_thesis: ( [x,y] in the InternalRel of (ComplRelStr (Necklace 4)) iff [(h . x),(h . y)] in the InternalRel of G )
thus ( [x,y] in the InternalRel of (ComplRelStr (Necklace 4)) implies [(h . x),(h . y)] in the InternalRel of G ) ::_thesis: ( [(h . x),(h . y)] in the InternalRel of G implies [x,y] in the InternalRel of (ComplRelStr (Necklace 4)) )
proof
assume [x,y] in the InternalRel of (ComplRelStr (Necklace 4)) ; ::_thesis: [(h . x),(h . y)] in the InternalRel of G
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 (Necklace 4) 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 [(h . x),(h . y)] in the InternalRel of G by FUNCT_2:15; ::_thesis: verum
end;
assume [(h . x),(h . y)] in the InternalRel of G ; ::_thesis: [x,y] in the InternalRel of (ComplRelStr (Necklace 4))
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 (Necklace 4) by A7;
then g . x <= g . y by ORDERS_2:def_5;
then x <= y by A5, WAYBEL_0:66;
hence [x,y] in the InternalRel of (ComplRelStr (Necklace 4)) by ORDERS_2:def_5; ::_thesis: verum
end;
A10: 0 in the carrier of (ComplRelStr (Necklace 4)) by A2, A4, ENUMSET1:def_2;
A11: 1 in the carrier of (ComplRelStr (Necklace 4)) by A2, A4, ENUMSET1:def_2;
A12: dom h = the carrier of (ComplRelStr (Necklace 4)) by FUNCT_2:def_1;
A13: [(h . 0),(h . 1)] in the InternalRel of (ComplRelStr G)
proof
assume A14: not [(h . 0),(h . 1)] in the InternalRel of (ComplRelStr G) ; ::_thesis: contradiction
[(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 (ComplRelStr G) 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 (ComplRelStr G) ) by XBOOLE_0:def_3;
assume not [(h . 0),(h . 1)] in the InternalRel of G ; ::_thesis: contradiction
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; ::_thesis: verum
end;
then A16: [0,1] in the InternalRel of (ComplRelStr (Necklace 4)) by A9, A10, A11;
[0,1] in the InternalRel of (Necklace 4) by ENUMSET1:def_4, NECKLA_2:2;
then [0,1] in the InternalRel of (Necklace 4) /\ the InternalRel of (ComplRelStr (Necklace 4)) by A16, XBOOLE_0:def_4;
then the InternalRel of (Necklace 4) meets the InternalRel of (ComplRelStr (Necklace 4)) by XBOOLE_0:def_7;
hence contradiction by Th12; ::_thesis: verum
end;
A17: 2 in the carrier of (ComplRelStr (Necklace 4)) by A2, A4, ENUMSET1:def_2;
A18: [(h . 1),(h . 2)] in the InternalRel of (ComplRelStr G)
proof
assume A19: not [(h . 1),(h . 2)] in the InternalRel of (ComplRelStr G) ; ::_thesis: contradiction
[(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 (ComplRelStr G) 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 (ComplRelStr G) ) by XBOOLE_0:def_3;
assume not [(h . 1),(h . 2)] in the InternalRel of G ; ::_thesis: contradiction
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; ::_thesis: verum
end;
then A21: [1,2] in the InternalRel of (ComplRelStr (Necklace 4)) by A9, A11, A17;
[1,2] in the InternalRel of (Necklace 4) by ENUMSET1:def_4, NECKLA_2:2;
then [1,2] in the InternalRel of (Necklace 4) /\ the InternalRel of (ComplRelStr (Necklace 4)) by A21, XBOOLE_0:def_4;
then the InternalRel of (Necklace 4) meets the InternalRel of (ComplRelStr (Necklace 4)) by XBOOLE_0:def_7;
hence contradiction by Th12; ::_thesis: verum
end;
A22: 3 in the carrier of (ComplRelStr (Necklace 4)) by A2, A4, ENUMSET1:def_2;
A23: [(h . 2),(h . 3)] in the InternalRel of (ComplRelStr G)
proof
assume A24: not [(h . 2),(h . 3)] in the InternalRel of (ComplRelStr G) ; ::_thesis: contradiction
[(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 (ComplRelStr G) 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 (ComplRelStr G) ) by XBOOLE_0:def_3;
assume not [(h . 2),(h . 3)] in the InternalRel of G ; ::_thesis: contradiction
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; ::_thesis: verum
end;
then A26: [2,3] in the InternalRel of (ComplRelStr (Necklace 4)) by A9, A17, A22;
[2,3] in the InternalRel of (Necklace 4) by ENUMSET1:def_4, NECKLA_2:2;
then [2,3] in the InternalRel of (Necklace 4) /\ the InternalRel of (ComplRelStr (Necklace 4)) by A26, XBOOLE_0:def_4;
then the InternalRel of (Necklace 4) meets the InternalRel of (ComplRelStr (Necklace 4)) by XBOOLE_0:def_7;
hence contradiction by Th12; ::_thesis: verum
end;
[3,1] in the InternalRel of (ComplRelStr (Necklace 4)) 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 (ComplRelStr (Necklace 4)) 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 (ComplRelStr (Necklace 4)) 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 (ComplRelStr (Necklace 4)) 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 (ComplRelStr G)
proof
assume A32: not [(h . 1),(h . 0)] in the InternalRel of (ComplRelStr G) ; ::_thesis: contradiction
[(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 (ComplRelStr G) 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 (ComplRelStr G) ) by XBOOLE_0:def_3;
assume not [(h . 1),(h . 0)] in the InternalRel of G ; ::_thesis: contradiction
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; ::_thesis: verum
end;
then A34: [1,0] in the InternalRel of (ComplRelStr (Necklace 4)) by A9, A10, A11;
[1,0] in the InternalRel of (Necklace 4) by ENUMSET1:def_4, NECKLA_2:2;
then [1,0] in the InternalRel of (Necklace 4) /\ the InternalRel of (ComplRelStr (Necklace 4)) by A34, XBOOLE_0:def_4;
then the InternalRel of (Necklace 4) meets the InternalRel of (ComplRelStr (Necklace 4)) by XBOOLE_0:def_7;
hence contradiction by Th12; ::_thesis: verum
end;
A35: [(h . 2),(h . 1)] in the InternalRel of (ComplRelStr G)
proof
assume A36: not [(h . 2),(h . 1)] in the InternalRel of (ComplRelStr G) ; ::_thesis: contradiction
[(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 (ComplRelStr G) 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 (ComplRelStr G) ) by XBOOLE_0:def_3;
assume not [(h . 2),(h . 1)] in the InternalRel of G ; ::_thesis: contradiction
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; ::_thesis: verum
end;
then A38: [2,1] in the InternalRel of (ComplRelStr (Necklace 4)) by A9, A11, A17;
[2,1] in the InternalRel of (Necklace 4) by ENUMSET1:def_4, NECKLA_2:2;
then [2,1] in the InternalRel of (Necklace 4) /\ the InternalRel of (ComplRelStr (Necklace 4)) by A38, XBOOLE_0:def_4;
then the InternalRel of (Necklace 4) meets the InternalRel of (ComplRelStr (Necklace 4)) by XBOOLE_0:def_7;
hence contradiction by Th12; ::_thesis: verum
end;
A39: [(h . 3),(h . 2)] in the InternalRel of (ComplRelStr G)
proof
assume A40: not [(h . 3),(h . 2)] in the InternalRel of (ComplRelStr G) ; ::_thesis: contradiction
[(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 (ComplRelStr G) 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 (ComplRelStr G) ) by XBOOLE_0:def_3;
assume not [(h . 3),(h . 2)] in the InternalRel of G ; ::_thesis: contradiction
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; ::_thesis: verum
end;
then A42: [3,2] in the InternalRel of (ComplRelStr (Necklace 4)) by A9, A17, A22;
[3,2] in the InternalRel of (Necklace 4) by ENUMSET1:def_4, NECKLA_2:2;
then [3,2] in the InternalRel of (Necklace 4) /\ the InternalRel of (ComplRelStr (Necklace 4)) by A42, XBOOLE_0:def_4;
then the InternalRel of (Necklace 4) meets the InternalRel of (ComplRelStr (Necklace 4)) by XBOOLE_0:def_7;
hence contradiction by Th12; ::_thesis: verum
end;
[2,0] in the InternalRel of (ComplRelStr (Necklace 4)) 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 (ComplRelStr (Necklace 4)) by Th11, ENUMSET1:def_4;
then A44: [(h . 0),(h . 2)] in the InternalRel of G by A9, A10, A17;
for x, y being Element of (Necklace 4) holds
( [x,y] in the InternalRel of (Necklace 4) iff [(h . x),(h . y)] in the InternalRel of (ComplRelStr G) )
proof
let x, y be Element of (Necklace 4); ::_thesis: ( [x,y] in the InternalRel of (Necklace 4) iff [(h . x),(h . y)] in the InternalRel of (ComplRelStr G) )
thus ( [x,y] in the InternalRel of (Necklace 4) implies [(h . x),(h . y)] in the InternalRel of (ComplRelStr G) ) ::_thesis: ( [(h . x),(h . y)] in the InternalRel of (ComplRelStr G) implies [x,y] in the InternalRel of (Necklace 4) )
proof
assume A45: [x,y] in the InternalRel of (Necklace 4) ; ::_thesis: [(h . x),(h . y)] in the InternalRel of (ComplRelStr G)
percases ( [x,y] = [0,1] or [x,y] = [1,0] or [x,y] = [1,2] or [x,y] = [2,1] or [x,y] = [2,3] or [x,y] = [3,2] ) by A45, ENUMSET1:def_4, NECKLA_2:2;
supposeA46: [x,y] = [0,1] ; ::_thesis: [(h . x),(h . y)] in the InternalRel of (ComplRelStr G)
then x = 0 by XTUPLE_0:1;
hence [(h . x),(h . y)] in the InternalRel of (ComplRelStr G) by A13, A46, XTUPLE_0:1; ::_thesis: verum
end;
supposeA47: [x,y] = [1,0] ; ::_thesis: [(h . x),(h . y)] in the InternalRel of (ComplRelStr G)
then x = 1 by XTUPLE_0:1;
hence [(h . x),(h . y)] in the InternalRel of (ComplRelStr G) by A31, A47, XTUPLE_0:1; ::_thesis: verum
end;
supposeA48: [x,y] = [1,2] ; ::_thesis: [(h . x),(h . y)] in the InternalRel of (ComplRelStr G)
then x = 1 by XTUPLE_0:1;
hence [(h . x),(h . y)] in the InternalRel of (ComplRelStr G) by A18, A48, XTUPLE_0:1; ::_thesis: verum
end;
supposeA49: [x,y] = [2,1] ; ::_thesis: [(h . x),(h . y)] in the InternalRel of (ComplRelStr G)
then x = 2 by XTUPLE_0:1;
hence [(h . x),(h . y)] in the InternalRel of (ComplRelStr G) by A35, A49, XTUPLE_0:1; ::_thesis: verum
end;
supposeA50: [x,y] = [2,3] ; ::_thesis: [(h . x),(h . y)] in the InternalRel of (ComplRelStr G)
then x = 2 by XTUPLE_0:1;
hence [(h . x),(h . y)] in the InternalRel of (ComplRelStr G) by A23, A50, XTUPLE_0:1; ::_thesis: verum
end;
supposeA51: [x,y] = [3,2] ; ::_thesis: [(h . x),(h . y)] in the InternalRel of (ComplRelStr G)
then x = 3 by XTUPLE_0:1;
hence [(h . x),(h . y)] in the InternalRel of (ComplRelStr G) by A39, A51, XTUPLE_0:1; ::_thesis: verum
end;
end;
end;
assume A52: [(h . x),(h . y)] in the InternalRel of (ComplRelStr G) ; ::_thesis: [x,y] in the InternalRel of (Necklace 4)
percases ( ( x = 0 & y = 0 ) or ( x = 0 & y = 1 ) or ( x = 0 & y = 2 ) or ( x = 0 & y = 3 ) or ( x = 1 & y = 0 ) or ( x = 1 & y = 1 ) or ( x = 1 & y = 2 ) or ( x = 1 & y = 3 ) or ( x = 2 & y = 0 ) or ( x = 2 & y = 1 ) or ( x = 2 & y = 2 ) or ( x = 2 & y = 3 ) or ( x = 3 & y = 0 ) or ( x = 3 & y = 1 ) or ( x = 3 & y = 2 ) or ( x = 3 & y = 3 ) ) by A2, ENUMSET1:def_2;
supposeA53: ( x = 0 & y = 0 ) ; ::_thesis: [x,y] in the InternalRel of (Necklace 4)
then h . 0 in the carrier of (ComplRelStr G) by A52, ZFMISC_1:87;
hence [x,y] in the InternalRel of (Necklace 4) by A52, A53, NECKLACE:def_5; ::_thesis: verum
end;
suppose ( x = 0 & y = 1 ) ; ::_thesis: [x,y] in the InternalRel of (Necklace 4)
hence [x,y] in the InternalRel of (Necklace 4) by ENUMSET1:def_4, NECKLA_2:2; ::_thesis: verum
end;
suppose ( x = 0 & y = 2 ) ; ::_thesis: [x,y] in the InternalRel of (Necklace 4)
then [(h . 0),(h . 2)] in the InternalRel of G /\ the InternalRel of (ComplRelStr G) by A44, A52, XBOOLE_0:def_4;
then the InternalRel of G meets the InternalRel of (ComplRelStr G) by XBOOLE_0:def_7;
hence [x,y] in the InternalRel of (Necklace 4) by Th12; ::_thesis: verum
end;
suppose ( x = 0 & y = 3 ) ; ::_thesis: [x,y] in the InternalRel of (Necklace 4)
then [(h . 0),(h . 3)] in the InternalRel of G /\ the InternalRel of (ComplRelStr G) by A30, A52, XBOOLE_0:def_4;
then the InternalRel of G meets the InternalRel of (ComplRelStr G) by XBOOLE_0:def_7;
hence [x,y] in the InternalRel of (Necklace 4) by Th12; ::_thesis: verum
end;
suppose ( x = 1 & y = 0 ) ; ::_thesis: [x,y] in the InternalRel of (Necklace 4)
hence [x,y] in the InternalRel of (Necklace 4) by ENUMSET1:def_4, NECKLA_2:2; ::_thesis: verum
end;
supposeA54: ( x = 1 & y = 1 ) ; ::_thesis: [x,y] in the InternalRel of (Necklace 4)
then h . 1 in the carrier of (ComplRelStr G) by A52, ZFMISC_1:87;
hence [x,y] in the InternalRel of (Necklace 4) by A52, A54, NECKLACE:def_5; ::_thesis: verum
end;
suppose ( x = 1 & y = 2 ) ; ::_thesis: [x,y] in the InternalRel of (Necklace 4)
hence [x,y] in the InternalRel of (Necklace 4) by ENUMSET1:def_4, NECKLA_2:2; ::_thesis: verum
end;
suppose ( x = 1 & y = 3 ) ; ::_thesis: [x,y] in the InternalRel of (Necklace 4)
then [(h . 1),(h . 3)] in the InternalRel of G /\ the InternalRel of (ComplRelStr G) by A28, A52, XBOOLE_0:def_4;
then the InternalRel of G meets the InternalRel of (ComplRelStr G) by XBOOLE_0:def_7;
hence [x,y] in the InternalRel of (Necklace 4) by Th12; ::_thesis: verum
end;
suppose ( x = 2 & y = 0 ) ; ::_thesis: [x,y] in the InternalRel of (Necklace 4)
then [(h . 2),(h . 0)] in the InternalRel of G /\ the InternalRel of (ComplRelStr G) by A43, A52, XBOOLE_0:def_4;
then the InternalRel of G meets the InternalRel of (ComplRelStr G) by XBOOLE_0:def_7;
hence [x,y] in the InternalRel of (Necklace 4) by Th12; ::_thesis: verum
end;
suppose ( x = 2 & y = 1 ) ; ::_thesis: [x,y] in the InternalRel of (Necklace 4)
hence [x,y] in the InternalRel of (Necklace 4) by ENUMSET1:def_4, NECKLA_2:2; ::_thesis: verum
end;
supposeA55: ( x = 2 & y = 2 ) ; ::_thesis: [x,y] in the InternalRel of (Necklace 4)
then h . 2 in the carrier of (ComplRelStr G) by A52, ZFMISC_1:87;
hence [x,y] in the InternalRel of (Necklace 4) by A52, A55, NECKLACE:def_5; ::_thesis: verum
end;
suppose ( x = 2 & y = 3 ) ; ::_thesis: [x,y] in the InternalRel of (Necklace 4)
hence [x,y] in the InternalRel of (Necklace 4) by ENUMSET1:def_4, NECKLA_2:2; ::_thesis: verum
end;
suppose ( x = 3 & y = 0 ) ; ::_thesis: [x,y] in the InternalRel of (Necklace 4)
then [(h . 3),(h . 0)] in the InternalRel of G /\ the InternalRel of (ComplRelStr G) by A29, A52, XBOOLE_0:def_4;
then the InternalRel of G meets the InternalRel of (ComplRelStr G) by XBOOLE_0:def_7;
hence [x,y] in the InternalRel of (Necklace 4) by Th12; ::_thesis: verum
end;
suppose ( x = 3 & y = 1 ) ; ::_thesis: [x,y] in the InternalRel of (Necklace 4)
then [(h . 3),(h . 1)] in the InternalRel of G /\ the InternalRel of (ComplRelStr G) by A27, A52, XBOOLE_0:def_4;
then the InternalRel of G meets the InternalRel of (ComplRelStr G) by XBOOLE_0:def_7;
hence [x,y] in the InternalRel of (Necklace 4) by Th12; ::_thesis: verum
end;
suppose ( x = 3 & y = 2 ) ; ::_thesis: [x,y] in the InternalRel of (Necklace 4)
hence [x,y] in the InternalRel of (Necklace 4) by ENUMSET1:def_4, NECKLA_2:2; ::_thesis: verum
end;
supposeA56: ( x = 3 & y = 3 ) ; ::_thesis: [x,y] in the InternalRel of (Necklace 4)
then h . 3 in the carrier of (ComplRelStr G) by A52, ZFMISC_1:87;
hence [x,y] in the InternalRel of (Necklace 4) by A52, A56, NECKLACE:def_5; ::_thesis: verum
end;
end;
end;
hence ComplRelStr G embeds Necklace 4 by A4, A1, A6, A8, NECKLACE:def_1; ::_thesis: verum
end;
assume ComplRelStr G embeds Necklace 4 ; ::_thesis: G embeds Necklace 4
then consider f being Function of (Necklace 4),(ComplRelStr G) such that
A57: f is V21() and
A58: for x, y being Element of (Necklace 4) holds
( [x,y] in the InternalRel of (Necklace 4) iff [(f . x),(f . y)] in the InternalRel of (ComplRelStr G) ) by NECKLACE:def_1;
consider g being Function of (Necklace 4),(ComplRelStr (Necklace 4)) such that
A59: g is isomorphic by NECKLACE:29, WAYBEL_1:def_8;
A60: 2 in the carrier of (Necklace 4) by A2, ENUMSET1:def_2;
A61: dom f = the carrier of (Necklace 4) by FUNCT_2:def_1;
A62: [(f . 0),(f . 2)] in the InternalRel of G
proof
assume A63: not [(f . 0),(f . 2)] in the InternalRel of G ; ::_thesis: contradiction
[(f . 0),(f . 2)] in the InternalRel of (ComplRelStr G)
proof
( f . 0 in the carrier of (ComplRelStr G) & f . 2 in the carrier of (ComplRelStr G) ) 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 (ComplRelStr G) 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 (ComplRelStr G) ) by XBOOLE_0:def_3;
assume not [(f . 0),(f . 2)] in the InternalRel of (ComplRelStr G) ; ::_thesis: contradiction
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; ::_thesis: verum
end;
then A65: [0,2] in the InternalRel of (Necklace 4) by A58, A3, A60;
[0,2] in the InternalRel of (ComplRelStr (Necklace 4)) by Th11, ENUMSET1:def_4;
then [0,2] in the InternalRel of (Necklace 4) /\ the InternalRel of (ComplRelStr (Necklace 4)) by A65, XBOOLE_0:def_4;
then the InternalRel of (Necklace 4) meets the InternalRel of (ComplRelStr (Necklace 4)) by XBOOLE_0:def_7;
hence contradiction by Th12; ::_thesis: verum
end;
A66: 3 in the carrier of (Necklace 4) 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 ; ::_thesis: contradiction
[(f . 0),(f . 3)] in the InternalRel of (ComplRelStr G)
proof
( f . 0 in the carrier of (ComplRelStr G) & f . 3 in the carrier of (ComplRelStr G) ) 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 (ComplRelStr G) 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 (ComplRelStr G) ) by XBOOLE_0:def_3;
assume not [(f . 0),(f . 3)] in the InternalRel of (ComplRelStr G) ; ::_thesis: contradiction
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; ::_thesis: verum
end;
then A70: [0,3] in the InternalRel of (Necklace 4) by A58, A3, A66;
[0,3] in the InternalRel of (ComplRelStr (Necklace 4)) by Th11, ENUMSET1:def_4;
then [0,3] in the InternalRel of (Necklace 4) /\ the InternalRel of (ComplRelStr (Necklace 4)) by A70, XBOOLE_0:def_4;
then the InternalRel of (Necklace 4) meets the InternalRel of (ComplRelStr (Necklace 4)) by XBOOLE_0:def_7;
hence contradiction by Th12; ::_thesis: verum
end;
A71: 1 in the carrier of (Necklace 4) 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 ; ::_thesis: contradiction
[(f . 1),(f . 3)] in the InternalRel of (ComplRelStr G)
proof
( f . 1 in the carrier of (ComplRelStr G) & f . 3 in the carrier of (ComplRelStr G) ) 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 (ComplRelStr G) 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 (ComplRelStr G) ) by XBOOLE_0:def_3;
assume not [(f . 1),(f . 3)] in the InternalRel of (ComplRelStr G) ; ::_thesis: contradiction
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; ::_thesis: verum
end;
then A75: [1,3] in the InternalRel of (Necklace 4) by A58, A71, A66;
[1,3] in the InternalRel of (ComplRelStr (Necklace 4)) by Th11, ENUMSET1:def_4;
then [1,3] in the InternalRel of (Necklace 4) /\ the InternalRel of (ComplRelStr (Necklace 4)) by A75, XBOOLE_0:def_4;
then the InternalRel of (Necklace 4) meets the InternalRel of (ComplRelStr (Necklace 4)) by XBOOLE_0:def_7;
hence contradiction by Th12; ::_thesis: verum
end;
[3,2] in the InternalRel of (Necklace 4) by ENUMSET1:def_4, NECKLA_2:2;
then A76: [(f . 3),(f . 2)] in the InternalRel of (ComplRelStr G) by A58, A60, A66;
[2,3] in the InternalRel of (Necklace 4) by ENUMSET1:def_4, NECKLA_2:2;
then A77: [(f . 2),(f . 3)] in the InternalRel of (ComplRelStr G) by A58, A60, A66;
[1,2] in the InternalRel of (Necklace 4) by ENUMSET1:def_4, NECKLA_2:2;
then A78: [(f . 1),(f . 2)] in the InternalRel of (ComplRelStr G) by A58, A71, A60;
[1,0] in the InternalRel of (Necklace 4) by ENUMSET1:def_4, NECKLA_2:2;
then A79: [(f . 1),(f . 0)] in the InternalRel of (ComplRelStr G) 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 ; ::_thesis: contradiction
[(f . 2),(f . 0)] in the InternalRel of (ComplRelStr G)
proof
( f . 0 in the carrier of (ComplRelStr G) & f . 2 in the carrier of (ComplRelStr G) ) 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 (ComplRelStr G) 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 (ComplRelStr G) ) by XBOOLE_0:def_3;
assume not [(f . 2),(f . 0)] in the InternalRel of (ComplRelStr G) ; ::_thesis: contradiction
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; ::_thesis: verum
end;
then A83: [2,0] in the InternalRel of (Necklace 4) by A58, A3, A60;
[2,0] in the InternalRel of (ComplRelStr (Necklace 4)) by Th11, ENUMSET1:def_4;
then [2,0] in the InternalRel of (Necklace 4) /\ the InternalRel of (ComplRelStr (Necklace 4)) by A83, XBOOLE_0:def_4;
then the InternalRel of (Necklace 4) meets the InternalRel of (ComplRelStr (Necklace 4)) by XBOOLE_0:def_7;
hence contradiction by Th12; ::_thesis: verum
end;
A84: [(f . 3),(f . 0)] in the InternalRel of G
proof
assume A85: not [(f . 3),(f . 0)] in the InternalRel of G ; ::_thesis: contradiction
[(f . 3),(f . 0)] in the InternalRel of (ComplRelStr G)
proof
( f . 0 in the carrier of (ComplRelStr G) & f . 3 in the carrier of (ComplRelStr G) ) 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 (ComplRelStr G) 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 (ComplRelStr G) ) by XBOOLE_0:def_3;
assume not [(f . 3),(f . 0)] in the InternalRel of (ComplRelStr G) ; ::_thesis: contradiction
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; ::_thesis: verum
end;
then A87: [3,0] in the InternalRel of (Necklace 4) by A58, A3, A66;
[3,0] in the InternalRel of (ComplRelStr (Necklace 4)) by Th11, ENUMSET1:def_4;
then [3,0] in the InternalRel of (Necklace 4) /\ the InternalRel of (ComplRelStr (Necklace 4)) by A87, XBOOLE_0:def_4;
then the InternalRel of (Necklace 4) meets the InternalRel of (ComplRelStr (Necklace 4)) by XBOOLE_0:def_7;
hence contradiction by Th12; ::_thesis: verum
end;
A88: [(f . 3),(f . 1)] in the InternalRel of G
proof
assume A89: not [(f . 3),(f . 1)] in the InternalRel of G ; ::_thesis: contradiction
[(f . 3),(f . 1)] in the InternalRel of (ComplRelStr G)
proof
( f . 1 in the carrier of (ComplRelStr G) & f . 3 in the carrier of (ComplRelStr G) ) 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 (ComplRelStr G) 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 (ComplRelStr G) ) by XBOOLE_0:def_3;
assume not [(f . 3),(f . 1)] in the InternalRel of (ComplRelStr G) ; ::_thesis: contradiction
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; ::_thesis: verum
end;
then A91: [3,1] in the InternalRel of (Necklace 4) by A58, A71, A66;
[3,1] in the InternalRel of (ComplRelStr (Necklace 4)) by Th11, ENUMSET1:def_4;
then [3,1] in the InternalRel of (Necklace 4) /\ the InternalRel of (ComplRelStr (Necklace 4)) by A91, XBOOLE_0:def_4;
then the InternalRel of (Necklace 4) meets the InternalRel of (ComplRelStr (Necklace 4)) by XBOOLE_0:def_7;
hence contradiction by Th12; ::_thesis: verum
end;
[2,1] in the InternalRel of (Necklace 4) by ENUMSET1:def_4, NECKLA_2:2;
then A92: [(f . 2),(f . 1)] in the InternalRel of (ComplRelStr G) by A58, A71, A60;
[0,1] in the InternalRel of (Necklace 4) by ENUMSET1:def_4, NECKLA_2:2;
then A93: [(f . 0),(f . 1)] in the InternalRel of (ComplRelStr G) by A58, A3, A71;
A94: for x, y being Element of (ComplRelStr (Necklace 4)) holds
( [x,y] in the InternalRel of (ComplRelStr (Necklace 4)) iff [(f . x),(f . y)] in the InternalRel of G )
proof
let x, y be Element of (ComplRelStr (Necklace 4)); ::_thesis: ( [x,y] in the InternalRel of (ComplRelStr (Necklace 4)) iff [(f . x),(f . y)] in the InternalRel of G )
A95: the carrier of (Necklace 4) = the carrier of (ComplRelStr (Necklace 4)) by NECKLACE:def_8;
thus ( [x,y] in the InternalRel of (ComplRelStr (Necklace 4)) implies [(f . x),(f . y)] in the InternalRel of G ) ::_thesis: ( [(f . x),(f . y)] in the InternalRel of G implies [x,y] in the InternalRel of (ComplRelStr (Necklace 4)) )
proof
assume A96: [x,y] in the InternalRel of (ComplRelStr (Necklace 4)) ; ::_thesis: [(f . x),(f . y)] in the InternalRel of G
percases ( [x,y] = [0,2] or [x,y] = [2,0] or [x,y] = [0,3] or [x,y] = [3,0] or [x,y] = [1,3] or [x,y] = [3,1] ) by A96, Th11, ENUMSET1:def_4;
supposeA97: [x,y] = [0,2] ; ::_thesis: [(f . x),(f . y)] in the InternalRel of G
then x = 0 by XTUPLE_0:1;
hence [(f . x),(f . y)] in the InternalRel of G by A62, A97, XTUPLE_0:1; ::_thesis: verum
end;
supposeA98: [x,y] = [2,0] ; ::_thesis: [(f . x),(f . y)] in the InternalRel of G
then x = 2 by XTUPLE_0:1;
hence [(f . x),(f . y)] in the InternalRel of G by A80, A98, XTUPLE_0:1; ::_thesis: verum
end;
supposeA99: [x,y] = [0,3] ; ::_thesis: [(f . x),(f . y)] in the InternalRel of G
then x = 0 by XTUPLE_0:1;
hence [(f . x),(f . y)] in the InternalRel of G by A67, A99, XTUPLE_0:1; ::_thesis: verum
end;
supposeA100: [x,y] = [3,0] ; ::_thesis: [(f . x),(f . y)] in the InternalRel of G
then x = 3 by XTUPLE_0:1;
hence [(f . x),(f . y)] in the InternalRel of G by A84, A100, XTUPLE_0:1; ::_thesis: verum
end;
supposeA101: [x,y] = [1,3] ; ::_thesis: [(f . x),(f . y)] in the InternalRel of G
then x = 1 by XTUPLE_0:1;
hence [(f . x),(f . y)] in the InternalRel of G by A72, A101, XTUPLE_0:1; ::_thesis: verum
end;
supposeA102: [x,y] = [3,1] ; ::_thesis: [(f . x),(f . y)] in the InternalRel of G
then x = 3 by XTUPLE_0:1;
hence [(f . x),(f . y)] in the InternalRel of G by A88, A102, XTUPLE_0:1; ::_thesis: verum
end;
end;
end;
assume A103: [(f . x),(f . y)] in the InternalRel of G ; ::_thesis: [x,y] in the InternalRel of (ComplRelStr (Necklace 4))
percases ( ( x = 0 & y = 0 ) or ( x = 0 & y = 1 ) or ( x = 0 & y = 2 ) or ( x = 0 & y = 3 ) or ( x = 1 & y = 0 ) or ( x = 2 & y = 0 ) or ( x = 3 & y = 0 ) or ( x = 1 & y = 1 ) or ( x = 1 & y = 2 ) or ( x = 1 & y = 3 ) or ( x = 2 & y = 1 ) or ( x = 2 & y = 2 ) or ( x = 2 & y = 3 ) or ( x = 3 & y = 1 ) or ( x = 3 & y = 2 ) or ( x = 3 & y = 3 ) ) by A2, A95, ENUMSET1:def_2;
supposeA104: ( x = 0 & y = 0 ) ; ::_thesis: [x,y] in the InternalRel of (ComplRelStr (Necklace 4))
then f . 0 in the carrier of G by A103, ZFMISC_1:87;
hence [x,y] in the InternalRel of (ComplRelStr (Necklace 4)) by A103, A104, NECKLACE:def_5; ::_thesis: verum
end;
suppose ( x = 0 & y = 1 ) ; ::_thesis: [x,y] in the InternalRel of (ComplRelStr (Necklace 4))
then [(f . 0),(f . 1)] in the InternalRel of G /\ the InternalRel of (ComplRelStr G) by A93, A103, XBOOLE_0:def_4;
then the InternalRel of G meets the InternalRel of (ComplRelStr G) by XBOOLE_0:def_7;
hence [x,y] in the InternalRel of (ComplRelStr (Necklace 4)) by Th12; ::_thesis: verum
end;
suppose ( x = 0 & y = 2 ) ; ::_thesis: [x,y] in the InternalRel of (ComplRelStr (Necklace 4))
hence [x,y] in the InternalRel of (ComplRelStr (Necklace 4)) by Th11, ENUMSET1:def_4; ::_thesis: verum
end;
suppose ( x = 0 & y = 3 ) ; ::_thesis: [x,y] in the InternalRel of (ComplRelStr (Necklace 4))
hence [x,y] in the InternalRel of (ComplRelStr (Necklace 4)) by Th11, ENUMSET1:def_4; ::_thesis: verum
end;
suppose ( x = 1 & y = 0 ) ; ::_thesis: [x,y] in the InternalRel of (ComplRelStr (Necklace 4))
then [(f . 1),(f . 0)] in the InternalRel of G /\ the InternalRel of (ComplRelStr G) by A79, A103, XBOOLE_0:def_4;
then the InternalRel of G meets the InternalRel of (ComplRelStr G) by XBOOLE_0:def_7;
hence [x,y] in the InternalRel of (ComplRelStr (Necklace 4)) by Th12; ::_thesis: verum
end;
suppose ( x = 2 & y = 0 ) ; ::_thesis: [x,y] in the InternalRel of (ComplRelStr (Necklace 4))
hence [x,y] in the InternalRel of (ComplRelStr (Necklace 4)) by Th11, ENUMSET1:def_4; ::_thesis: verum
end;
suppose ( x = 3 & y = 0 ) ; ::_thesis: [x,y] in the InternalRel of (ComplRelStr (Necklace 4))
hence [x,y] in the InternalRel of (ComplRelStr (Necklace 4)) by Th11, ENUMSET1:def_4; ::_thesis: verum
end;
supposeA105: ( x = 1 & y = 1 ) ; ::_thesis: [x,y] in the InternalRel of (ComplRelStr (Necklace 4))
then f . 1 in the carrier of G by A103, ZFMISC_1:87;
hence [x,y] in the InternalRel of (ComplRelStr (Necklace 4)) by A103, A105, NECKLACE:def_5; ::_thesis: verum
end;
suppose ( x = 1 & y = 2 ) ; ::_thesis: [x,y] in the InternalRel of (ComplRelStr (Necklace 4))
then [(f . 1),(f . 2)] in the InternalRel of G /\ the InternalRel of (ComplRelStr G) by A78, A103, XBOOLE_0:def_4;
then the InternalRel of G meets the InternalRel of (ComplRelStr G) by XBOOLE_0:def_7;
hence [x,y] in the InternalRel of (ComplRelStr (Necklace 4)) by Th12; ::_thesis: verum
end;
suppose ( x = 1 & y = 3 ) ; ::_thesis: [x,y] in the InternalRel of (ComplRelStr (Necklace 4))
hence [x,y] in the InternalRel of (ComplRelStr (Necklace 4)) by Th11, ENUMSET1:def_4; ::_thesis: verum
end;
suppose ( x = 2 & y = 1 ) ; ::_thesis: [x,y] in the InternalRel of (ComplRelStr (Necklace 4))
then [(f . 2),(f . 1)] in the InternalRel of G /\ the InternalRel of (ComplRelStr G) by A92, A103, XBOOLE_0:def_4;
then the InternalRel of G meets the InternalRel of (ComplRelStr G) by XBOOLE_0:def_7;
hence [x,y] in the InternalRel of (ComplRelStr (Necklace 4)) by Th12; ::_thesis: verum
end;
supposeA106: ( x = 2 & y = 2 ) ; ::_thesis: [x,y] in the InternalRel of (ComplRelStr (Necklace 4))
then f . 2 in the carrier of G by A103, ZFMISC_1:87;
hence [x,y] in the InternalRel of (ComplRelStr (Necklace 4)) by A103, A106, NECKLACE:def_5; ::_thesis: verum
end;
suppose ( x = 2 & y = 3 ) ; ::_thesis: [x,y] in the InternalRel of (ComplRelStr (Necklace 4))
then [(f . 2),(f . 3)] in the InternalRel of G /\ the InternalRel of (ComplRelStr G) by A77, A103, XBOOLE_0:def_4;
then the InternalRel of G meets the InternalRel of (ComplRelStr G) by XBOOLE_0:def_7;
hence [x,y] in the InternalRel of (ComplRelStr (Necklace 4)) by Th12; ::_thesis: verum
end;
suppose ( x = 3 & y = 1 ) ; ::_thesis: [x,y] in the InternalRel of (ComplRelStr (Necklace 4))
hence [x,y] in the InternalRel of (ComplRelStr (Necklace 4)) by Th11, ENUMSET1:def_4; ::_thesis: verum
end;
suppose ( x = 3 & y = 2 ) ; ::_thesis: [x,y] in the InternalRel of (ComplRelStr (Necklace 4))
then [(f . 3),(f . 2)] in the InternalRel of G /\ the InternalRel of (ComplRelStr G) by A76, A103, XBOOLE_0:def_4;
then the InternalRel of G meets the InternalRel of (ComplRelStr G) by XBOOLE_0:def_7;
hence [x,y] in the InternalRel of (ComplRelStr (Necklace 4)) by Th12; ::_thesis: verum
end;
supposeA107: ( x = 3 & y = 3 ) ; ::_thesis: [x,y] in the InternalRel of (ComplRelStr (Necklace 4))
then f . 3 in the carrier of G by A103, ZFMISC_1:87;
hence [x,y] in the InternalRel of (ComplRelStr (Necklace 4)) by A103, A107, NECKLACE:def_5; ::_thesis: verum
end;
end;
end;
reconsider f = f as Function of (ComplRelStr (Necklace 4)),G by A4, NECKLACE:def_8;
reconsider h = f * g as Function of (Necklace 4),G ;
A108: ( g is V21() & g is monotone ) by A59, WAYBEL_0:def_38;
for x, y being Element of (Necklace 4) holds
( [x,y] in the InternalRel of (Necklace 4) iff [(h . x),(h . y)] in the InternalRel of G )
proof
let x, y be Element of (Necklace 4); ::_thesis: ( [x,y] in the InternalRel of (Necklace 4) iff [(h . x),(h . y)] in the InternalRel of G )
thus ( [x,y] in the InternalRel of (Necklace 4) implies [(h . x),(h . y)] in the InternalRel of G ) ::_thesis: ( [(h . x),(h . y)] in the InternalRel of G implies [x,y] in the InternalRel of (Necklace 4) )
proof
assume [x,y] in the InternalRel of (Necklace 4) ; ::_thesis: [(h . x),(h . y)] in the InternalRel of G
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 (ComplRelStr (Necklace 4)) 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 [(h . x),(h . y)] in the InternalRel of G by FUNCT_2:15; ::_thesis: verum
end;
assume [(h . x),(h . y)] in the InternalRel of G ; ::_thesis: [x,y] in the InternalRel of (Necklace 4)
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 (ComplRelStr (Necklace 4)) by A94;
then g . x <= g . y by ORDERS_2:def_5;
then x <= y by A59, WAYBEL_0:66;
hence [x,y] in the InternalRel of (Necklace 4) by ORDERS_2:def_5; ::_thesis: verum
end;
hence G embeds Necklace 4 by A57, A108, NECKLACE:def_1; ::_thesis: verum
end;
theorem Th25: :: NECKLA_3:25
for G being non empty irreflexive RelStr holds
( G is N-free iff ComplRelStr G is N-free )
proof
let G be non empty irreflexive RelStr ; ::_thesis: ( G is N-free iff ComplRelStr G is N-free )
thus ( G is N-free implies ComplRelStr G is N-free ) ::_thesis: ( ComplRelStr G is N-free implies G is N-free )
proof
assume A1: G is N-free ; ::_thesis: ComplRelStr G is N-free
assume not ComplRelStr G is N-free ; ::_thesis: contradiction
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; ::_thesis: verum
end;
assume A2: ComplRelStr G is N-free ; ::_thesis: G is N-free
assume not G is N-free ; ::_thesis: contradiction
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; ::_thesis: verum
end;
begin
definition
let R be RelStr ;
mode path of R is RedSequence of the InternalRel of R;
end;
definition
let R be RelStr ;
attrR is path-connected means :Def1: :: NECKLA_3:def 1
for x, y being set st x in the carrier of R & y in the carrier of R & x <> y & not the InternalRel of R reduces x,y holds
the InternalRel of R reduces y,x;
end;
:: deftheorem Def1 defines path-connected NECKLA_3:def_1_:_
for R being RelStr holds
( R is path-connected iff for x, y being set st x in the carrier of R & y in the carrier of R & x <> y & not the InternalRel of R reduces x,y holds
the InternalRel of R reduces y,x );
registration
cluster empty -> path-connected for RelStr ;
correctness
coherence
for b1 being RelStr st b1 is empty holds
b1 is path-connected ;
proof
let R be RelStr ; ::_thesis: ( R is empty implies R is path-connected )
assume A1: R is empty ; ::_thesis: R is path-connected
let x, y be set ; :: according to NECKLA_3:def_1 ::_thesis: ( x in the carrier of R & y in the carrier of R & x <> y & not the InternalRel of R reduces x,y implies the InternalRel of R reduces y,x )
assume that
A2: x in the carrier of R and
y in the carrier of R and
x <> y ; ::_thesis: ( the InternalRel of R reduces x,y or the InternalRel of R reduces y,x )
thus ( the InternalRel of R reduces x,y or the InternalRel of R reduces y,x ) by A1, A2; ::_thesis: verum
end;
end;
registration
cluster non empty connected -> non empty path-connected for RelStr ;
correctness
coherence
for b1 being non empty RelStr st b1 is connected holds
b1 is path-connected ;
proof
let R be non empty RelStr ; ::_thesis: ( R is connected implies R is path-connected )
set cR = the carrier of R;
set IR = the InternalRel of R;
assume A1: R is connected ; ::_thesis: R is path-connected
for x, y being set st x in the carrier of R & y in the carrier of R & x <> y & not the InternalRel of R reduces x,y holds
the InternalRel of R reduces y,x
proof
let x, y be set ; ::_thesis: ( x in the carrier of R & y in the carrier of R & x <> y & not the InternalRel of R reduces x,y implies the InternalRel of R reduces y,x )
assume that
A2: ( x in the carrier of R & y in the carrier of R ) and
x <> y ; ::_thesis: ( the InternalRel of R reduces x,y or the InternalRel of R reduces y,x )
reconsider x = x, y = y as Element of R by A2;
A3: ( x <= y or y <= x ) by A1, WAYBEL_0:def_29;
percases ( [x,y] in the InternalRel of R or [y,x] in the InternalRel of R ) by A3, ORDERS_2:def_5;
supposeA4: [x,y] in the InternalRel of R ; ::_thesis: ( the InternalRel of R reduces x,y or the InternalRel of R reduces y,x )
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 the InternalRel of R by A4, REWRITE1:7;
hence ( the InternalRel of R reduces x,y or the InternalRel of R reduces y,x ) by A5, A6, REWRITE1:def_3; ::_thesis: verum
end;
supposeA7: [y,x] in the InternalRel of R ; ::_thesis: ( the InternalRel of R reduces x,y or the InternalRel of R reduces y,x )
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 the InternalRel of R by A7, REWRITE1:7;
hence ( the InternalRel of R reduces x,y or the InternalRel of R reduces y,x ) by A8, A9, REWRITE1:def_3; ::_thesis: verum
end;
end;
end;
hence R is path-connected by Def1; ::_thesis: verum
end;
end;
theorem Th26: :: NECKLA_3:26
for R being non empty reflexive transitive RelStr
for x, y being Element of R st the InternalRel of R reduces x,y holds
[x,y] in the InternalRel of R
proof
let R be non empty reflexive transitive RelStr ; ::_thesis: for x, y being Element of R st the InternalRel of R reduces x,y holds
[x,y] in the InternalRel of R
let x, y be Element of R; ::_thesis: ( the InternalRel of R reduces x,y implies [x,y] in the InternalRel of R )
set cR = the carrier of R;
set IR = the InternalRel of R;
assume the InternalRel of R reduces x,y ; ::_thesis: [x,y] in the InternalRel of R
then consider p being RedSequence of the InternalRel of R such that
A1: p . 1 = x and
A2: p . (len p) = y by REWRITE1:def_3;
reconsider p = p as FinSequence ;
defpred S1[ Nat] means ( $1 in dom p implies [(p . 1),(p . $1)] in the InternalRel of R );
A3: the InternalRel of R is_transitive_in the carrier of R by ORDERS_2:def_3;
A4: for k being non empty Nat st S1[k] holds
S1[k + 1]
proof
let k be non empty Nat; ::_thesis: ( S1[k] implies S1[k + 1] )
assume A5: S1[k] ; ::_thesis: S1[k + 1]
assume A6: k + 1 in dom p ; ::_thesis: [(p . 1),(p . (k + 1))] in the InternalRel of R
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 the carrier of R 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 the InternalRel of R by A6, REWRITE1:def_2;
then ( p . k in the carrier of R & p . (k + 1) in the carrier of R ) by ZFMISC_1:87;
hence [(p . 1),(p . (k + 1))] in the InternalRel of R by A3, A5, A7, A9, A8, FINSEQ_3:25, RELAT_2:def_8; ::_thesis: verum
end;
the InternalRel of R is_reflexive_in the carrier of R by ORDERS_2:def_2;
then A10: S1[1] by A1, RELAT_2:def_1;
A11: for k being non empty Nat holds S1[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 [x,y] in the InternalRel of R by A1, A2, A12, A11; ::_thesis: verum
end;
registration
cluster non empty reflexive transitive path-connected -> non empty reflexive transitive connected for RelStr ;
correctness
coherence
for b1 being non empty reflexive transitive RelStr st b1 is path-connected holds
b1 is connected ;
proof
let R be non empty reflexive transitive RelStr ; ::_thesis: ( R is path-connected implies R is connected )
set IR = the InternalRel of R;
assume A1: R is path-connected ; ::_thesis: R is connected
for x, y being Element of R holds
( x <= y or y <= x )
proof
let x, y be Element of R; ::_thesis: ( x <= y or y <= x )
percases ( x = y or x <> y ) ;
suppose x = y ; ::_thesis: ( x <= y or y <= x )
hence ( x <= y or y <= x ) ; ::_thesis: verum
end;
suppose x <> y ; ::_thesis: ( x <= y or y <= x )
then ( the InternalRel of R reduces x,y or the InternalRel of R reduces y,x ) by A1, Def1;
then ( [x,y] in the InternalRel of R or [y,x] in the InternalRel of R ) by Th26;
hence ( x <= y or y <= x ) by ORDERS_2:def_5; ::_thesis: verum
end;
end;
end;
hence R is connected by WAYBEL_0:def_29; ::_thesis: verum
end;
end;
theorem Th27: :: NECKLA_3:27
for R being symmetric RelStr
for x, y being set st the InternalRel of R reduces x,y holds
the InternalRel of R reduces y,x
proof
let R be symmetric RelStr ; ::_thesis: for x, y being set st the InternalRel of R reduces x,y holds
the InternalRel of R reduces y,x
set IR = the InternalRel of R;
let x, y be set ; ::_thesis: ( the InternalRel of R reduces x,y implies the InternalRel of R reduces y,x )
A1: the InternalRel of R = the InternalRel of R ~ by RELAT_2:13;
assume the InternalRel of R reduces x,y ; ::_thesis: the InternalRel of R reduces y,x
then consider p being RedSequence of the InternalRel of R such that
A2: p . 1 = x and
A3: p . (len p) = y by REWRITE1:def_3;
reconsider p = p as FinSequence ;
A4: (Rev p) . (len p) = x by A2, FINSEQ_5:62;
the InternalRel of R reduces y,x
proof
reconsider q = Rev p as RedSequence of the InternalRel of R by A1, REWRITE1:9;
( q . 1 = y & q . (len q) = x ) by A3, A4, FINSEQ_5:62, FINSEQ_5:def_3;
hence the InternalRel of R reduces y,x by REWRITE1:def_3; ::_thesis: verum
end;
hence the InternalRel of R reduces y,x ; ::_thesis: verum
end;
definition
let R be symmetric RelStr ;
redefine attr R is path-connected means :Def2: :: NECKLA_3:def 2
for x, y being 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
( R is path-connected iff for x, y being 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
set IR = the InternalRel of R;
set cR = the carrier of R;
thus ( R is path-connected implies for x, y being set st x in the carrier of R & y in the carrier of R & x <> y holds
the InternalRel of R reduces x,y ) ::_thesis: ( ( for x, y being set st x in the carrier of R & y in the carrier of R & x <> y holds
the InternalRel of R reduces x,y ) implies R is path-connected )
proof
assume A1: R is path-connected ; ::_thesis: for x, y being set st x in the carrier of R & y in the carrier of R & x <> y holds
the InternalRel of R reduces x,y
let x, y be set ; ::_thesis: ( x in the carrier of R & y in the carrier of R & x <> y implies the InternalRel of R reduces x,y )
assume A2: ( x in the carrier of R & y in the carrier of R & x <> y ) ; ::_thesis: the InternalRel of R reduces x,y
percases ( the InternalRel of R reduces x,y or the InternalRel of R reduces y,x ) by A1, A2, Def1;
suppose the InternalRel of R reduces x,y ; ::_thesis: the InternalRel of R reduces x,y
hence the InternalRel of R reduces x,y ; ::_thesis: verum
end;
suppose the InternalRel of R reduces y,x ; ::_thesis: the InternalRel of R reduces x,y
hence the InternalRel of R reduces x,y by Th27; ::_thesis: verum
end;
end;
end;
assume for x, y being set st x in the carrier of R & y in the carrier of R & x <> y holds
the InternalRel of R reduces x,y ; ::_thesis: R is path-connected
then for x, y being set st x in the carrier of R & y in the carrier of R & x <> y & not the InternalRel of R reduces x,y holds
the InternalRel of R reduces y,x ;
hence R is path-connected by Def1; ::_thesis: verum
end;
end;
:: deftheorem Def2 defines path-connected NECKLA_3:def_2_:_
for R being symmetric RelStr holds
( R is path-connected iff for x, y being set st x in the carrier of R & y in the carrier of R & x <> y holds
the InternalRel of R reduces x,y );
definition
let R be RelStr ;
let x be Element of R;
func component x -> Subset of R equals :: NECKLA_3:def 3
Class ((EqCl the InternalRel of R),x);
coherence
Class ((EqCl the InternalRel of R),x) is Subset of R ;
end;
:: deftheorem defines component NECKLA_3:def_3_:_
for R being RelStr
for x being Element of R holds component x = Class ((EqCl the InternalRel of R),x);
registration
let R be non empty RelStr ;
let x be Element of R;
cluster component x -> non empty ;
correctness
coherence
not component x is empty ;
by EQREL_1:20;
end;
theorem Th28: :: NECKLA_3:28
for R being RelStr
for x being Element of R
for y being set st y in component x holds
[x,y] in EqCl the InternalRel of R
proof
let R be RelStr ; ::_thesis: for x being Element of R
for y being set st y in component x holds
[x,y] in EqCl the InternalRel of R
let x be Element of R; ::_thesis: for y being set st y in component x holds
[x,y] in EqCl the InternalRel of R
let y be set ; ::_thesis: ( y in component x implies [x,y] in EqCl the InternalRel of R )
set IR = the InternalRel of R;
assume y in component x ; ::_thesis: [x,y] in EqCl the InternalRel of R
then [y,x] in EqCl the InternalRel of R by EQREL_1:19;
hence [x,y] in EqCl the InternalRel of R by EQREL_1:6; ::_thesis: verum
end;
theorem Th29: :: NECKLA_3:29
for R being RelStr
for x being Element of R
for A being set holds
( A = component x iff for y being set holds
( y in A iff [x,y] in EqCl the InternalRel of R ) )
proof
let R be RelStr ; ::_thesis: for x being Element of R
for A being set holds
( A = component x iff for y being set holds
( y in A iff [x,y] in EqCl the InternalRel of R ) )
let x be Element of R; ::_thesis: for A being set holds
( A = component x iff for y being set holds
( y in A iff [x,y] in EqCl the InternalRel of R ) )
let A be set ; ::_thesis: ( A = component x iff for y being set holds
( y in A iff [x,y] in EqCl the InternalRel of R ) )
set IR = the InternalRel of R;
A1: ( ( for y being set holds
( y in A iff [x,y] in EqCl the InternalRel of R ) ) implies A = component x )
proof
assume A2: for y being set holds
( y in A iff [x,y] in EqCl the InternalRel of R ) ; ::_thesis: A = component x
A3: component x c= A
proof
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in component x or a in A )
assume a in component x ; ::_thesis: a in A
then [a,x] in EqCl the InternalRel of R by EQREL_1:19;
then [x,a] in EqCl the InternalRel of R by EQREL_1:6;
hence a in A by A2; ::_thesis: verum
end;
A c= component x
proof
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in A or a in component x )
assume a in A ; ::_thesis: a in component x
then [x,a] in EqCl the InternalRel of R by A2;
then [a,x] in EqCl the InternalRel of R by EQREL_1:6;
hence a in component x by EQREL_1:19; ::_thesis: verum
end;
hence A = component x by A3, XBOOLE_0:def_10; ::_thesis: verum
end;
( A = component x implies for y being set st [x,y] in EqCl the InternalRel of R holds
y in A )
proof
assume A4: A = component x ; ::_thesis: for y being set st [x,y] in EqCl the InternalRel of R holds
y in A
let y be set ; ::_thesis: ( [x,y] in EqCl the InternalRel of R implies y in A )
assume [x,y] in EqCl the InternalRel of R ; ::_thesis: y in A
then [y,x] in EqCl the InternalRel of R by EQREL_1:6;
hence y in A by A4, EQREL_1:19; ::_thesis: verum
end;
hence ( A = component x iff for y being set holds
( y in A iff [x,y] in EqCl the InternalRel of R ) ) by A1, Th28; ::_thesis: verum
end;
theorem Th30: :: NECKLA_3:30
for R being non empty symmetric irreflexive RelStr st not R is path-connected holds
ex G1, G2 being non empty strict symmetric irreflexive RelStr st
( the carrier of G1 misses the carrier of G2 & RelStr(# the carrier of R, the InternalRel of R #) = union_of (G1,G2) )
proof
let R be non empty symmetric irreflexive RelStr ; ::_thesis: ( not R is path-connected implies ex G1, G2 being non empty strict symmetric irreflexive RelStr st
( the carrier of G1 misses the carrier of G2 & RelStr(# the carrier of R, the InternalRel of R #) = union_of (G1,G2) ) )
set cR = the carrier of R;
set IR = the InternalRel of R;
assume not R is path-connected ; ::_thesis: ex G1, G2 being non empty strict symmetric irreflexive RelStr st
( the carrier of G1 misses the carrier of G2 & RelStr(# the carrier of R, the InternalRel of R #) = union_of (G1,G2) )
then consider x, y being set such that
A1: ( x in the carrier of R & y in the carrier of R ) and
x <> y and
A2: not the InternalRel of R reduces x,y by Def2;
reconsider x = x, y = y as Element of R by A1;
set A1 = component x;
set A2 = the carrier of R \ (component x);
reconsider A2 = the carrier of R \ (component x) as Subset of R ;
set G1 = subrelstr (component x);
set G2 = subrelstr A2;
A3: the carrier of (subrelstr A2) = A2 by YELLOW_0:def_15;
the carrier of R c= (component x) \/ A2
proof
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in the carrier of R or a in (component x) \/ A2 )
assume A4: a in the carrier of R ; ::_thesis: a in (component x) \/ A2
assume not a in (component x) \/ A2 ; ::_thesis: contradiction
then ( not a in component x & not a in A2 ) by XBOOLE_0:def_3;
hence contradiction by A4, XBOOLE_0:def_5; ::_thesis: verum
end;
then A5: the carrier of R = (component x) \/ A2 by XBOOLE_0:def_10;
A6: the carrier of (subrelstr (component x)) = component x by YELLOW_0:def_15;
then A7: the carrier of (subrelstr (component x)) misses the carrier of (subrelstr A2) by A3, XBOOLE_1:79;
A8: the InternalRel of (subrelstr (component x)) misses the InternalRel of (subrelstr A2)
proof
set IG1 = the InternalRel of (subrelstr (component x));
set IG2 = the InternalRel of (subrelstr A2);
assume not the InternalRel of (subrelstr (component x)) misses the InternalRel of (subrelstr A2) ; ::_thesis: contradiction
then the InternalRel of (subrelstr (component x)) /\ the InternalRel of (subrelstr A2) <> {} by XBOOLE_0:def_7;
then consider a being set such that
A9: a in the InternalRel of (subrelstr (component x)) /\ the InternalRel of (subrelstr A2) by XBOOLE_0:def_1;
a in the InternalRel of (subrelstr (component x)) by A9, XBOOLE_0:def_4;
then consider c1, c2 being set such that
A10: a = [c1,c2] and
A11: c1 in component x and
c2 in component x by A6, RELSET_1:2;
ex g1, g2 being set 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 (component x) /\ A2 by A11, XBOOLE_0:def_4;
hence contradiction by A6, A3, A7, XBOOLE_0:def_7; ::_thesis: verum
end;
A12: the InternalRel of (subrelstr A2) = the InternalRel of R \ the InternalRel of (subrelstr (component x))
proof
set IG1 = the InternalRel of (subrelstr (component x));
set IG2 = the InternalRel of (subrelstr A2);
thus the InternalRel of (subrelstr A2) c= the InternalRel of R \ the InternalRel of (subrelstr (component x)) :: according to XBOOLE_0:def_10 ::_thesis: the InternalRel of R \ the InternalRel of (subrelstr (component x)) c= the InternalRel of (subrelstr A2)
proof
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in the InternalRel of (subrelstr A2) or a in the InternalRel of R \ the InternalRel of (subrelstr (component x)) )
assume A13: a in the InternalRel of (subrelstr A2) ; ::_thesis: a in the InternalRel of R \ the InternalRel of (subrelstr (component x))
then consider g1, g2 being set such that
A14: a = [g1,g2] and
A15: ( g1 in A2 & g2 in A2 ) by A3, RELSET_1:2;
reconsider g1 = g1, g2 = g2 as Element of (subrelstr A2) by A15, YELLOW_0:def_15;
reconsider u1 = g1, u2 = g2 as Element of R by A15;
A16: not a in the InternalRel of (subrelstr (component x))
proof
assume a in the InternalRel of (subrelstr (component x)) ; ::_thesis: contradiction
then a in the InternalRel of (subrelstr (component x)) /\ the InternalRel of (subrelstr A2) by A13, XBOOLE_0:def_4;
hence contradiction by A8, XBOOLE_0:def_7; ::_thesis: verum
end;
g1 <= g2 by A13, A14, ORDERS_2:def_5;
then u1 <= u2 by YELLOW_0:59;
then a in the InternalRel of R by A14, ORDERS_2:def_5;
hence a in the InternalRel of R \ the InternalRel of (subrelstr (component x)) by A16, XBOOLE_0:def_5; ::_thesis: verum
end;
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in the InternalRel of R \ the InternalRel of (subrelstr (component x)) or a in the InternalRel of (subrelstr A2) )
assume A17: a in the InternalRel of R \ the InternalRel of (subrelstr (component x)) ; ::_thesis: a in the InternalRel of (subrelstr A2)
then A18: a in the InternalRel of R by XBOOLE_0:def_5;
A19: not a in the InternalRel of (subrelstr (component x)) by A17, XBOOLE_0:def_5;
consider c1, c2 being set such that
A20: a = [c1,c2] and
A21: ( c1 in the carrier of R & c2 in the carrier of R ) by A17, RELSET_1:2;
reconsider c1 = c1, c2 = c2 as Element of R by A21;
A22: c1 <= c2 by A18, A20, ORDERS_2:def_5;
percases ( ( c1 in component x & c2 in component x ) or ( c1 in component x & c2 in A2 ) or ( c1 in A2 & c2 in component x ) or ( c1 in A2 & c2 in A2 ) ) by A5, XBOOLE_0:def_3;
supposeA23: ( c1 in component x & c2 in component x ) ; ::_thesis: a in the InternalRel of (subrelstr A2)
then reconsider d2 = c2 as Element of (subrelstr (component x)) by YELLOW_0:def_15;
reconsider d1 = c1 as Element of (subrelstr (component x)) by A23, YELLOW_0:def_15;
d1 <= d2 by A6, A22, YELLOW_0:60;
hence a in the InternalRel of (subrelstr A2) by A19, A20, ORDERS_2:def_5; ::_thesis: verum
end;
supposeA24: ( c1 in component x & c2 in A2 ) ; ::_thesis: a in the InternalRel of (subrelstr A2)
A25: [:(component x),A2:] misses the InternalRel of R
proof
assume not [:(component x),A2:] misses the InternalRel of R ; ::_thesis: contradiction
then [:(component x),A2:] /\ the InternalRel of R <> {} by XBOOLE_0:def_7;
then consider b being set such that
A26: b in [:(component x),A2:] /\ the InternalRel of R by XBOOLE_0:def_1;
A27: b in the InternalRel of R by A26, XBOOLE_0:def_4;
b in [:(component x),A2:] by A26, XBOOLE_0:def_4;
then consider b1, b2 being set such that
A28: b1 in component x and
A29: b2 in A2 and
A30: b = [b1,b2] by ZFMISC_1:def_2;
reconsider b2 = b2 as Element of R by A29;
reconsider b1 = b1 as Element of R by A28;
( the InternalRel of R c= EqCl the InternalRel of R & [x,b1] in EqCl the InternalRel of R ) by A28, Th29, MSUALG_5:def_1;
then [x,b2] in EqCl the InternalRel of R by A27, A30, EQREL_1:7;
then b2 in component x by Th29;
then b2 in (component x) /\ A2 by A29, XBOOLE_0:def_4;
hence contradiction by A6, A3, A7, XBOOLE_0:def_7; ::_thesis: verum
end;
a in [:(component x),A2:] by A20, A24, ZFMISC_1:def_2;
then a in [:(component x),A2:] /\ the InternalRel of R by A18, XBOOLE_0:def_4;
hence a in the InternalRel of (subrelstr A2) by A25, XBOOLE_0:def_7; ::_thesis: verum
end;
supposeA31: ( c1 in A2 & c2 in component x ) ; ::_thesis: a in the InternalRel of (subrelstr A2)
A32: [:A2,(component x):] misses the InternalRel of R
proof
assume not [:A2,(component x):] misses the InternalRel of R ; ::_thesis: contradiction
then [:A2,(component x):] /\ the InternalRel of R <> {} by XBOOLE_0:def_7;
then consider b being set such that
A33: b in [:A2,(component x):] /\ the InternalRel of R by XBOOLE_0:def_1;
b in [:A2,(component x):] by A33, XBOOLE_0:def_4;
then consider b1, b2 being set such that
A34: b1 in A2 and
A35: b2 in component x and
A36: b = [b1,b2] by ZFMISC_1:def_2;
reconsider b1 = b1 as Element of R by A34;
reconsider b2 = b2 as Element of R by A35;
A37: [x,b2] in EqCl the InternalRel of R by A35, Th29;
A38: the InternalRel of R c= EqCl the InternalRel of R by MSUALG_5:def_1;
b in the InternalRel of R by A33, XBOOLE_0:def_4;
then [b2,b1] in EqCl the InternalRel of R by A36, A38, EQREL_1:6;
then [x,b1] in EqCl the InternalRel of R by A37, EQREL_1:7;
then b1 in component x by Th29;
then b1 in (component x) /\ A2 by A34, XBOOLE_0:def_4;
hence contradiction by A6, A3, A7, XBOOLE_0:def_7; ::_thesis: verum
end;
a in [:A2,(component x):] by A20, A31, ZFMISC_1:def_2;
then a in [:A2,(component x):] /\ the InternalRel of R by A18, XBOOLE_0:def_4;
hence a in the InternalRel of (subrelstr A2) by A32, XBOOLE_0:def_7; ::_thesis: verum
end;
supposeA39: ( c1 in A2 & c2 in A2 ) ; ::_thesis: a in the InternalRel of (subrelstr A2)
then reconsider d2 = c2 as Element of (subrelstr A2) by YELLOW_0:def_15;
reconsider d1 = c1 as Element of (subrelstr A2) by A39, YELLOW_0:def_15;
d1 <= d2 by A3, A22, A39, YELLOW_0:60;
hence a in the InternalRel of (subrelstr A2) by A20, ORDERS_2:def_5; ::_thesis: verum
end;
end;
end;
the InternalRel of R = the InternalRel of (subrelstr (component x)) \/ the InternalRel of (subrelstr A2)
proof
set IG1 = the InternalRel of (subrelstr (component x));
set IG2 = the InternalRel of (subrelstr A2);
thus the InternalRel of R c= the InternalRel of (subrelstr (component x)) \/ the InternalRel of (subrelstr A2) :: according to XBOOLE_0:def_10 ::_thesis: the InternalRel of (subrelstr (component x)) \/ the InternalRel of (subrelstr A2) c= the InternalRel of R
proof
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in the InternalRel of R or a in the InternalRel of (subrelstr (component x)) \/ the InternalRel of (subrelstr A2) )
assume A40: a in the InternalRel of R ; ::_thesis: a in the InternalRel of (subrelstr (component x)) \/ the InternalRel of (subrelstr A2)
assume not a in the InternalRel of (subrelstr (component x)) \/ the InternalRel of (subrelstr A2) ; ::_thesis: contradiction
then ( not a in the InternalRel of (subrelstr (component x)) & not a in the InternalRel of (subrelstr A2) ) by XBOOLE_0:def_3;
hence contradiction by A12, A40, XBOOLE_0:def_5; ::_thesis: verum
end;
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in the InternalRel of (subrelstr (component x)) \/ the InternalRel of (subrelstr A2) or a in the InternalRel of R )
assume A41: a in the InternalRel of (subrelstr (component x)) \/ the InternalRel of (subrelstr A2) ; ::_thesis: a in the InternalRel of R
percases ( a in the InternalRel of (subrelstr (component x)) or a in the InternalRel of (subrelstr A2) ) by A41, XBOOLE_0:def_3;
supposeA42: a in the InternalRel of (subrelstr (component x)) ; ::_thesis: a in the InternalRel of R
then consider v, w being set such that
A43: a = [v,w] and
A44: ( v in component x & w in component x ) by A6, RELSET_1:2;
reconsider v = v, w = w as Element of (subrelstr (component x)) 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 a in the InternalRel of R by A43, ORDERS_2:def_5; ::_thesis: verum
end;
supposeA45: a in the InternalRel of (subrelstr A2) ; ::_thesis: a in the InternalRel of R
then consider v, w being set such that
A46: a = [v,w] and
A47: ( v in A2 & w in A2 ) by A3, RELSET_1:2;
reconsider v = v, w = w as Element of (subrelstr A2) 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 a in the InternalRel of R by A46, ORDERS_2:def_5; ::_thesis: verum
end;
end;
end;
then A48: the InternalRel of R = the InternalRel of (union_of ((subrelstr (component x)),(subrelstr A2))) by NECKLA_2:def_2;
the InternalRel of R = the InternalRel of R ~ by RELAT_2:13;
then not the InternalRel of R \/ ( the InternalRel of R ~) reduces x,y by A2;
then not x,y are_convertible_wrt the InternalRel of R by REWRITE1:def_4;
then not [x,y] in EqCl the InternalRel of R by MSUALG_6:41;
then not y in component x by Th29;
then A49: subrelstr A2 is non empty strict RelStr by A3, XBOOLE_0:def_5;
the carrier of R = the carrier of (union_of ((subrelstr (component x)),(subrelstr A2))) by A6, A3, A5, NECKLA_2:def_2;
hence ex G1, G2 being non empty strict symmetric irreflexive RelStr st
( the carrier of G1 misses the carrier of G2 & RelStr(# the carrier of R, the InternalRel of R #) = union_of (G1,G2) ) by A6, A7, A49, A48; ::_thesis: verum
end;
theorem Th31: :: NECKLA_3:31
for R being non empty symmetric irreflexive RelStr st not ComplRelStr R is path-connected holds
ex G1, G2 being non empty strict symmetric irreflexive RelStr st
( the carrier of G1 misses the carrier of G2 & RelStr(# the carrier of R, the InternalRel of R #) = sum_of (G1,G2) )
proof
let R be non empty symmetric irreflexive RelStr ; ::_thesis: ( not ComplRelStr R is path-connected implies ex G1, G2 being non empty strict symmetric irreflexive RelStr st
( the carrier of G1 misses the carrier of G2 & RelStr(# the carrier of R, the InternalRel of R #) = sum_of (G1,G2) ) )
set cR = the carrier of R;
set IR = the InternalRel of R;
set CR = ComplRelStr R;
set ICR = the InternalRel of (ComplRelStr R);
set cCR = the carrier of (ComplRelStr R);
assume not ComplRelStr R is path-connected ; ::_thesis: ex G1, G2 being non empty strict symmetric irreflexive RelStr st
( the carrier of G1 misses the carrier of G2 & RelStr(# the carrier of R, the InternalRel of R #) = sum_of (G1,G2) )
then consider x, y being set such that
A1: x in the carrier of (ComplRelStr R) and
A2: y in the carrier of (ComplRelStr R) and
x <> y and
A3: not the InternalRel of (ComplRelStr R) reduces x,y by Def2;
reconsider x = x, y = y as Element of (ComplRelStr R) by A1, A2;
set A1 = component x;
set A2 = the carrier of R \ (component x);
reconsider A1 = component x as Subset of R by NECKLACE:def_8;
the InternalRel of (ComplRelStr R) = the InternalRel of (ComplRelStr R) ~ by RELAT_2:13;
then not the InternalRel of (ComplRelStr R) \/ ( the InternalRel of (ComplRelStr R) ~) reduces x,y by A3;
then not x,y are_convertible_wrt the InternalRel of (ComplRelStr R) by REWRITE1:def_4;
then not [x,y] in EqCl the InternalRel of (ComplRelStr R) by MSUALG_6:41;
then A4: not y in A1 by Th29;
reconsider A2 = the carrier of R \ (component x) as Subset of R ;
set G1 = subrelstr A1;
set G2 = subrelstr A2;
A5: the carrier of (subrelstr A1) = A1 by YELLOW_0:def_15;
set IG1 = the InternalRel of (subrelstr A1);
set IG2 = the InternalRel of (subrelstr A2);
set G1G2 = [: the carrier of (subrelstr A1), the carrier of (subrelstr A2):];
set G2G1 = [: the carrier of (subrelstr A2), the carrier of (subrelstr A1):];
A6: the carrier of R = A1 \/ A2
proof
thus the carrier of R c= A1 \/ A2 :: according to XBOOLE_0:def_10 ::_thesis: A1 \/ A2 c= the carrier of R
proof
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in the carrier of R or a in A1 \/ A2 )
assume A7: a in the carrier of R ; ::_thesis: a in A1 \/ A2
assume not a in A1 \/ A2 ; ::_thesis: contradiction
then ( not a in A1 & not a in A2 ) by XBOOLE_0:def_3;
hence contradiction by A7, XBOOLE_0:def_5; ::_thesis: verum
end;
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in A1 \/ A2 or a in the carrier of R )
assume A8: a in A1 \/ A2 ; ::_thesis: a in the carrier of R
percases ( a in A1 or a in A2 ) by A8, XBOOLE_0:def_3;
suppose a in A1 ; ::_thesis: a in the carrier of R
hence a in the carrier of R ; ::_thesis: verum
end;
suppose a in A2 ; ::_thesis: a in the carrier of R
hence a in the carrier of R ; ::_thesis: verum
end;
end;
end;
A9: the carrier of (subrelstr A2) = A2 by YELLOW_0:def_15;
then A10: the carrier of (subrelstr A1) misses the carrier of (subrelstr A2) by A5, XBOOLE_1:79;
A11: the InternalRel of (subrelstr A1) misses the InternalRel of (subrelstr A2)
proof
assume not the InternalRel of (subrelstr A1) misses the InternalRel of (subrelstr A2) ; ::_thesis: contradiction
then the InternalRel of (subrelstr A1) /\ the InternalRel of (subrelstr A2) <> {} by XBOOLE_0:def_7;
then consider a being set such that
A12: a in the InternalRel of (subrelstr A1) /\ the InternalRel of (subrelstr A2) by XBOOLE_0:def_1;
a in the InternalRel of (subrelstr A1) by A12, XBOOLE_0:def_4;
then consider c1, c2 being set such that
A13: a = [c1,c2] and
A14: c1 in A1 and
c2 in A1 by A5, RELSET_1:2;
ex g1, g2 being set 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, XBOOLE_0:def_7; ::_thesis: verum
end;
A15: the InternalRel of (subrelstr A2) = (( the InternalRel of R \ the InternalRel of (subrelstr A1)) \ [: the carrier of (subrelstr A1), the carrier of (subrelstr A2):]) \ [: the carrier of (subrelstr A2), the carrier of (subrelstr A1):]
proof
thus the InternalRel of (subrelstr A2) c= (( the InternalRel of R \ the InternalRel of (subrelstr A1)) \ [: the carrier of (subrelstr A1), the carrier of (subrelstr A2):]) \ [: the carrier of (subrelstr A2), the carrier of (subrelstr A1):] :: according to XBOOLE_0:def_10 ::_thesis: (( the InternalRel of R \ the InternalRel of (subrelstr A1)) \ [: the carrier of (subrelstr A1), the carrier of (subrelstr A2):]) \ [: the carrier of (subrelstr A2), the carrier of (subrelstr A1):] c= the InternalRel of (subrelstr A2)
proof
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in the InternalRel of (subrelstr A2) or a in (( the InternalRel of R \ the InternalRel of (subrelstr A1)) \ [: the carrier of (subrelstr A1), the carrier of (subrelstr A2):]) \ [: the carrier of (subrelstr A2), the carrier of (subrelstr A1):] )
assume A16: a in the InternalRel of (subrelstr A2) ; ::_thesis: a in (( the InternalRel of R \ the InternalRel of (subrelstr A1)) \ [: the carrier of (subrelstr A1), the carrier of (subrelstr A2):]) \ [: the carrier of (subrelstr A2), the carrier of (subrelstr A1):]
then consider g1, g2 being set such that
A17: a = [g1,g2] and
A18: g1 in A2 and
A19: g2 in A2 by A9, RELSET_1:2;
reconsider g1 = g1, g2 = g2 as Element of (subrelstr A2) by A18, A19, YELLOW_0:def_15;
reconsider u1 = g1, u2 = g2 as Element of R by A18, A19;
A20: not a in the InternalRel of (subrelstr A1)
proof
assume a in the InternalRel of (subrelstr A1) ; ::_thesis: contradiction
then a in the InternalRel of (subrelstr A1) /\ the InternalRel of (subrelstr A2) by A16, XBOOLE_0:def_4;
hence contradiction by A11, XBOOLE_0:def_7; ::_thesis: verum
end;
A21: not a in [: the carrier of (subrelstr A2), the carrier of (subrelstr A1):]
proof
assume a in [: the carrier of (subrelstr A2), the carrier of (subrelstr A1):] ; ::_thesis: contradiction
then g2 in A1 by A5, A17, ZFMISC_1:87;
then g2 in A1 /\ A2 by A19, XBOOLE_0:def_4;
hence contradiction by A5, A9, A10, XBOOLE_0:def_7; ::_thesis: verum
end;
A22: not a in [: the carrier of (subrelstr A1), the carrier of (subrelstr A2):]
proof
assume a in [: the carrier of (subrelstr A1), the carrier of (subrelstr A2):] ; ::_thesis: contradiction
then g1 in A1 by A5, A17, ZFMISC_1:87;
then g1 in A1 /\ A2 by A18, XBOOLE_0:def_4;
hence contradiction by A5, A9, A10, XBOOLE_0:def_7; ::_thesis: verum
end;
g1 <= g2 by A16, A17, ORDERS_2:def_5;
then u1 <= u2 by YELLOW_0:59;
then a in the InternalRel of R by A17, ORDERS_2:def_5;
then a in the InternalRel of R \ the InternalRel of (subrelstr A1) by A20, XBOOLE_0:def_5;
then a in ( the InternalRel of R \ the InternalRel of (subrelstr A1)) \ [: the carrier of (subrelstr A1), the carrier of (subrelstr A2):] by A22, XBOOLE_0:def_5;
hence a in (( the InternalRel of R \ the InternalRel of (subrelstr A1)) \ [: the carrier of (subrelstr A1), the carrier of (subrelstr A2):]) \ [: the carrier of (subrelstr A2), the carrier of (subrelstr A1):] by A21, XBOOLE_0:def_5; ::_thesis: verum
end;
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in (( the InternalRel of R \ the InternalRel of (subrelstr A1)) \ [: the carrier of (subrelstr A1), the carrier of (subrelstr A2):]) \ [: the carrier of (subrelstr A2), the carrier of (subrelstr A1):] or a in the InternalRel of (subrelstr A2) )
assume A23: a in (( the InternalRel of R \ the InternalRel of (subrelstr A1)) \ [: the carrier of (subrelstr A1), the carrier of (subrelstr A2):]) \ [: the carrier of (subrelstr A2), the carrier of (subrelstr A1):] ; ::_thesis: a in the InternalRel of (subrelstr A2)
then A24: not a in [: the carrier of (subrelstr A2), the carrier of (subrelstr A1):] by XBOOLE_0:def_5;
A25: a in ( the InternalRel of R \ the InternalRel of (subrelstr A1)) \ [: the carrier of (subrelstr A1), the carrier of (subrelstr A2):] by A23, XBOOLE_0:def_5;
then A26: a in the InternalRel of R \ the InternalRel of (subrelstr A1) by XBOOLE_0:def_5;
then A27: not a in the InternalRel of (subrelstr A1) by XBOOLE_0:def_5;
A28: not a in [: the carrier of (subrelstr A1), the carrier of (subrelstr A2):] by A25, XBOOLE_0:def_5;
consider c1, c2 being set such that
A29: a = [c1,c2] and
A30: ( c1 in the carrier of R & c2 in the carrier of R ) by A23, RELSET_1:2;
reconsider c1 = c1, c2 = c2 as Element of R by A30;
a in the InternalRel of R by A26, XBOOLE_0:def_5;
then A31: c1 <= c2 by A29, ORDERS_2:def_5;
percases ( ( c1 in A1 & c2 in A1 ) or ( c1 in A1 & c2 in A2 ) or ( c1 in A2 & c2 in A1 ) or ( c1 in A2 & c2 in A2 ) ) by A6, XBOOLE_0:def_3;
supposeA32: ( c1 in A1 & c2 in A1 ) ; ::_thesis: a in the InternalRel of (subrelstr A2)
then reconsider d2 = c2 as Element of (subrelstr A1) by YELLOW_0:def_15;
reconsider d1 = c1 as Element of (subrelstr A1) by A32, YELLOW_0:def_15;
d1 <= d2 by A5, A31, YELLOW_0:60;
hence a in the InternalRel of (subrelstr A2) by A27, A29, ORDERS_2:def_5; ::_thesis: verum
end;
suppose ( c1 in A1 & c2 in A2 ) ; ::_thesis: a in the InternalRel of (subrelstr A2)
hence a in the InternalRel of (subrelstr A2) by A5, A9, A28, A29, ZFMISC_1:87; ::_thesis: verum
end;
suppose ( c1 in A2 & c2 in A1 ) ; ::_thesis: a in the InternalRel of (subrelstr A2)
hence a in the InternalRel of (subrelstr A2) by A5, A9, A24, A29, ZFMISC_1:87; ::_thesis: verum
end;
supposeA33: ( c1 in A2 & c2 in A2 ) ; ::_thesis: a in the InternalRel of (subrelstr A2)
then reconsider d1 = c1, d2 = c2 as Element of (subrelstr A2) by YELLOW_0:def_15;
d1 <= d2 by A9, A31, A33, YELLOW_0:60;
hence a in the InternalRel of (subrelstr A2) by A29, ORDERS_2:def_5; ::_thesis: verum
end;
end;
end;
the InternalRel of R = (( the InternalRel of (subrelstr A1) \/ the InternalRel of (subrelstr A2)) \/ [: the carrier of (subrelstr A1), the carrier of (subrelstr A2):]) \/ [: the carrier of (subrelstr A2), the carrier of (subrelstr A1):]
proof
set G1G2 = [: the carrier of (subrelstr A1), the carrier of (subrelstr A2):];
set G2G1 = [: the carrier of (subrelstr A2), the carrier of (subrelstr A1):];
thus the InternalRel of R c= (( the InternalRel of (subrelstr A1) \/ the InternalRel of (subrelstr A2)) \/ [: the carrier of (subrelstr A1), the carrier of (subrelstr A2):]) \/ [: the carrier of (subrelstr A2), the carrier of (subrelstr A1):] :: according to XBOOLE_0:def_10 ::_thesis: (( the InternalRel of (subrelstr A1) \/ the InternalRel of (subrelstr A2)) \/ [: the carrier of (subrelstr A1), the carrier of (subrelstr A2):]) \/ [: the carrier of (subrelstr A2), the carrier of (subrelstr A1):] c= the InternalRel of R
proof
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in the InternalRel of R or a in (( the InternalRel of (subrelstr A1) \/ the InternalRel of (subrelstr A2)) \/ [: the carrier of (subrelstr A1), the carrier of (subrelstr A2):]) \/ [: the carrier of (subrelstr A2), the carrier of (subrelstr A1):] )
assume A34: a in the InternalRel of R ; ::_thesis: a in (( the InternalRel of (subrelstr A1) \/ the InternalRel of (subrelstr A2)) \/ [: the carrier of (subrelstr A1), the carrier of (subrelstr A2):]) \/ [: the carrier of (subrelstr A2), the carrier of (subrelstr A1):]
assume A35: not a in (( the InternalRel of (subrelstr A1) \/ the InternalRel of (subrelstr A2)) \/ [: the carrier of (subrelstr A1), the carrier of (subrelstr A2):]) \/ [: the carrier of (subrelstr A2), the carrier of (subrelstr A1):] ; ::_thesis: contradiction
then A36: not a in ( the InternalRel of (subrelstr A1) \/ the InternalRel of (subrelstr A2)) \/ [: the carrier of (subrelstr A1), the carrier of (subrelstr A2):] by XBOOLE_0:def_3;
then A37: not a in the InternalRel of (subrelstr A1) \/ the InternalRel of (subrelstr A2) by XBOOLE_0:def_3;
then not a in the InternalRel of (subrelstr A2) by XBOOLE_0:def_3;
then ( not a in ( the InternalRel of R \ the InternalRel of (subrelstr A1)) \ [: the carrier of (subrelstr A1), the carrier of (subrelstr A2):] or a in [: the carrier of (subrelstr A2), the carrier of (subrelstr A1):] ) by A15, XBOOLE_0:def_5;
then A38: ( not a in the InternalRel of R \ the InternalRel of (subrelstr A1) or a in [: the carrier of (subrelstr A1), the carrier of (subrelstr A2):] or a in [: the carrier of (subrelstr A2), the carrier of (subrelstr A1):] ) by XBOOLE_0:def_5;
not a in the InternalRel of (subrelstr A1) by A37, XBOOLE_0:def_3;
hence contradiction by A34, A35, A36, A38, XBOOLE_0:def_3, XBOOLE_0:def_5; ::_thesis: verum
end;
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in (( the InternalRel of (subrelstr A1) \/ the InternalRel of (subrelstr A2)) \/ [: the carrier of (subrelstr A1), the carrier of (subrelstr A2):]) \/ [: the carrier of (subrelstr A2), the carrier of (subrelstr A1):] or a in the InternalRel of R )
assume a in (( the InternalRel of (subrelstr A1) \/ the InternalRel of (subrelstr A2)) \/ [: the carrier of (subrelstr A1), the carrier of (subrelstr A2):]) \/ [: the carrier of (subrelstr A2), the carrier of (subrelstr A1):] ; ::_thesis: a in the InternalRel of R
then ( a in ( the InternalRel of (subrelstr A1) \/ the InternalRel of (subrelstr A2)) \/ [: the carrier of (subrelstr A1), the carrier of (subrelstr A2):] or a in [: the carrier of (subrelstr A2), the carrier of (subrelstr A1):] ) by XBOOLE_0:def_3;
then A39: ( a in the InternalRel of (subrelstr A1) \/ the InternalRel of (subrelstr A2) or a in [: the carrier of (subrelstr A1), the carrier of (subrelstr A2):] or a in [: the carrier of (subrelstr A2), the carrier of (subrelstr A1):] ) by XBOOLE_0:def_3;
percases ( a in the InternalRel of (subrelstr A1) or a in the InternalRel of (subrelstr A2) or a in [: the carrier of (subrelstr A1), the carrier of (subrelstr A2):] or a in [: the carrier of (subrelstr A2), the carrier of (subrelstr A1):] ) by A39, XBOOLE_0:def_3;
supposeA40: a in the InternalRel of (subrelstr A1) ; ::_thesis: a in the InternalRel of R
then consider v, w being set such that
A41: a = [v,w] and
A42: ( v in A1 & w in A1 ) by A5, RELSET_1:2;
reconsider v = v, w = w as Element of (subrelstr A1) 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 a in the InternalRel of R by A41, ORDERS_2:def_5; ::_thesis: verum
end;
supposeA43: a in the InternalRel of (subrelstr A2) ; ::_thesis: a in the InternalRel of R
then consider v, w being set such that
A44: a = [v,w] and
A45: ( v in A2 & w in A2 ) by A9, RELSET_1:2;
reconsider v = v, w = w as Element of (subrelstr A2) 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 a in the InternalRel of R by A44, ORDERS_2:def_5; ::_thesis: verum
end;
supposeA46: a in [: the carrier of (subrelstr A1), the carrier of (subrelstr A2):] ; ::_thesis: a in the InternalRel of R
assume A47: not a in the InternalRel of R ; ::_thesis: contradiction
consider v, w being set 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 = v, w = w as Element of (ComplRelStr R) by A49, NECKLACE:def_8;
v <> w
proof
assume A51: v = w ; ::_thesis: contradiction
A1 /\ A2 = {} by A5, A9, A10, XBOOLE_0:def_7;
hence contradiction by A50, A49, A51, XBOOLE_0:def_4; ::_thesis: verum
end;
then A52: not a in id the carrier of R by A48, RELAT_1:def_10;
[v,w] in [: the carrier of R, the carrier of R:] by A50, A49, ZFMISC_1:87;
then a in [: the carrier of R, the carrier of R:] \ the InternalRel of R by A48, A47, XBOOLE_0:def_5;
then a in the InternalRel of R ` by SUBSET_1:def_4;
then a in ( the InternalRel of R `) \ (id the carrier of R) by A52, XBOOLE_0:def_5;
then [v,w] in the InternalRel of (ComplRelStr R) by A48, NECKLACE:def_8;
then v,w are_convertible_wrt the InternalRel of (ComplRelStr R) by REWRITE1:29;
then A53: [v,w] in EqCl the InternalRel of (ComplRelStr R) by MSUALG_6:41;
[x,v] in EqCl the InternalRel of (ComplRelStr R) by A50, Th29;
then [x,w] in EqCl the InternalRel of (ComplRelStr R) by A53, EQREL_1:7;
then w in component x by Th29;
then w in A1 /\ A2 by A49, XBOOLE_0:def_4;
hence contradiction by A5, A9, A10, XBOOLE_0:def_7; ::_thesis: verum
end;
supposeA54: a in [: the carrier of (subrelstr A2), the carrier of (subrelstr A1):] ; ::_thesis: a in the InternalRel of R
assume A55: not a in the InternalRel of R ; ::_thesis: contradiction
consider v, w being set such that
A56: a = [v,w] by A54, RELAT_1:def_1;
A57: w in A1 by A5, A54, A56, ZFMISC_1:87;
A58: v in A2 by A9, A54, A56, ZFMISC_1:87;
then reconsider v = v, w = w as Element of (ComplRelStr R) by A57, NECKLACE:def_8;
v <> w
proof
assume A59: v = w ; ::_thesis: contradiction
A1 /\ A2 = {} by A5, A9, A10, XBOOLE_0:def_7;
hence contradiction by A58, A57, A59, XBOOLE_0:def_4; ::_thesis: verum
end;
then A60: not a in id the carrier of R by A56, RELAT_1:def_10;
[v,w] in [: the carrier of R, the carrier of R:] by A58, A57, ZFMISC_1:87;
then a in [: the carrier of R, the carrier of R:] \ the InternalRel of R by A56, A55, XBOOLE_0:def_5;
then a in the InternalRel of R ` by SUBSET_1:def_4;
then a in ( the InternalRel of R `) \ (id the carrier of R) by A60, XBOOLE_0:def_5;
then [v,w] in the InternalRel of (ComplRelStr R) by A56, NECKLACE:def_8;
then v,w are_convertible_wrt the InternalRel of (ComplRelStr R) by REWRITE1:29;
then [v,w] in EqCl the InternalRel of (ComplRelStr R) by MSUALG_6:41;
then A61: [w,v] in EqCl the InternalRel of (ComplRelStr R) by EQREL_1:6;
[x,w] in EqCl the InternalRel of (ComplRelStr R) by A57, Th29;
then [x,v] in EqCl the InternalRel of (ComplRelStr R) by A61, EQREL_1:7;
then v in component x by Th29;
then v in A1 /\ A2 by A58, XBOOLE_0:def_4;
hence contradiction by A5, A9, A10, XBOOLE_0:def_7; ::_thesis: verum
end;
end;
end;
then A62: the InternalRel of R = the InternalRel of (sum_of ((subrelstr A1),(subrelstr A2))) by NECKLA_2:def_3;
y in the carrier of R by A2, NECKLACE:def_8;
then A63: subrelstr A2 is non empty strict RelStr by A9, A4, XBOOLE_0:def_5;
the carrier of R = the carrier of (sum_of ((subrelstr A1),(subrelstr A2))) by A5, A9, A6, NECKLA_2:def_3;
hence ex G1, G2 being non empty strict symmetric irreflexive RelStr st
( the carrier of G1 misses the carrier of G2 & RelStr(# the carrier of R, the InternalRel of R #) = sum_of (G1,G2) ) by A5, A10, A63, A62; ::_thesis: verum
end;
Lm1: for X being non empty finite set
for A, B being non empty set st X = A \/ B & A misses B holds
card A in card X
proof
let X be non empty finite set ; ::_thesis: for A, B being non empty set st X = A \/ B & A misses B holds
card A in card X
let A, B be non empty set ; ::_thesis: ( X = A \/ B & A misses B implies card A in card X )
set n = card X;
assume that
A1: X = A \/ B and
A2: A misses B ; ::_thesis: card A in card X
card B c= card X by A1, CARD_1:11, XBOOLE_1:7;
then reconsider B = B as finite set ;
card A c= card X by A1, CARD_1:11, XBOOLE_1:7;
then reconsider A = A as finite set ;
A3: card B >= 1 by NAT_1:14;
A4: card X = (card A) + (card B) by A1, A2, CARD_2:40;
card A < card X
proof
assume not card A < card X ; ::_thesis: contradiction
then (card A) + (card B) >= (card X) + 1 by A3, XREAL_1:7;
hence contradiction by A4, NAT_1:13; ::_thesis: verum
end;
hence card A in card X by NAT_1:44; ::_thesis: verum
end;
theorem :: NECKLA_3:32
for G being irreflexive RelStr st G in fin_RelStr_sp holds
ComplRelStr G in fin_RelStr_sp
proof
defpred S1[ Nat] means for G being 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 ; ::_thesis: ( G in fin_RelStr_sp implies ComplRelStr G in fin_RelStr_sp )
A1: for k being Nat st ( for n being Nat st n < k holds
S1[n] ) holds
S1[k]
proof
let k be Nat; ::_thesis: ( ( for n being Nat st n < k holds
S1[n] ) implies S1[k] )
assume A2: for n being Nat st n < k holds
S1[n] ; ::_thesis: S1[k]
let G be irreflexive RelStr ; ::_thesis: ( card the carrier of G = k & G in fin_RelStr_sp implies ComplRelStr G in fin_RelStr_sp )
assume that
A3: card the carrier of G = k and
A4: G in fin_RelStr_sp ; ::_thesis: ComplRelStr G in fin_RelStr_sp
percases ( G is 1 -element strict RelStr or ex G1, G2 being 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) ) ) ) by A4, NECKLA_2:6;
suppose G is 1 -element strict RelStr ; ::_thesis: ComplRelStr G in fin_RelStr_sp
hence ComplRelStr G in fin_RelStr_sp by A4, Th15; ::_thesis: verum
end;
suppose ex G1, G2 being 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) ) ) ; ::_thesis: ComplRelStr G in fin_RelStr_sp
then consider G1, G2 being 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: ( not G2 is empty & G2 is finite ) by A7, NECKLA_2:4;
then reconsider n2 = card the carrier of G2 as Nat ;
A10: ( not G1 is empty & G1 is finite ) by A6, NECKLA_2:4;
then reconsider n1 = card the carrier of G1 as Nat ;
thus ComplRelStr G in fin_RelStr_sp ::_thesis: verum
proof
percases ( G = union_of (G1,G2) or G = sum_of (G1,G2) ) by A8;
supposeA11: G = union_of (G1,G2) ; ::_thesis: ComplRelStr G in fin_RelStr_sp
then reconsider G2 = G2 as irreflexive RelStr by Th9;
reconsider G1 = 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 n2 < k ; ::_thesis: contradiction
then k + 0 < n1 + n2 by A13, XREAL_1:8;
hence contradiction by A3, A12; ::_thesis: verum
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 n1 < k ; ::_thesis: contradiction
then k + 0 < n2 + n1 by A16, XREAL_1:8;
hence contradiction by A3, A12; ::_thesis: verum
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 ComplRelStr G in fin_RelStr_sp by A5, A17, A14, A15, NECKLA_2:def_5; ::_thesis: verum
end;
supposeA18: G = sum_of (G1,G2) ; ::_thesis: ComplRelStr G in fin_RelStr_sp
then reconsider G2 = G2 as irreflexive RelStr by Th9;
reconsider G1 = 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 n2 < k ; ::_thesis: contradiction
then k + 0 < n1 + n2 by A20, XREAL_1:8;
hence contradiction by A3, A19; ::_thesis: verum
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 n1 < k ; ::_thesis: contradiction
then k + 0 < n2 + n1 by A23, XREAL_1:8;
hence contradiction by A3, A19; ::_thesis: verum
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 ComplRelStr G in fin_RelStr_sp by A5, A24, A21, A22, NECKLA_2:def_5; ::_thesis: verum
end;
end;
end;
end;
end;
end;
A25: for k being Nat holds S1[k] from NAT_1:sch_4(A1);
assume A26: G in fin_RelStr_sp ; ::_thesis: ComplRelStr G in fin_RelStr_sp
then G is finite by NECKLA_2:4;
then card the carrier of G is Nat ;
hence ComplRelStr G in fin_RelStr_sp by A26, A25; ::_thesis: verum
end;
theorem Th33: :: NECKLA_3:33
for R being symmetric irreflexive RelStr st card the carrier of R = 2 & the carrier of R in FinSETS holds
RelStr(# the carrier of R, the InternalRel of R #) in fin_RelStr_sp
proof
let R be symmetric irreflexive RelStr ; ::_thesis: ( card the carrier of R = 2 & the carrier of R in FinSETS implies RelStr(# the carrier of R, the InternalRel of R #) in fin_RelStr_sp )
assume that
A1: card the carrier of R = 2 and
A2: the carrier of R in FinSETS ; ::_thesis: RelStr(# the carrier of R, the InternalRel of R #) in fin_RelStr_sp
consider a, b being set 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};
set B = {b};
A5: {a} c= the carrier of R
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {a} or x in the carrier of R )
assume x in {a} ; ::_thesis: x in the carrier of R
then x = a by TARSKI:def_1;
hence x in the carrier of R by A3, TARSKI:def_2; ::_thesis: verum
end;
A6: {b} c= the carrier of R
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {b} or x in the carrier of R )
assume x in {b} ; ::_thesis: x in the carrier of R
then x = b by TARSKI:def_1;
hence x in the carrier of R by A3, TARSKI:def_2; ::_thesis: verum
end;
then reconsider B = {b} as Subset of R ;
reconsider A = {a} as Subset of R by A5;
set H1 = subrelstr A;
set H2 = subrelstr B;
reconsider H2 = subrelstr B as non empty strict symmetric irreflexive 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 the InternalRel of H2 = {} ; ::_thesis: contradiction
hence contradiction by A8, A10, TARSKI:def_1; ::_thesis: verum
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 = subrelstr A as non empty strict symmetric irreflexive 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 a <> b ; ::_thesis: contradiction
then the carrier of R = {a} by A3, ENUMSET1:29;
hence contradiction by A1, CARD_1:30; ::_thesis: verum
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 the InternalRel of H1 = {} ; ::_thesis: contradiction
hence contradiction by A16, A18, TARSKI:def_1; ::_thesis: verum
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;
percases ( the InternalRel of R = {[a,b],[b,a]} or the InternalRel of R = {} ) by A4;
supposeA20: the InternalRel of R = {[a,b],[b,a]} ; ::_thesis: RelStr(# the carrier of R, the InternalRel of R #) in fin_RelStr_sp
set S = sum_of (H1,H2);
the InternalRel of (sum_of (H1,H2)) = (( the InternalRel of H1 \/ the InternalRel of H2) \/ [:A,B:]) \/ [:B,A:] by A12, A7, NECKLA_2:def_3;
then the InternalRel of (sum_of (H1,H2)) = {[a,b]} \/ [:{b},{a}:] by A17, A9, ZFMISC_1:29;
then the InternalRel of (sum_of (H1,H2)) = {[a,b]} \/ {[b,a]} by ZFMISC_1:29;
then A21: the InternalRel of (sum_of (H1,H2)) = the InternalRel of R by A20, ENUMSET1:1;
the carrier of (sum_of (H1,H2)) = the carrier of R by A13, NECKLA_2:def_3;
hence RelStr(# the carrier of R, the InternalRel of R #) in fin_RelStr_sp by A12, A19, A7, A11, A14, A21, NECKLA_2:def_5; ::_thesis: verum
end;
supposeA22: the InternalRel of R = {} ; ::_thesis: RelStr(# the carrier of R, the InternalRel of R #) in fin_RelStr_sp
set U = union_of (H1,H2);
( the InternalRel of (union_of (H1,H2)) = the InternalRel of H1 \/ the InternalRel of H2 & the carrier of (union_of (H1,H2)) = the carrier of R ) by A13, NECKLA_2:def_2;
hence RelStr(# the carrier of R, the InternalRel of R #) in fin_RelStr_sp by A19, A11, A15, A17, A9, A22, NECKLA_2:def_5; ::_thesis: verum
end;
end;
end;
theorem :: NECKLA_3:34
for R being RelStr st R in fin_RelStr_sp holds
R is symmetric
proof
let X be RelStr ; ::_thesis: ( X in fin_RelStr_sp implies X is symmetric )
assume A1: X in fin_RelStr_sp ; ::_thesis: X is symmetric
percases ( X is trivial or not X is trivial ) ;
supposeA2: X is trivial ; ::_thesis: X is symmetric
thus X is symmetric ::_thesis: verum
proof
percases ( the carrier of X is empty or ex x being set st the carrier of X = {x} ) by A2, ZFMISC_1:131;
supposeA3: the carrier of X is empty ; ::_thesis: X is symmetric
let a, b be set ; :: according to NECKLACE:def_3,RELAT_2:def_3 ::_thesis: ( not a in the carrier of X or not b in the carrier of X or not [a,b] in the InternalRel of X or [b,a] in the InternalRel of X )
assume that
A4: a in the carrier of X and
b in the carrier of X and
[a,b] in the InternalRel of X ; ::_thesis: [b,a] in the InternalRel of X
thus [b,a] in the InternalRel of X by A3, A4; ::_thesis: verum
end;
suppose ex x being set st the carrier of X = {x} ; ::_thesis: X is symmetric
then consider x being set 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 X is symmetric ::_thesis: verum
proof
percases ( the InternalRel of X = {} or the InternalRel of X = {[x,x]} ) by A6, ZFMISC_1:33;
supposeA7: the InternalRel of X = {} ; ::_thesis: X is symmetric
let a, b be set ; :: according to NECKLACE:def_3,RELAT_2:def_3 ::_thesis: ( not a in the carrier of X or not b in the carrier of X or not [a,b] in the InternalRel of X or [b,a] in the InternalRel of X )
assume that
a in the carrier of X and
b in the carrier of X and
A8: [a,b] in the InternalRel of X ; ::_thesis: [b,a] in the InternalRel of X
thus [b,a] in the InternalRel of X by A7, A8; ::_thesis: verum
end;
supposeA9: the InternalRel of X = {[x,x]} ; ::_thesis: X is symmetric
let a, b be set ; :: according to NECKLACE:def_3,RELAT_2:def_3 ::_thesis: ( not a in the carrier of X or not b in the carrier of X or not [a,b] in the InternalRel of X or [b,a] in the InternalRel of X )
assume that
a in the carrier of X and
b in the carrier of X and
A10: [a,b] in the InternalRel of X ; ::_thesis: [b,a] in the InternalRel of X
A11: [a,b] = [x,x] by A9, A10, TARSKI:def_1;
then a = x by XTUPLE_0:1;
hence [b,a] in the InternalRel of X by A10, A11, XTUPLE_0:1; ::_thesis: verum
end;
end;
end;
end;
end;
end;
end;
supposeA12: not X is trivial ; ::_thesis: X is symmetric
defpred S1[ Nat] means for X being non empty RelStr st not X is trivial & X in fin_RelStr_sp & card the carrier of X c= $1 holds
X is symmetric ;
A13: ex R being strict RelStr st
( X = R & the carrier of R in FinSETS ) by A1, NECKLA_2:def_4;
reconsider X = X as non empty RelStr by A1, NECKLA_2:4;
A14: card the carrier of X is Nat by A13;
A15: for k being Nat st S1[k] holds
S1[k + 1]
proof
let k be Nat; ::_thesis: ( S1[k] implies S1[k + 1] )
assume A16: S1[k] ; ::_thesis: S1[k + 1]
reconsider k1 = k as Element of NAT by ORDINAL1:def_12;
let Y be non empty RelStr ; ::_thesis: ( not Y is trivial & Y in fin_RelStr_sp & card the carrier of Y c= k + 1 implies Y is symmetric )
assume that
A17: not Y is trivial and
A18: Y in fin_RelStr_sp ; ::_thesis: ( not card the carrier of Y c= k + 1 or Y is symmetric )
consider H1, H2 being 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;
set cY = the carrier of Y;
ex R being 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 ; ::_thesis: Y is symmetric
then card cY c= card (k1 + 1) by CARD_1:def_2;
then card cY <= card (k1 + 1) by NAT_1:39;
then A23: card cY <= k + 1 by CARD_1:def_2;
set cH1 = the carrier of H1;
set cH2 = the carrier of H2;
A24: card cY = card ( the carrier of H1 \/ the carrier of H2) by A22, NECKLA_2:def_2, NECKLA_2:def_3;
ex R2 being strict RelStr st
( H2 = R2 & the carrier of R2 in FinSETS ) by A21, NECKLA_2:def_4;
then reconsider cH2 = the carrier of H2 as finite set ;
ex R1 being strict RelStr st
( H1 = R1 & the carrier of R1 in FinSETS ) by A20, NECKLA_2:def_4;
then reconsider cH1 = the carrier of H1 as finite set ;
A25: card cY = (card cH1) + (card cH2) by A19, A24, CARD_2:40;
not H1 is empty by A20, NECKLA_2:4;
then A26: card cH1 >= 1 by NAT_1:14;
not H2 is empty by A21, NECKLA_2:4;
then A27: card cH2 >= 1 by NAT_1:14;
percases ( card cY <= k or ( card cY = k + 1 & k = 0 ) or ( (card cH1) + (card cH2) = k + 1 & k > 0 & card cH1 = 1 & card cH2 = 1 ) or ( (card cH1) + (card cH2) = k + 1 & k > 0 & card cH1 = 1 & card cH2 > 1 ) or ( (card cH1) + (card cH2) = k + 1 & k > 0 & card cH1 > 1 & card cH2 = 1 ) or ( (card cH1) + (card cH2) = k + 1 & k > 0 & card cH1 > 1 & card cH2 > 1 ) ) by A25, A23, A26, A27, NAT_1:8, XXREAL_0:1;
suppose card cY <= k ; ::_thesis: Y is symmetric
then card cY <= card k1 by CARD_1:def_2;
then card the carrier of Y c= card k by NAT_1:39;
then card the carrier of Y c= k1 by CARD_1:def_2;
hence Y is symmetric by A16, A17, A18; ::_thesis: verum
end;
supposeA28: ( card cY = k + 1 & k = 0 ) ; ::_thesis: Y is symmetric
set x = the set ;
card cY = card { the set } by A28, CARD_1:30;
then cY,{ the set } are_equipotent by CARD_1:5;
then ex y being set st cY = {y} by CARD_1:28;
hence Y is symmetric by A17; ::_thesis: verum
end;
supposeA29: ( (card cH1) + (card cH2) = k + 1 & k > 0 & card cH1 = 1 & card cH2 = 1 ) ; ::_thesis: Y is symmetric
then ex x being set st cH1 = {x} by CARD_2:42;
then the InternalRel of H1 is_symmetric_in cH1 by Th5;
then reconsider H1 = H1 as symmetric RelStr by NECKLACE:def_3;
ex y being set st cH2 = {y} by A29, CARD_2:42;
then the InternalRel of H2 is_symmetric_in cH2 by Th5;
then reconsider H2 = H2 as symmetric RelStr by NECKLACE:def_3;
union_of (H1,H2) is symmetric ;
hence Y is symmetric by A22; ::_thesis: verum
end;
supposeA30: ( (card cH1) + (card cH2) = k + 1 & k > 0 & card cH1 = 1 & card cH2 > 1 ) ; ::_thesis: Y is symmetric
then ex x being set st cH1 = {x} by CARD_2:42;
then the InternalRel of H1 is_symmetric_in cH1 by Th5;
then reconsider H1 = H1 as symmetric RelStr by NECKLACE:def_3;
not card cH2 is trivial by A30, NAT_2:28;
then card cH2 >= 2 by NAT_2:29;
then ( not H2 is empty & not H2 is trivial ) by NAT_D:60;
then reconsider H2 = H2 as symmetric RelStr by A16, A21, A30;
union_of (H1,H2) is symmetric ;
hence Y is symmetric by A22; ::_thesis: verum
end;
supposeA31: ( (card cH1) + (card cH2) = k + 1 & k > 0 & card cH1 > 1 & card cH2 = 1 ) ; ::_thesis: Y is symmetric
then ex x being set st cH2 = {x} by CARD_2:42;
then the InternalRel of H2 is_symmetric_in cH2 by Th5;
then reconsider H2 = H2 as symmetric RelStr by NECKLACE:def_3;
not card cH1 is trivial by A31, NAT_2:28;
then card cH1 >= 2 by NAT_2:29;
then ( not H1 is empty & not H1 is trivial ) by NAT_D:60;
then reconsider H1 = H1 as symmetric RelStr by A16, A20, A31;
union_of (H1,H2) is symmetric ;
hence Y is symmetric by A22; ::_thesis: verum
end;
supposeA32: ( (card cH1) + (card cH2) = k + 1 & k > 0 & card cH1 > 1 & card cH2 > 1 ) ; ::_thesis: Y is symmetric
then not card cH2 is trivial by NAT_2:28;
then card cH2 >= 2 by NAT_2:29;
then A33: ( not H2 is empty & not H2 is trivial ) by NAT_D:60;
card cH2 < k + 1
proof
assume not card cH2 < k + 1 ; ::_thesis: contradiction
then (card cH1) + (card cH2) >= (k + 1) + 1 by A26, XREAL_1:7;
hence contradiction by A32, NAT_1:13; ::_thesis: verum
end;
then card cH2 <= k by NAT_1:13;
then card cH2 <= card k1 by CARD_1:def_2;
then card cH2 c= card k by NAT_1:39;
then card cH2 c= k1 by CARD_1:def_2;
then reconsider H2 = H2 as symmetric RelStr by A16, A21, A33;
not card cH1 is trivial by A32, NAT_2:28;
then card cH1 >= 2 by NAT_2:29;
then A34: ( not H1 is empty & not H1 is trivial ) by NAT_D:60;
card cH1 < k + 1
proof
assume not card cH1 < k + 1 ; ::_thesis: contradiction
then (card cH1) + (card cH2) >= (k + 1) + 1 by A27, XREAL_1:7;
hence contradiction by A32, NAT_1:13; ::_thesis: verum
end;
then card cH1 <= k by NAT_1:13;
then card cH1 <= card k1 by CARD_1:def_2;
then card cH1 c= card k by NAT_1:39;
then card cH1 c= k1 by CARD_1:def_2;
then reconsider H1 = H1 as symmetric RelStr by A16, A20, A34;
union_of (H1,H2) is symmetric ;
hence Y is symmetric by A22; ::_thesis: verum
end;
end;
end;
A35: S1[ 0 ] ;
for k being Nat holds S1[k] from NAT_1:sch_2(A35, A15);
hence X is symmetric by A1, A12, A14; ::_thesis: verum
end;
end;
end;
theorem Th35: :: NECKLA_3:35
for G being RelStr
for H1, H2 being non empty RelStr
for x being Element of H1
for y being 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 ; ::_thesis: for H1, H2 being non empty RelStr
for x being Element of H1
for y being 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
let H1, H2 be non empty RelStr ; ::_thesis: for x being Element of H1
for y being 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
let x be Element of H1; ::_thesis: for y being 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
let y be Element of H2; ::_thesis: ( G = union_of (H1,H2) & the carrier of H1 misses the carrier of H2 implies not [x,y] in the InternalRel of G )
assume that
A1: G = union_of (H1,H2) and
A2: the carrier of H1 misses the carrier of H2 ; ::_thesis: not [x,y] in the InternalRel of G
assume [x,y] in the InternalRel of G ; ::_thesis: contradiction
then A3: [x,y] in the InternalRel of H1 \/ the InternalRel of H2 by A1, NECKLA_2:def_2;
percases ( [x,y] in the InternalRel of H1 or [x,y] in the InternalRel of H2 ) by A3, XBOOLE_0:def_3;
suppose [x,y] in the InternalRel of H1 ; ::_thesis: contradiction
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 contradiction by A2, XBOOLE_0:def_7; ::_thesis: verum
end;
suppose [x,y] in the InternalRel of H2 ; ::_thesis: contradiction
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 contradiction by A2, XBOOLE_0:def_7; ::_thesis: verum
end;
end;
end;
theorem :: NECKLA_3:36
for G being RelStr
for H1, H2 being non empty RelStr
for x being Element of H1
for y being Element of H2 st G = sum_of (H1,H2) holds
not [x,y] in the InternalRel of (ComplRelStr G)
proof
let G be RelStr ; ::_thesis: for H1, H2 being non empty RelStr
for x being Element of H1
for y being Element of H2 st G = sum_of (H1,H2) holds
not [x,y] in the InternalRel of (ComplRelStr G)
let H1, H2 be non empty RelStr ; ::_thesis: for x being Element of H1
for y being Element of H2 st G = sum_of (H1,H2) holds
not [x,y] in the InternalRel of (ComplRelStr G)
let x be Element of H1; ::_thesis: for y being Element of H2 st G = sum_of (H1,H2) holds
not [x,y] in the InternalRel of (ComplRelStr G)
let y be Element of H2; ::_thesis: ( G = sum_of (H1,H2) implies not [x,y] in the InternalRel of (ComplRelStr G) )
set cH1 = the carrier of H1;
set cH2 = the carrier of H2;
set IH1 = the InternalRel of H1;
set IH2 = the InternalRel of H2;
[x,y] in [: the carrier of H1, the carrier of H2:] \/ [: the carrier of H2, the carrier of H1:] by XBOOLE_0:def_3;
then [x,y] in the InternalRel of H2 \/ ([: the carrier of H1, the carrier of H2:] \/ [: the carrier of H2, the carrier of H1:]) by XBOOLE_0:def_3;
then [x,y] in the InternalRel of H1 \/ ( the InternalRel of H2 \/ ([: the carrier of H1, the carrier of H2:] \/ [: the carrier of H2, the carrier of H1:])) by XBOOLE_0:def_3;
then [x,y] in the InternalRel of H1 \/ (( the InternalRel of H2 \/ [: the carrier of H1, the carrier of H2:]) \/ [: the carrier of H2, the carrier of H1:]) by XBOOLE_1:4;
then A1: [x,y] in (( the InternalRel of H1 \/ the InternalRel of H2) \/ [: the carrier of H1, the carrier of H2:]) \/ [: the carrier of H2, the carrier of H1:] by XBOOLE_1:113;
assume G = sum_of (H1,H2) ; ::_thesis: not [x,y] in the InternalRel of (ComplRelStr G)
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 [x,y] in the InternalRel of (ComplRelStr G) ; ::_thesis: contradiction
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) by XBOOLE_0:def_7;
hence contradiction by Th12; ::_thesis: verum
end;
hence not [x,y] in the InternalRel of (ComplRelStr G) ; ::_thesis: verum
end;
theorem Th37: :: NECKLA_3:37
for G being non empty symmetric RelStr
for x being Element of G
for R1, R2 being 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 being Element of R1 st [b,x] in the InternalRel of G
proof
let G be non empty symmetric RelStr ; ::_thesis: for x being Element of G
for R1, R2 being 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 being Element of R1 st [b,x] in the InternalRel of G
let x be Element of G; ::_thesis: for R1, R2 being 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 being Element of R1 st [b,x] in the InternalRel of G
let R1, R2 be non empty RelStr ; ::_thesis: ( the carrier of R1 misses the carrier of R2 & subrelstr (([#] G) \ {x}) = union_of (R1,R2) & G is path-connected implies ex b being Element of R1 st [b,x] in the InternalRel of G )
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 ; ::_thesis: ex b being Element of R1 st [b,x] in the InternalRel of G
set R = subrelstr (([#] G) \ {x});
set A = the carrier of (subrelstr (([#] G) \ {x}));
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 (subrelstr (([#] G) \ {x})) by A2, NECKLA_2:def_2;
set a = the Element of R1;
A5: the carrier of (subrelstr (([#] G) \ {x})) = ([#] G) \ {x} by YELLOW_0:def_15;
A6: x <> the Element of R1
proof
assume not x <> the Element of R1 ; ::_thesis: contradiction
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 contradiction by TARSKI:def_1; ::_thesis: verum
end;
reconsider A = the carrier of (subrelstr (([#] G) \ {x})) as Subset of G by YELLOW_0:def_15;
A7: the carrier of (subrelstr (([#] G) \ {x})) = A ;
then the carrier of R1 c= the carrier of G by A4, XBOOLE_1:1;
then the Element of R1 in the carrier of G by TARSKI:def_3;
then the InternalRel of G reduces x, the Element of R1 by A3, A6, Def2;
then consider p being FinSequence such that
A8: len p > 0 and
A9: p . 1 = x and
A10: p . (len p) = the Element of R1 and
A11: for i being Element of 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 S1[ Nat] means ( p . $1 in the carrier of R1 & $1 in dom p & ( for k being Nat st k > $1 & k in dom p holds
p . k in the carrier of R1 ) );
S1[ len p] by A8, A10, CARD_1:27, FINSEQ_3:25, FINSEQ_5:6;
then A12: ex k being Nat st S1[k] ;
ex n0 being Nat st
( S1[n0] & ( for n being Nat st S1[n] holds
n >= n0 ) ) from NAT_1:sch_5(A12);
then consider n0 being Element of NAT such that
A13: S1[n0] and
A14: for n being Nat st S1[n] holds
n >= n0 ;
n0 <> 0
proof
assume not n0 <> 0 ; ::_thesis: contradiction
then 0 in Seg (len p) by A13, FINSEQ_1:def_3;
hence contradiction by FINSEQ_1:1; ::_thesis: verum
end;
then consider k0 being Nat such that
A15: n0 = k0 + 1 by NAT_1:6;
A16: n0 <> 1
proof
assume not n0 <> 1 ; ::_thesis: contradiction
then not x in {x} by A5, A4, A9, A13, XBOOLE_0:def_5;
hence contradiction by TARSKI:def_1; ::_thesis: verum
end;
A17: k0 >= 1
proof
assume not k0 >= 1 ; ::_thesis: contradiction
then k0 = 0 by NAT_1:25;
hence contradiction by A15, A16; ::_thesis: verum
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 being Nat st k > k0 & k in dom p holds
p . k in the carrier of R1
proof
assume ex k being Nat st
( k > k0 & k in dom p & not p . k in the carrier of R1 ) ; ::_thesis: contradiction
then consider k being 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
percases ( k < n0 or n0 < k or n0 = k ) by XXREAL_0:1;
suppose k < n0 ; ::_thesis: k > n0
hence k > n0 by A15, A21, NAT_1:13; ::_thesis: verum
end;
suppose n0 < k ; ::_thesis: k > n0
hence k > n0 ; ::_thesis: verum
end;
suppose n0 = k ; ::_thesis: k > n0
hence k > n0 by A13, A23; ::_thesis: verum
end;
end;
end;
hence contradiction by A13, A22, A23; ::_thesis: verum
end;
A24: the carrier of G = the carrier of (subrelstr (([#] G) \ {x})) \/ {x}
proof
thus the carrier of G c= the carrier of (subrelstr (([#] G) \ {x})) \/ {x} :: according to XBOOLE_0:def_10 ::_thesis: the carrier of (subrelstr (([#] G) \ {x})) \/ {x} c= the carrier of G
proof
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in the carrier of G or a in the carrier of (subrelstr (([#] G) \ {x})) \/ {x} )
assume A25: a in the carrier of G ; ::_thesis: a in the carrier of (subrelstr (([#] G) \ {x})) \/ {x}
percases ( a = x or a <> x ) ;
suppose a = x ; ::_thesis: a in the carrier of (subrelstr (([#] G) \ {x})) \/ {x}
then a in {x} by TARSKI:def_1;
hence a in the carrier of (subrelstr (([#] G) \ {x})) \/ {x} by XBOOLE_0:def_3; ::_thesis: verum
end;
suppose a <> x ; ::_thesis: a in the carrier of (subrelstr (([#] G) \ {x})) \/ {x}
then not a in {x} by TARSKI:def_1;
then a in A by A5, A25, XBOOLE_0:def_5;
hence a in the carrier of (subrelstr (([#] G) \ {x})) \/ {x} by XBOOLE_0:def_3; ::_thesis: verum
end;
end;
end;
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in the carrier of (subrelstr (([#] G) \ {x})) \/ {x} or a in the carrier of G )
assume A26: a in the carrier of (subrelstr (([#] G) \ {x})) \/ {x} ; ::_thesis: a in the carrier of G
percases ( a in the carrier of (subrelstr (([#] G) \ {x})) or a in {x} ) by A26, XBOOLE_0:def_3;
suppose a in the carrier of (subrelstr (([#] G) \ {x})) ; ::_thesis: a in the carrier of G
hence a in the carrier of G by A5; ::_thesis: verum
end;
suppose a in {x} ; ::_thesis: a in the carrier of G
hence a in the carrier of G ; ::_thesis: verum
end;
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 ex b being Element of R1 st [b,x] in the InternalRel of G ::_thesis: verum
proof
percases ( p . k0 in the carrier of (subrelstr (([#] G) \ {x})) or p . k0 in {x} ) by A29, A24, XBOOLE_0:def_3;
supposeA30: p . k0 in the carrier of (subrelstr (([#] G) \ {x})) ; ::_thesis: ex b being Element of R1 st [b,x] in the InternalRel of G
set u = p . k0;
set v = p . n0;
[(p . k0),(p . n0)] in [: the carrier of (subrelstr (([#] G) \ {x})), the carrier of (subrelstr (([#] G) \ {x})):] by A4, A13, A30, ZFMISC_1:87;
then A31: [(p . k0),(p . n0)] in the InternalRel of G |_2 the carrier of (subrelstr (([#] G) \ {x})) 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 = p . k0 as Element of R2 by A14, A27, A19, A20;
reconsider v = p . n0 as Element of R1 by A13;
not [u,v] in the InternalRel of (subrelstr (([#] G) \ {x}))
proof
u in the carrier of R1 \/ the carrier of R2 by XBOOLE_0:def_3;
then A32: u in the carrier of (subrelstr (([#] G) \ {x})) by A2, NECKLA_2:def_2;
A33: ( v in the carrier of R1 & the InternalRel of (subrelstr (([#] G) \ {x})) is_symmetric_in the carrier of (subrelstr (([#] G) \ {x})) ) by NECKLACE:def_3;
assume [u,v] in the InternalRel of (subrelstr (([#] G) \ {x})) ; ::_thesis: contradiction
then [v,u] in the InternalRel of (subrelstr (([#] G) \ {x})) by A4, A32, A33, RELAT_2:def_3;
hence contradiction by A1, A2, Th35; ::_thesis: verum
end;
hence ex b being Element of R1 st [b,x] in the InternalRel of G by A31, YELLOW_0:def_14; ::_thesis: verum
end;
supposeA34: p . k0 in {x} ; ::_thesis: ex b being Element of R1 st [b,x] in the InternalRel of G
set b = p . n0;
reconsider b = p . n0 as Element of R1 by A13;
A35: ( b in the carrier of (subrelstr (([#] G) \ {x})) & the InternalRel of G is_symmetric_in the carrier of G ) by A4, NECKLACE:def_3, TARSKI:def_3;
p . k0 = x by A34, TARSKI:def_1;
then [b,x] in the InternalRel of G by A7, A15, A28, A35, RELAT_2:def_3;
hence ex b being Element of R1 st [b,x] in the InternalRel of G ; ::_thesis: verum
end;
end;
end;
end;
theorem Th38: :: NECKLA_3:38
for G being non empty symmetric irreflexive RelStr
for a, b, c, d being Element of G
for Z being Subset of G st Z = {a,b,c,d} & a,b,c,d are_mutually_different & [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 RelStr ; ::_thesis: for a, b, c, d being Element of G
for Z being Subset of G st Z = {a,b,c,d} & a,b,c,d are_mutually_different & [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
let a, b, c, d be Element of G; ::_thesis: for Z being Subset of G st Z = {a,b,c,d} & a,b,c,d are_mutually_different & [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
let Z be Subset of G; ::_thesis: ( Z = {a,b,c,d} & a,b,c,d are_mutually_different & [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 implies subrelstr Z embeds Necklace 4 )
assume that
A1: Z = {a,b,c,d} and
A2: a,b,c,d are_mutually_different 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 ; ::_thesis: subrelstr Z embeds Necklace 4
set g = (0,1) --> (a,b);
set h = (2,3) --> (c,d);
set f = ((0,1) --> (a,b)) +* ((2,3) --> (c,d));
A9: rng ((2,3) --> (c,d)) = {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) ; ::_thesis: contradiction
then consider x being set 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; ::_thesis: verum
end;
set H = subrelstr Z;
set N4 = Necklace 4;
set IH = the InternalRel of (subrelstr Z);
set cH = the carrier of (subrelstr Z);
set IG = the InternalRel of G;
set X = {[a,a],[a,b],[b,a],[b,b],[a,c],[a,d],[b,c],[b,d]};
set Y = {[c,a],[c,b],[d,a],[d,b],[c,c],[c,d],[d,c],[d,d]};
A14: not the carrier of (subrelstr Z) is empty by A1, YELLOW_0:def_15;
A15: (2,3) --> (c,d) = (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) ; ::_thesis: contradiction
then consider x being set 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; ::_thesis: verum
end;
then A19: (2,3) --> (c,d) is one-to-one by A15, FUNCT_4:92;
A20: rng ((0,1) --> (a,b)) = {a,b} by FUNCT_4:64;
A21: rng ((0,1) --> (a,b)) misses rng ((2,3) --> (c,d))
proof
assume not rng ((0,1) --> (a,b)) misses rng ((2,3) --> (c,d)) ; ::_thesis: contradiction
then consider x being set such that
A22: x in rng ((0,1) --> (a,b)) and
A23: x in rng ((2,3) --> (c,d)) 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; ::_thesis: verum
end;
dom (((0,1) --> (a,b)) +* ((2,3) --> (c,d))) = (dom ((0,1) --> (a,b))) \/ (dom ((2,3) --> (c,d))) by FUNCT_4:def_1
.= {0,1} \/ (dom ((2,3) --> (c,d))) by FUNCT_4:62
.= {0,1} \/ {2,3} by FUNCT_4:62
.= {0,1,2,3} by ENUMSET1:5 ;
then A25: dom (((0,1) --> (a,b)) +* ((2,3) --> (c,d))) = the carrier of (Necklace 4) by NECKLACE:1, NECKLACE:20;
A26: dom ((0,1) --> (a,b)) misses dom ((2,3) --> (c,d))
proof
assume not dom ((0,1) --> (a,b)) misses dom ((2,3) --> (c,d)) ; ::_thesis: contradiction
then consider x being set such that
A27: x in dom ((0,1) --> (a,b)) and
A28: x in dom ((2,3) --> (c,d)) by XBOOLE_0:3;
( x = 0 or x = 1 ) by A27, TARSKI:def_2;
hence contradiction by A28, TARSKI:def_2; ::_thesis: verum
end;
then rng (((0,1) --> (a,b)) +* ((2,3) --> (c,d))) = (rng ((0,1) --> (a,b))) \/ (rng ((2,3) --> (c,d))) by NECKLACE:6;
then rng (((0,1) --> (a,b)) +* ((2,3) --> (c,d))) = {a,b,c,d} by A20, A9, ENUMSET1:5;
then A29: rng (((0,1) --> (a,b)) +* ((2,3) --> (c,d))) = the carrier of (subrelstr Z) by A1, YELLOW_0:def_15;
then reconsider f = ((0,1) --> (a,b)) +* ((2,3) --> (c,d)) as Function of (Necklace 4),(subrelstr Z) by A25, FUNCT_2:def_1, RELSET_1:4;
(0,1) --> (a,b) = (0 .--> a) +* (1 .--> b) by FUNCT_4:def_4;
then A30: (0,1) --> (a,b) 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 (subrelstr Z) = {[a,b],[b,a],[b,c],[c,b],[c,d],[d,c]}
proof
thus the InternalRel of (subrelstr Z) c= {[a,b],[b,a],[b,c],[c,b],[c,d],[d,c]} :: according to XBOOLE_0:def_10 ::_thesis: {[a,b],[b,a],[b,c],[c,b],[c,d],[d,c]} c= the InternalRel of (subrelstr Z)
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in the InternalRel of (subrelstr Z) or x in {[a,b],[b,a],[b,c],[c,b],[c,d],[d,c]} )
A33: the carrier of (subrelstr Z) = Z by YELLOW_0:def_15;
assume A34: x in the InternalRel of (subrelstr Z) ; ::_thesis: x in {[a,b],[b,a],[b,c],[c,b],[c,d],[d,c]}
then A35: x in the InternalRel of G |_2 the carrier of (subrelstr Z) by YELLOW_0:def_14;
then A36: x in the InternalRel of G by XBOOLE_0:def_4;
x in [: the carrier of (subrelstr Z), the carrier of (subrelstr Z):] by A34;
then A37: x in {[a,a],[a,b],[b,a],[b,b],[a,c],[a,d],[b,c],[b,d]} \/ {[c,a],[c,b],[d,a],[d,b],[c,c],[c,d],[d,c],[d,d]} by A1, A33, Th3;
percases ( x in {[a,a],[a,b],[b,a],[b,b],[a,c],[a,d],[b,c],[b,d]} or x in {[c,a],[c,b],[d,a],[d,b],[c,c],[c,d],[d,c],[d,d]} ) by A37, XBOOLE_0:def_3;
supposeA38: x in {[a,a],[a,b],[b,a],[b,b],[a,c],[a,d],[b,c],[b,d]} ; ::_thesis: x in {[a,b],[b,a],[b,c],[c,b],[c,d],[d,c]}
thus x in {[a,b],[b,a],[b,c],[c,b],[c,d],[d,c]} ::_thesis: verum
proof
percases ( x = [a,a] or x = [a,b] or x = [b,a] or x = [b,b] or x = [a,c] or x = [a,d] or x = [b,c] or x = [b,d] ) by A38, ENUMSET1:def_6;
supposeA39: x = [a,a] ; ::_thesis: x in {[a,b],[b,a],[b,c],[c,b],[c,d],[d,c]}
not [a,a] in the InternalRel of G by NECKLACE:def_5;
hence x in {[a,b],[b,a],[b,c],[c,b],[c,d],[d,c]} by A35, A39, XBOOLE_0:def_4; ::_thesis: verum
end;
suppose x = [a,b] ; ::_thesis: x in {[a,b],[b,a],[b,c],[c,b],[c,d],[d,c]}
hence x in {[a,b],[b,a],[b,c],[c,b],[c,d],[d,c]} by ENUMSET1:def_4; ::_thesis: verum
end;
suppose x = [b,a] ; ::_thesis: x in {[a,b],[b,a],[b,c],[c,b],[c,d],[d,c]}
hence x in {[a,b],[b,a],[b,c],[c,b],[c,d],[d,c]} by ENUMSET1:def_4; ::_thesis: verum
end;
supposeA40: x = [b,b] ; ::_thesis: x in {[a,b],[b,a],[b,c],[c,b],[c,d],[d,c]}
not [b,b] in the InternalRel of G by NECKLACE:def_5;
hence x in {[a,b],[b,a],[b,c],[c,b],[c,d],[d,c]} by A35, A40, XBOOLE_0:def_4; ::_thesis: verum
end;
suppose x = [a,c] ; ::_thesis: x in {[a,b],[b,a],[b,c],[c,b],[c,d],[d,c]}
hence x in {[a,b],[b,a],[b,c],[c,b],[c,d],[d,c]} by A6, A35, XBOOLE_0:def_4; ::_thesis: verum
end;
suppose x = [a,d] ; ::_thesis: x in {[a,b],[b,a],[b,c],[c,b],[c,d],[d,c]}
hence x in {[a,b],[b,a],[b,c],[c,b],[c,d],[d,c]} by A7, A35, XBOOLE_0:def_4; ::_thesis: verum
end;
suppose x = [b,c] ; ::_thesis: x in {[a,b],[b,a],[b,c],[c,b],[c,d],[d,c]}
hence x in {[a,b],[b,a],[b,c],[c,b],[c,d],[d,c]} by ENUMSET1:def_4; ::_thesis: verum
end;
suppose x = [b,d] ; ::_thesis: x in {[a,b],[b,a],[b,c],[c,b],[c,d],[d,c]}
hence x in {[a,b],[b,a],[b,c],[c,b],[c,d],[d,c]} by A8, A35, XBOOLE_0:def_4; ::_thesis: verum
end;
end;
end;
end;
supposeA41: x in {[c,a],[c,b],[d,a],[d,b],[c,c],[c,d],[d,c],[d,d]} ; ::_thesis: x in {[a,b],[b,a],[b,c],[c,b],[c,d],[d,c]}
A42: the InternalRel of G is_symmetric_in the carrier of G by NECKLACE:def_3;
thus x in {[a,b],[b,a],[b,c],[c,b],[c,d],[d,c]} ::_thesis: verum
proof
percases ( x = [c,a] or x = [c,b] or x = [d,a] or x = [d,b] or x = [c,c] or x = [c,d] or x = [d,c] or x = [d,d] ) by A41, ENUMSET1:def_6;
suppose x = [c,a] ; ::_thesis: x in {[a,b],[b,a],[b,c],[c,b],[c,d],[d,c]}
hence x in {[a,b],[b,a],[b,c],[c,b],[c,d],[d,c]} by A6, A36, A42, RELAT_2:def_3; ::_thesis: verum
end;
suppose x = [c,b] ; ::_thesis: x in {[a,b],[b,a],[b,c],[c,b],[c,d],[d,c]}
hence x in {[a,b],[b,a],[b,c],[c,b],[c,d],[d,c]} by ENUMSET1:def_4; ::_thesis: verum
end;
suppose x = [d,a] ; ::_thesis: x in {[a,b],[b,a],[b,c],[c,b],[c,d],[d,c]}
hence x in {[a,b],[b,a],[b,c],[c,b],[c,d],[d,c]} by A7, A36, A42, RELAT_2:def_3; ::_thesis: verum
end;
suppose x = [d,b] ; ::_thesis: x in {[a,b],[b,a],[b,c],[c,b],[c,d],[d,c]}
hence x in {[a,b],[b,a],[b,c],[c,b],[c,d],[d,c]} by A8, A36, A42, RELAT_2:def_3; ::_thesis: verum
end;
supposeA43: x = [c,c] ; ::_thesis: x in {[a,b],[b,a],[b,c],[c,b],[c,d],[d,c]}
not [c,c] in the InternalRel of G by NECKLACE:def_5;
hence x in {[a,b],[b,a],[b,c],[c,b],[c,d],[d,c]} by A35, A43, XBOOLE_0:def_4; ::_thesis: verum
end;
suppose x = [c,d] ; ::_thesis: x in {[a,b],[b,a],[b,c],[c,b],[c,d],[d,c]}
hence x in {[a,b],[b,a],[b,c],[c,b],[c,d],[d,c]} by ENUMSET1:def_4; ::_thesis: verum
end;
suppose x = [d,c] ; ::_thesis: x in {[a,b],[b,a],[b,c],[c,b],[c,d],[d,c]}
hence x in {[a,b],[b,a],[b,c],[c,b],[c,d],[d,c]} by ENUMSET1:def_4; ::_thesis: verum
end;
supposeA44: x = [d,d] ; ::_thesis: x in {[a,b],[b,a],[b,c],[c,b],[c,d],[d,c]}
not [d,d] in the InternalRel of G by NECKLACE:def_5;
hence x in {[a,b],[b,a],[b,c],[c,b],[c,d],[d,c]} by A35, A44, XBOOLE_0:def_4; ::_thesis: verum
end;
end;
end;
end;
end;
end;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {[a,b],[b,a],[b,c],[c,b],[c,d],[d,c]} or x in the InternalRel of (subrelstr Z) )
assume A45: x in {[a,b],[b,a],[b,c],[c,b],[c,d],[d,c]} ; ::_thesis: x in the InternalRel of (subrelstr Z)
percases ( x = [a,b] or x = [b,a] or x = [b,c] or x = [c,b] or x = [c,d] or x = [d,c] ) by A45, ENUMSET1:def_4;
supposeA46: x = [a,b] ; ::_thesis: x in the InternalRel of (subrelstr Z)
b in Z by A1, ENUMSET1:def_2;
then A47: b in the carrier of (subrelstr Z) by YELLOW_0:def_15;
a in Z by A1, ENUMSET1:def_2;
then a in the carrier of (subrelstr Z) by YELLOW_0:def_15;
then [a,b] in [: the carrier of (subrelstr Z), the carrier of (subrelstr Z):] by A47, ZFMISC_1:87;
then x in the InternalRel of G |_2 the carrier of (subrelstr Z) by A3, A46, XBOOLE_0:def_4;
hence x in the InternalRel of (subrelstr Z) by YELLOW_0:def_14; ::_thesis: verum
end;
supposeA48: x = [b,a] ; ::_thesis: x in the InternalRel of (subrelstr Z)
the InternalRel of G is_symmetric_in the carrier of G by NECKLACE:def_3;
then A49: [b,a] in the InternalRel of G by A3, RELAT_2:def_3;
a in Z by A1, ENUMSET1:def_2;
then A50: a in the carrier of (subrelstr Z) by YELLOW_0:def_15;
b in Z by A1, ENUMSET1:def_2;
then b in the carrier of (subrelstr Z) by YELLOW_0:def_15;
then [b,a] in [: the carrier of (subrelstr Z), the carrier of (subrelstr Z):] by A50, ZFMISC_1:87;
then x in the InternalRel of G |_2 the carrier of (subrelstr Z) by A48, A49, XBOOLE_0:def_4;
hence x in the InternalRel of (subrelstr Z) by YELLOW_0:def_14; ::_thesis: verum
end;
supposeA51: x = [b,c] ; ::_thesis: x in the InternalRel of (subrelstr Z)
c in Z by A1, ENUMSET1:def_2;
then A52: c in the carrier of (subrelstr Z) by YELLOW_0:def_15;
b in Z by A1, ENUMSET1:def_2;
then b in the carrier of (subrelstr Z) by YELLOW_0:def_15;
then [b,c] in [: the carrier of (subrelstr Z), the carrier of (subrelstr Z):] by A52, ZFMISC_1:87;
then x in the InternalRel of G |_2 the carrier of (subrelstr Z) by A4, A51, XBOOLE_0:def_4;
hence x in the InternalRel of (subrelstr Z) by YELLOW_0:def_14; ::_thesis: verum
end;
supposeA53: x = [c,b] ; ::_thesis: x in the InternalRel of (subrelstr Z)
the InternalRel of G is_symmetric_in the carrier of G by NECKLACE:def_3;
then A54: [c,b] in the InternalRel of G by A4, RELAT_2:def_3;
c in Z by A1, ENUMSET1:def_2;
then A55: c in the carrier of (subrelstr Z) by YELLOW_0:def_15;
b in Z by A1, ENUMSET1:def_2;
then b in the carrier of (subrelstr Z) by YELLOW_0:def_15;
then [c,b] in [: the carrier of (subrelstr Z), the carrier of (subrelstr Z):] by A55, ZFMISC_1:87;
then x in the InternalRel of G |_2 the carrier of (subrelstr Z) by A53, A54, XBOOLE_0:def_4;
hence x in the InternalRel of (subrelstr Z) by YELLOW_0:def_14; ::_thesis: verum
end;
supposeA56: x = [c,d] ; ::_thesis: x in the InternalRel of (subrelstr Z)
d in Z by A1, ENUMSET1:def_2;
then A57: d in the carrier of (subrelstr Z) by YELLOW_0:def_15;
c in Z by A1, ENUMSET1:def_2;
then c in the carrier of (subrelstr Z) by YELLOW_0:def_15;
then [c,d] in [: the carrier of (subrelstr Z), the carrier of (subrelstr Z):] by A57, ZFMISC_1:87;
then x in the InternalRel of G |_2 the carrier of (subrelstr Z) by A5, A56, XBOOLE_0:def_4;
hence x in the InternalRel of (subrelstr Z) by YELLOW_0:def_14; ::_thesis: verum
end;
supposeA58: x = [d,c] ; ::_thesis: x in the InternalRel of (subrelstr Z)
the InternalRel of G is_symmetric_in the carrier of G by NECKLACE:def_3;
then A59: [d,c] in the InternalRel of G by A5, RELAT_2:def_3;
d in Z by A1, ENUMSET1:def_2;
then A60: d in the carrier of (subrelstr Z) by YELLOW_0:def_15;
c in Z by A1, ENUMSET1:def_2;
then c in the carrier of (subrelstr Z) by YELLOW_0:def_15;
then [d,c] in [: the carrier of (subrelstr Z), the carrier of (subrelstr Z):] by A60, ZFMISC_1:87;
then x in the InternalRel of G |_2 the carrier of (subrelstr Z) by A58, A59, XBOOLE_0:def_4;
hence x in the InternalRel of (subrelstr Z) by YELLOW_0:def_14; ::_thesis: verum
end;
end;
end;
for x, y being Element of (Necklace 4) holds
( [x,y] in the InternalRel of (Necklace 4) iff [(f . x),(f . y)] in the InternalRel of (subrelstr Z) )
proof
let x, y be Element of (Necklace 4); ::_thesis: ( [x,y] in the InternalRel of (Necklace 4) iff [(f . x),(f . y)] in the InternalRel of (subrelstr Z) )
thus ( [x,y] in the InternalRel of (Necklace 4) implies [(f . x),(f . y)] in the InternalRel of (subrelstr Z) ) ::_thesis: ( [(f . x),(f . y)] in the InternalRel of (subrelstr Z) implies [x,y] in the InternalRel of (Necklace 4) )
proof
assume A61: [x,y] in the InternalRel of (Necklace 4) ; ::_thesis: [(f . x),(f . y)] in the InternalRel of (subrelstr Z)
percases ( [x,y] = [0,1] or [x,y] = [1,0] or [x,y] = [1,2] or [x,y] = [2,1] or [x,y] = [2,3] or [x,y] = [3,2] ) by A61, ENUMSET1:def_4, NECKLA_2:2;
supposeA62: [x,y] = [0,1] ; ::_thesis: [(f . x),(f . y)] in the InternalRel of (subrelstr Z)
then A63: y = 1 by XTUPLE_0:1;
then y in {0,1} by TARSKI:def_2;
then y in dom ((0,1) --> (a,b)) by FUNCT_4:62;
then A64: f . y = ((0,1) --> (a,b)) . 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 ((0,1) --> (a,b)) by FUNCT_4:62;
then f . x = ((0,1) --> (a,b)) . 0 by A26, A65, FUNCT_4:16
.= a by FUNCT_4:63 ;
hence [(f . x),(f . y)] in the InternalRel of (subrelstr Z) by A32, A64, ENUMSET1:def_4; ::_thesis: verum
end;
supposeA66: [x,y] = [1,0] ; ::_thesis: [(f . x),(f . y)] in the InternalRel of (subrelstr Z)
then A67: y = 0 by XTUPLE_0:1;
then y in {0,1} by TARSKI:def_2;
then y in dom ((0,1) --> (a,b)) by FUNCT_4:62;
then A68: f . y = ((0,1) --> (a,b)) . 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 ((0,1) --> (a,b)) by FUNCT_4:62;
then f . x = ((0,1) --> (a,b)) . 1 by A26, A69, FUNCT_4:16
.= b by FUNCT_4:63 ;
hence [(f . x),(f . y)] in the InternalRel of (subrelstr Z) by A32, A68, ENUMSET1:def_4; ::_thesis: verum
end;
supposeA70: [x,y] = [1,2] ; ::_thesis: [(f . x),(f . y)] in the InternalRel of (subrelstr Z)
then A71: x = 1 by XTUPLE_0:1;
then x in {0,1} by TARSKI:def_2;
then x in dom ((0,1) --> (a,b)) by FUNCT_4:62;
then A72: f . x = ((0,1) --> (a,b)) . 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 ((2,3) --> (c,d)) by FUNCT_4:62;
((0,1) --> (a,b)) +* ((2,3) --> (c,d)) = ((2,3) --> (c,d)) +* ((0,1) --> (a,b)) by A26, FUNCT_4:35;
then f . y = ((2,3) --> (c,d)) . 2 by A26, A73, A74, FUNCT_4:16
.= c by FUNCT_4:63 ;
hence [(f . x),(f . y)] in the InternalRel of (subrelstr Z) by A32, A72, ENUMSET1:def_4; ::_thesis: verum
end;
supposeA75: [x,y] = [2,1] ; ::_thesis: [(f . x),(f . y)] in the InternalRel of (subrelstr Z)
then A76: y = 1 by XTUPLE_0:1;
then y in {0,1} by TARSKI:def_2;
then y in dom ((0,1) --> (a,b)) by FUNCT_4:62;
then A77: f . y = ((0,1) --> (a,b)) . 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 ((2,3) --> (c,d)) by FUNCT_4:62;
((0,1) --> (a,b)) +* ((2,3) --> (c,d)) = ((2,3) --> (c,d)) +* ((0,1) --> (a,b)) by A26, FUNCT_4:35;
then f . x = ((2,3) --> (c,d)) . 2 by A26, A78, A79, FUNCT_4:16
.= c by FUNCT_4:63 ;
hence [(f . x),(f . y)] in the InternalRel of (subrelstr Z) by A32, A77, ENUMSET1:def_4; ::_thesis: verum
end;
supposeA80: [x,y] = [2,3] ; ::_thesis: [(f . x),(f . y)] in the InternalRel of (subrelstr Z)
A81: ((0,1) --> (a,b)) +* ((2,3) --> (c,d)) = ((2,3) --> (c,d)) +* ((0,1) --> (a,b)) 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 ((2,3) --> (c,d)) by FUNCT_4:62;
then A83: f . y = ((2,3) --> (c,d)) . 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 ((2,3) --> (c,d)) by FUNCT_4:62;
then f . x = ((2,3) --> (c,d)) . 2 by A26, A84, A81, FUNCT_4:16
.= c by FUNCT_4:63 ;
hence [(f . x),(f . y)] in the InternalRel of (subrelstr Z) by A32, A83, ENUMSET1:def_4; ::_thesis: verum
end;
supposeA85: [x,y] = [3,2] ; ::_thesis: [(f . x),(f . y)] in the InternalRel of (subrelstr Z)
A86: ((0,1) --> (a,b)) +* ((2,3) --> (c,d)) = ((2,3) --> (c,d)) +* ((0,1) --> (a,b)) 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 ((2,3) --> (c,d)) by FUNCT_4:62;
then A88: f . y = ((2,3) --> (c,d)) . 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 ((2,3) --> (c,d)) by FUNCT_4:62;
then f . x = ((2,3) --> (c,d)) . 3 by A26, A89, A86, FUNCT_4:16
.= d by FUNCT_4:63 ;
hence [(f . x),(f . y)] in the InternalRel of (subrelstr Z) by A32, A88, ENUMSET1:def_4; ::_thesis: verum
end;
end;
end;
thus ( [(f . x),(f . y)] in the InternalRel of (subrelstr Z) implies [x,y] in the InternalRel of (Necklace 4) ) ::_thesis: verum
proof
reconsider F = f " as Function of the carrier of (subrelstr Z), the carrier of (Necklace 4) by A29, A31, FUNCT_2:25;
A90: dom ((0,1) --> (a,b)) = {0,1} by FUNCT_4:62;
A91: rng ((0,1) --> (a,b)) = {a,b} by FUNCT_4:64;
then reconsider g = (0,1) --> (a,b) 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 ((2,3) --> (c,d)) = {2,3} by FUNCT_4:62;
A93: rng ((2,3) --> (c,d)) = {c,d} by FUNCT_4:64;
then reconsider h = (2,3) --> (c,d) 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 (subrelstr Z) ; ::_thesis: [x,y] in the InternalRel of (Necklace 4)
percases ( [(f . x),(f . y)] = [a,b] or [(f . x),(f . y)] = [b,a] or [(f . x),(f . y)] = [b,c] or [(f . x),(f . y)] = [c,b] or [(f . x),(f . y)] = [c,d] or [(f . x),(f . y)] = [d,c] ) by A32, A100, ENUMSET1:def_4;
supposeA101: [(f . x),(f . y)] = [a,b] ; ::_thesis: [x,y] in the InternalRel of (Necklace 4)
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 [x,y] in the InternalRel of (Necklace 4) by A103, A105, ENUMSET1:def_4, NECKLA_2:2; ::_thesis: verum
end;
supposeA106: [(f . x),(f . y)] = [b,a] ; ::_thesis: [x,y] in the InternalRel of (Necklace 4)
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 [x,y] in the InternalRel of (Necklace 4) by A108, A110, ENUMSET1:def_4, NECKLA_2:2; ::_thesis: verum
end;
supposeA111: [(f . x),(f . y)] = [b,c] ; ::_thesis: [x,y] in the InternalRel of (Necklace 4)
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 [x,y] in the InternalRel of (Necklace 4) by A113, A115, ENUMSET1:def_4, NECKLA_2:2; ::_thesis: verum
end;
supposeA116: [(f . x),(f . y)] = [c,b] ; ::_thesis: [x,y] in the InternalRel of (Necklace 4)
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 [x,y] in the InternalRel of (Necklace 4) by A118, A120, ENUMSET1:def_4, NECKLA_2:2; ::_thesis: verum
end;
supposeA121: [(f . x),(f . y)] = [c,d] ; ::_thesis: [x,y] in the InternalRel of (Necklace 4)
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 [x,y] in the InternalRel of (Necklace 4) by A123, A125, ENUMSET1:def_4, NECKLA_2:2; ::_thesis: verum
end;
supposeA126: [(f . x),(f . y)] = [d,c] ; ::_thesis: [x,y] in the InternalRel of (Necklace 4)
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 [x,y] in the InternalRel of (Necklace 4) by A128, A130, ENUMSET1:def_4, NECKLA_2:2; ::_thesis: verum
end;
end;
end;
end;
hence subrelstr Z embeds Necklace 4 by A31, NECKLACE:def_1; ::_thesis: verum
end;
theorem Th39: :: NECKLA_3:39
for G being non empty symmetric irreflexive RelStr
for x being Element of G
for R1, R2 being non empty RelStr st the carrier of R1 misses the carrier of R2 & subrelstr (([#] G) \ {x}) = union_of (R1,R2) & not G is trivial & G is path-connected & ComplRelStr G is path-connected holds
G embeds Necklace 4
proof
let G be non empty symmetric irreflexive RelStr ; ::_thesis: for x being Element of G
for R1, R2 being non empty RelStr st the carrier of R1 misses the carrier of R2 & subrelstr (([#] G) \ {x}) = union_of (R1,R2) & not G is trivial & G is path-connected & ComplRelStr G is path-connected holds
G embeds Necklace 4
let x be Element of G; ::_thesis: for R1, R2 being non empty RelStr st the carrier of R1 misses the carrier of R2 & subrelstr (([#] G) \ {x}) = union_of (R1,R2) & not G is trivial & G is path-connected & ComplRelStr G is path-connected holds
G embeds Necklace 4
let R1, R2 be non empty RelStr ; ::_thesis: ( the carrier of R1 misses the carrier of R2 & subrelstr (([#] G) \ {x}) = union_of (R1,R2) & not G is trivial & G is path-connected & ComplRelStr G is path-connected implies G embeds Necklace 4 )
assume that
A1: the carrier of R1 misses the carrier of R2 and
A2: subrelstr (([#] G) \ {x}) = union_of (R1,R2) and
A3: not G is trivial and
A4: G is path-connected and
A5: ComplRelStr G is path-connected ; ::_thesis: G embeds Necklace 4
consider a being 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};
set X = {x};
reconsider A = the carrier of G \ {x} as Subset of G ;
set R = subrelstr A;
reconsider R = subrelstr A as non empty symmetric irreflexive RelStr by A3, YELLOW_0:def_15;
( R = subrelstr (([#] G) \ {x}) & R = union_of (R2,R1) ) by A2, Th8;
then consider b being Element of R2 such that
A7: [b,x] in the InternalRel of G by A1, A4, Th37;
reconsider X1 = { y where y is Element of R1 : [y,x] in the InternalRel of G } , Y1 = { y where y is Element of R1 : not [y,x] in the InternalRel of G } , X2 = { y where y is Element of R2 : [y,x] in the InternalRel of G } , Y2 = { y where y is Element of R2 : not [y,x] in the InternalRel of G } as set ;
reconsider X = {x} as Subset of G ;
set H = subrelstr X;
A8: X1 misses Y1
proof
assume not X1 misses Y1 ; ::_thesis: contradiction
then consider a being set such that
A9: ( a in X1 & a in Y1 ) by XBOOLE_0:3;
( ex y1 being Element of R1 st
( y1 = a & [y1,x] in the InternalRel of G ) & ex y2 being Element of R1 st
( y2 = a & not [y2,x] in the InternalRel of G ) ) by A9;
hence contradiction ; ::_thesis: verum
end;
A10: a in X1 by A6;
A11: the carrier of R1 = X1 \/ Y1
proof
thus the carrier of R1 c= X1 \/ Y1 :: according to XBOOLE_0:def_10 ::_thesis: X1 \/ Y1 c= the carrier of R1
proof
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in the carrier of R1 or a in X1 \/ Y1 )
assume A12: a in the carrier of R1 ; ::_thesis: a in X1 \/ Y1
percases ( [a,x] in the InternalRel of G or not [a,x] in the InternalRel of G ) ;
suppose [a,x] in the InternalRel of G ; ::_thesis: a in X1 \/ Y1
then a in X1 by A12;
hence a in X1 \/ Y1 by XBOOLE_0:def_3; ::_thesis: verum
end;
suppose not [a,x] in the InternalRel of G ; ::_thesis: a in X1 \/ Y1
then a in Y1 by A12;
hence a in X1 \/ Y1 by XBOOLE_0:def_3; ::_thesis: verum
end;
end;
end;
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in X1 \/ Y1 or a in the carrier of R1 )
assume A13: a in X1 \/ Y1 ; ::_thesis: a in the carrier of R1
percases ( a in X1 or a in Y1 ) by A13, XBOOLE_0:def_3;
suppose a in X1 ; ::_thesis: a in the carrier of R1
then ex y being Element of R1 st
( a = y & [y,x] in the InternalRel of G ) ;
hence a in the carrier of R1 ; ::_thesis: verum
end;
suppose a in Y1 ; ::_thesis: a in the carrier of R1
then ex y being Element of R1 st
( a = y & not [y,x] in the InternalRel of G ) ;
hence a in the carrier of R1 ; ::_thesis: verum
end;
end;
end;
A14: X2 misses Y2
proof
assume not X2 misses Y2 ; ::_thesis: contradiction
then consider a being set such that
A15: ( a in X2 & a in Y2 ) by XBOOLE_0:3;
( ex y1 being Element of R2 st
( y1 = a & [y1,x] in the InternalRel of G ) & ex y2 being Element of R2 st
( y2 = a & not [y2,x] in the InternalRel of G ) ) by A15;
hence contradiction ; ::_thesis: verum
end;
A16: the carrier of (subrelstr X) misses the carrier of R
proof
assume not the carrier of (subrelstr X) misses the carrier of R ; ::_thesis: contradiction
then the carrier of (subrelstr X) /\ the carrier of R <> {} by XBOOLE_0:def_7;
then X /\ the carrier of R <> {} by YELLOW_0:def_15;
then X /\ A <> {} by YELLOW_0:def_15;
then consider a being set 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; ::_thesis: verum
end;
reconsider H = subrelstr X as non empty symmetric irreflexive 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} :: according to XBOOLE_0:def_10 ::_thesis: the carrier of R \/ {x} c= the carrier of G
proof
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in the carrier of G or a in the carrier of R \/ {x} )
assume A20: a in the carrier of G ; ::_thesis: a in the carrier of R \/ {x}
percases ( a = x or a <> x ) ;
suppose a = x ; ::_thesis: a in the carrier of R \/ {x}
then a in {x} by TARSKI:def_1;
hence a in the carrier of R \/ {x} by XBOOLE_0:def_3; ::_thesis: verum
end;
suppose a <> x ; ::_thesis: a in the carrier of R \/ {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 a in the carrier of R \/ {x} by XBOOLE_0:def_3; ::_thesis: verum
end;
end;
end;
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in the carrier of R \/ {x} or a in the carrier of G )
assume A21: a in the carrier of R \/ {x} ; ::_thesis: a in the carrier of G
percases ( a in the carrier of R or a in {x} ) by A21, XBOOLE_0:def_3;
suppose a in the carrier of R ; ::_thesis: a in the carrier of G
then a in the carrier of G \ {x} by YELLOW_0:def_15;
hence a in the carrier of G ; ::_thesis: verum
end;
suppose a in {x} ; ::_thesis: a in the carrier of G
hence a in the carrier of G ; ::_thesis: verum
end;
end;
end;
A22: the carrier of R2 = X2 \/ Y2
proof
thus the carrier of R2 c= X2 \/ Y2 :: according to XBOOLE_0:def_10 ::_thesis: X2 \/ Y2 c= the carrier of R2
proof
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in the carrier of R2 or a in X2 \/ Y2 )
assume A23: a in the carrier of R2 ; ::_thesis: a in X2 \/ Y2
percases ( [a,x] in the InternalRel of G or not [a,x] in the InternalRel of G ) ;
suppose [a,x] in the InternalRel of G ; ::_thesis: a in X2 \/ Y2
then a in X2 by A23;
hence a in X2 \/ Y2 by XBOOLE_0:def_3; ::_thesis: verum
end;
suppose not [a,x] in the InternalRel of G ; ::_thesis: a in X2 \/ Y2
then a in Y2 by A23;
hence a in X2 \/ Y2 by XBOOLE_0:def_3; ::_thesis: verum
end;
end;
end;
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in X2 \/ Y2 or a in the carrier of R2 )
assume A24: a in X2 \/ Y2 ; ::_thesis: a in the carrier of R2
percases ( a in X2 or a in Y2 ) by A24, XBOOLE_0:def_3;
suppose a in X2 ; ::_thesis: a in the carrier of R2
then ex y being Element of R2 st
( a = y & [y,x] in the InternalRel of G ) ;
hence a in the carrier of R2 ; ::_thesis: verum
end;
suppose a in Y2 ; ::_thesis: a in the carrier of R2
then ex y being Element of R2 st
( a = y & not [y,x] in the InternalRel of G ) ;
hence a in the carrier of R2 ; ::_thesis: verum
end;
end;
end;
A25: not Y1 \/ Y2 is empty
proof
assume A26: Y1 \/ Y2 is empty ; ::_thesis: contradiction
then A27: Y2 is empty ;
A28: Y1 is empty by A26;
A29: for a being Element of R holds [a,x] in the InternalRel of G
proof
let a be Element of R; ::_thesis: [a,x] in the InternalRel of G
A30: the carrier of R = the carrier of R1 \/ the carrier of R2 by A2, NECKLA_2:def_2;
percases ( a in the carrier of R1 or a in the carrier of R2 ) by A30, XBOOLE_0:def_3;
suppose a in the carrier of R1 ; ::_thesis: [a,x] in the InternalRel of G
then ex y being Element of R1 st
( a = y & [y,x] in the InternalRel of G ) by A11, A28;
hence [a,x] in the InternalRel of G ; ::_thesis: verum
end;
suppose a in the carrier of R2 ; ::_thesis: [a,x] in the InternalRel of G
then ex y being Element of R2 st
( a = y & [y,x] in the InternalRel of G ) by A22, A27;
hence [a,x] in the InternalRel of G ; ::_thesis: verum
end;
end;
end;
not ComplRelStr G is path-connected
proof
A31: a <> x
proof
assume not a <> x ; ::_thesis: contradiction
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, XBOOLE_0:def_7; ::_thesis: verum
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 ComplRelStr G is path-connected ; ::_thesis: contradiction
then the InternalRel of (ComplRelStr G) reduces x,a by A31, A34, A35, Def2;
then consider p being FinSequence such that
A36: len p > 0 and
A37: p . 1 = x and
A38: p . (len p) = a and
A39: for i being Element of 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 p . 2 <> x ; ::_thesis: contradiction
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 by XBOOLE_0:def_7;
hence contradiction by Th13; ::_thesis: verum
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 p . 2 in the carrier of R ; ::_thesis: contradiction
then p . 2 in {x} by A19, A45, XBOOLE_0:def_3;
hence contradiction by A43, TARSKI:def_1; ::_thesis: verum
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, RELAT_2:def_3;
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 by XBOOLE_0:def_7;
hence contradiction by Th12; ::_thesis: verum
end;
hence contradiction by A5; ::_thesis: verum
end;
thus G embeds Necklace 4 ::_thesis: verum
proof
percases ( not Y1 is empty or not Y2 is empty ) by A25;
supposeA48: not Y1 is empty ; ::_thesis: G embeds Necklace 4
ex b being Element of Y1 ex c being 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;
the Element of Y1 in Y1 by A48;
then ex y being Element of R1 st
( y = the Element of Y1 & not [y,x] in the InternalRel of G ) ;
then the Element of Y1 in the carrier of R1 \/ the carrier of R2 by XBOOLE_0:def_3;
then A50: the Element of Y1 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 = a as Element of G by A49;
reconsider b = the Element of Y1 as Element of G by A51, A50;
a <> b
proof
assume A52: not a <> b ; ::_thesis: contradiction
a in X1 by A6;
then a in X1 /\ Y1 by A48, A52, XBOOLE_0:def_4;
hence contradiction by A8, XBOOLE_0:def_7; ::_thesis: verum
end;
then the InternalRel of G reduces a,b by A4, Def2;
then consider p being FinSequence such that
A53: len p > 0 and
A54: p . 1 = a and
A55: p . (len p) = b and
A56: for i being Element of 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 S1[ Nat] means ( p . $1 in Y1 & $1 in dom p & ( for k being Nat st k > $1 & k in dom p holds
p . k in Y1 ) );
for k being Nat st k > len p & k in dom p holds
p . k in Y1
proof
let k be Nat; ::_thesis: ( k > len p & k in dom p implies p . k in Y1 )
assume A57: k > len p ; ::_thesis: ( not k in dom p or p . k in Y1 )
assume k in dom p ; ::_thesis: p . k in Y1
then k in Seg (len p) by FINSEQ_1:def_3;
hence p . k in Y1 by A57, FINSEQ_1:1; ::_thesis: verum
end;
then S1[ len p] by A48, A53, A55, CARD_1:27, FINSEQ_5:6;
then A58: ex k being Nat st S1[k] ;
ex n0 being Nat st
( S1[n0] & ( for n being Nat st S1[n] holds
n >= n0 ) ) from NAT_1:sch_5(A58);
then consider n0 being Nat such that
A59: S1[n0] and
A60: for n being Nat st S1[n] holds
n >= n0 ;
n0 <> 0
proof
assume not n0 <> 0 ; ::_thesis: contradiction
then 0 in Seg (len p) by A59, FINSEQ_1:def_3;
hence contradiction by FINSEQ_1:1; ::_thesis: verum
end;
then consider k0 being Nat such that
A61: n0 = k0 + 1 by NAT_1:6;
A62: n0 <> 1
proof
assume A63: not n0 <> 1 ; ::_thesis: contradiction
a in X1 by A6;
then not X1 /\ Y1 is empty by A54, A59, A63, XBOOLE_0:def_4;
hence contradiction by A8, XBOOLE_0:def_7; ::_thesis: verum
end;
A64: k0 >= 1
proof
assume not k0 >= 1 ; ::_thesis: contradiction
then k0 = 0 by NAT_1:25;
hence contradiction by A61, A62; ::_thesis: verum
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, RELAT_2:def_3;
A70: for k being Nat st k > k0 & k in dom p holds
p . k in Y1
proof
assume ex k being Nat st
( k > k0 & k in dom p & not p . k in Y1 ) ; ::_thesis: contradiction
then consider k being Nat such that
A71: k > k0 and
A72: k in dom p and
A73: not p . k in Y1 ;
k > n0
proof
percases ( k < n0 or n0 < k or n0 = k ) by XXREAL_0:1;
suppose k < n0 ; ::_thesis: k > n0
hence k > n0 by A61, A71, NAT_1:13; ::_thesis: verum
end;
suppose n0 < k ; ::_thesis: k > n0
hence k > n0 ; ::_thesis: verum
end;
suppose n0 = k ; ::_thesis: k > n0
hence k > n0 by A59, A73; ::_thesis: verum
end;
end;
end;
hence contradiction by A59, A72, A73; ::_thesis: verum
end;
k0 < n0 by A61, NAT_1:13;
then A74: not S1[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 ex b being Element of Y1 ex c being Element of X1 st [b,c] in the InternalRel of G ::_thesis: verum
proof
percases ( ( p . k0 in the carrier of R1 & p . n0 in the carrier of G ) or ( p . k0 in the carrier of R2 & p . n0 in the carrier of G ) or ( p . k0 in {x} & p . n0 in the carrier of G ) ) by A61, A67, A75, XBOOLE_0:def_3, ZFMISC_1:87;
supposeA76: ( p . k0 in the carrier of R1 & p . n0 in the carrier of G ) ; ::_thesis: ex b being Element of Y1 ex c being Element of X1 st [b,c] in the InternalRel 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, RELAT_2:def_3;
hence ex b being Element of Y1 ex c being Element of X1 st [b,c] in the InternalRel of G ; ::_thesis: verum
end;
suppose ( p . k0 in the carrier of R2 & p . n0 in the carrier of G ) ; ::_thesis: ex b being Element of Y1 ex c being Element of X1 st [b,c] in the InternalRel 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 ex b being Element of Y1 ex c being Element of X1 st [b,c] in the InternalRel of G by A1, A2, Th35; ::_thesis: verum
end;
supposeA81: ( p . k0 in {x} & p . n0 in the carrier of G ) ; ::_thesis: ex b being Element of Y1 ex c being Element of X1 st [b,c] in the InternalRel of G
ex y1 being Element of R1 st
( p . n0 = y1 & not [y1,x] in the InternalRel of G ) by A59;
hence ex b being Element of Y1 ex c being Element of X1 st [b,c] in the InternalRel of G by A69, A81, TARSKI:def_1; ::_thesis: verum
end;
end;
end;
end;
then consider u being Element of Y1, v being Element of X1 such that
A82: [u,v] in the InternalRel of G ;
set w = the Element of X2;
the Element of X2 in X2 by A18;
then A83: ex y being Element of R2 st
( y = the Element of X2 & [y,x] in the InternalRel of G ) ;
set Z = {u,v,x, the Element of X2};
{u,v,x, the Element of X2} c= the carrier of G
proof
the Element of X2 in X2 by A18;
then ex y2 being Element of R2 st
( y2 = the Element of X2 & [y2,x] in the InternalRel of G ) ;
then the Element of X2 in the carrier of R1 \/ the carrier of R2 by XBOOLE_0:def_3;
then A84: the Element of X2 in the carrier of R by A2, NECKLA_2:def_2;
v in X1 by A10;
then ex y1 being 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 set ; :: according to TARSKI:def_3 ::_thesis: ( not q in {u,v,x, the Element of X2} or q in the carrier of G )
assume q in {u,v,x, the Element of X2} ; ::_thesis: q in the carrier of G
then A87: ( q = u or q = v or q = x or q = the Element of X2 ) by ENUMSET1:def_2;
the carrier of R c= the carrier of G by A19, XBOOLE_1:7;
hence q in the carrier of G by A87, A86, A85, A84; ::_thesis: verum
end;
then reconsider Z = {u,v,x, the Element of X2} as Subset of G ;
reconsider H = subrelstr Z as non empty full SubRelStr of G by YELLOW_0:def_15;
A88: the Element of X2 in X2 by A18;
reconsider w = the Element of X2 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 being 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 [x,w] in the InternalRel of G by RELAT_2:def_3; ::_thesis: verum
end;
A91: u in Y1 by A48;
reconsider u = u, v = v as Element of G by A82, ZFMISC_1:87;
A92: [v,x] in the InternalRel of G
proof
ex y1 being Element of R1 st
( v = y1 & [y1,x] in the InternalRel of G ) by A89;
hence [v,x] in the InternalRel of G ; ::_thesis: verum
end;
A93: w <> u
proof
assume A94: not w <> u ; ::_thesis: contradiction
( ex y1 being Element of R2 st
( w = y1 & [y1,x] in the InternalRel of G ) & ex y2 being Element of R1 st
( u = y2 & not [y2,x] in the InternalRel of G ) ) by A91, A88;
hence contradiction by A94; ::_thesis: verum
end;
A95: not [u,x] in the InternalRel of G
proof
ex y1 being Element of R1 st
( u = y1 & not [y1,x] in the InternalRel of G ) by A91;
hence not [u,x] in the InternalRel of G ; ::_thesis: verum
end;
A96: not [v,w] in the InternalRel of G
proof
A97: ex y2 being 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 = w as Element of R by A2, NECKLA_2:def_2;
A98: ex y1 being 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 = v as Element of R by A2, NECKLA_2:def_2;
assume [v,w] in the InternalRel of G ; ::_thesis: contradiction
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;
percases ( [v,w] in the InternalRel of R1 or [v,w] in the InternalRel of R2 ) by A99, XBOOLE_0:def_3;
suppose [v,w] in the InternalRel of R1 ; ::_thesis: contradiction
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, XBOOLE_0:def_7; ::_thesis: verum
end;
suppose [v,w] in the InternalRel of R2 ; ::_thesis: contradiction
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, XBOOLE_0:def_7; ::_thesis: verum
end;
end;
end;
A100: w <> x
proof
assume A101: not w <> x ; ::_thesis: contradiction
ex y1 being 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; ::_thesis: verum
end;
A102: not [u,w] in the InternalRel of G
proof
A103: ex y2 being 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 = w as Element of R by A2, NECKLA_2:def_2;
A104: ex y1 being 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 = u as Element of R by A2, NECKLA_2:def_2;
assume [u,w] in the InternalRel of G ; ::_thesis: contradiction
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;
percases ( [u,w] in the InternalRel of R1 or [u,w] in the InternalRel of R2 ) by A105, XBOOLE_0:def_3;
suppose [u,w] in the InternalRel of R1 ; ::_thesis: contradiction
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, XBOOLE_0:def_7; ::_thesis: verum
end;
suppose [u,w] in the InternalRel of R2 ; ::_thesis: contradiction
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, XBOOLE_0:def_7; ::_thesis: verum
end;
end;
end;
A106: x <> u
proof
assume A107: not x <> u ; ::_thesis: contradiction
ex y1 being 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; ::_thesis: verum
end;
A108: w <> v
proof
consider y1 being Element of R2 such that
A109: w = y1 and
[y1,x] in the InternalRel of G by A88;
assume A110: not w <> v ; ::_thesis: contradiction
ex y2 being 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, XBOOLE_0:def_7; ::_thesis: verum
end;
A111: v <> x
proof
assume A112: not v <> x ; ::_thesis: contradiction
ex y1 being 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; ::_thesis: verum
end;
u <> v
proof
assume A113: not u <> v ; ::_thesis: contradiction
( ex y1 being Element of R1 st
( u = y1 & not [y1,x] in the InternalRel of G ) & ex y2 being Element of R1 st
( v = y2 & [y2,x] in the InternalRel of G ) ) by A91, A89;
hence contradiction by A113; ::_thesis: verum
end;
then u,v,x,w are_mutually_different 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 G embeds Necklace 4 ; ::_thesis: contradiction
then G is N-free by NECKLA_2:def_1;
then H is N-free by Th23;
hence contradiction by A114, NECKLA_2:def_1; ::_thesis: verum
end;
hence G embeds Necklace 4 ; ::_thesis: verum
end;
supposeA115: not Y2 is empty ; ::_thesis: G embeds Necklace 4
ex c being Element of Y2 ex d being 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;
the Element of Y2 in Y2 by A115;
then ex y being Element of R2 st
( y = the Element of Y2 & not [y,x] in the InternalRel of G ) ;
then the Element of Y2 in the carrier of R1 \/ the carrier of R2 by XBOOLE_0:def_3;
then A117: the Element of Y2 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 = b as Element of G by A116;
reconsider c = the Element of Y2 as Element of G by A118, A117;
b <> c
proof
assume not b <> c ; ::_thesis: contradiction
then c in X2 by A7;
then c in X2 /\ Y2 by A115, XBOOLE_0:def_4;
hence contradiction by A14, XBOOLE_0:def_7; ::_thesis: verum
end;
then the InternalRel of G reduces b,c by A4, Def2;
then consider p being FinSequence such that
A119: len p > 0 and
A120: p . 1 = b and
A121: p . (len p) = c and
A122: for i being Element of 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 S1[ Nat] means ( p . $1 in Y2 & $1 in dom p & ( for k being Nat st k > $1 & k in dom p holds
p . k in Y2 ) );
for k being Nat st k > len p & k in dom p holds
p . k in Y2
proof
let k be Nat; ::_thesis: ( k > len p & k in dom p implies p . k in Y2 )
assume A123: k > len p ; ::_thesis: ( not k in dom p or p . k in Y2 )
assume k in dom p ; ::_thesis: p . k in Y2
then k in Seg (len p) by FINSEQ_1:def_3;
hence p . k in Y2 by A123, FINSEQ_1:1; ::_thesis: verum
end;
then S1[ len p] by A115, A119, A121, CARD_1:27, FINSEQ_5:6;
then A124: ex k being Nat st S1[k] ;
ex n0 being Nat st
( S1[n0] & ( for n being Nat st S1[n] holds
n >= n0 ) ) from NAT_1:sch_5(A124);
then consider n0 being Nat such that
A125: S1[n0] and
A126: for n being Nat st S1[n] holds
n >= n0 ;
n0 <> 0
proof
assume not n0 <> 0 ; ::_thesis: contradiction
then 0 in Seg (len p) by A125, FINSEQ_1:def_3;
hence contradiction by FINSEQ_1:1; ::_thesis: verum
end;
then consider k0 being Nat such that
A127: n0 = k0 + 1 by NAT_1:6;
A128: n0 <> 1
proof
assume A129: not n0 <> 1 ; ::_thesis: contradiction
b in X2 by A7;
then not X2 /\ Y2 is empty by A120, A125, A129, XBOOLE_0:def_4;
hence contradiction by A14, XBOOLE_0:def_7; ::_thesis: verum
end;
A130: k0 >= 1
proof
assume not k0 >= 1 ; ::_thesis: contradiction
then k0 = 0 by NAT_1:25;
hence contradiction by A127, A128; ::_thesis: verum
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, RELAT_2:def_3;
A136: for k being Nat st k > k0 & k in dom p holds
p . k in Y2
proof
assume ex k being Nat st
( k > k0 & k in dom p & not p . k in Y2 ) ; ::_thesis: contradiction
then consider k being Nat such that
A137: k > k0 and
A138: k in dom p and
A139: not p . k in Y2 ;
k > n0
proof
percases ( k < n0 or n0 < k or n0 = k ) by XXREAL_0:1;
suppose k < n0 ; ::_thesis: k > n0
hence k > n0 by A127, A137, NAT_1:13; ::_thesis: verum
end;
suppose n0 < k ; ::_thesis: k > n0
hence k > n0 ; ::_thesis: verum
end;
suppose n0 = k ; ::_thesis: k > n0
hence k > n0 by A125, A139; ::_thesis: verum
end;
end;
end;
hence contradiction by A125, A138, A139; ::_thesis: verum
end;
k0 < n0 by A127, NAT_1:13;
then A140: not S1[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 ex c being Element of Y2 ex d being Element of X2 st [c,d] in the InternalRel of G ::_thesis: verum
proof
percases ( ( p . k0 in the carrier of R2 & p . n0 in the carrier of G ) or ( p . k0 in the carrier of R1 & p . n0 in the carrier of G ) or ( p . k0 in {x} & p . n0 in the carrier of G ) ) 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 ) ; ::_thesis: ex c being Element of Y2 ex d being Element of X2 st [c,d] in the InternalRel 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 ex c being Element of Y2 ex d being Element of X2 st [c,d] in the InternalRel of G by A135; ::_thesis: verum
end;
suppose ( p . k0 in the carrier of R1 & p . n0 in the carrier of G ) ; ::_thesis: ex c being Element of Y2 ex d being Element of X2 st [c,d] in the InternalRel 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, RELAT_2:def_3;
hence ex c being Element of Y2 ex d being Element of X2 st [c,d] in the InternalRel of G by A1, A2, Th35; ::_thesis: verum
end;
supposeA145: ( p . k0 in {x} & p . n0 in the carrier of G ) ; ::_thesis: ex c being Element of Y2 ex d being Element of X2 st [c,d] in the InternalRel of G
ex y1 being Element of R2 st
( p . n0 = y1 & not [y1,x] in the InternalRel of G ) by A125;
hence ex c being Element of Y2 ex d being Element of X2 st [c,d] in the InternalRel of G by A135, A145, TARSKI:def_1; ::_thesis: verum
end;
end;
end;
end;
then consider u being Element of Y2, v being Element of X2 such that
A146: [u,v] in the InternalRel of G ;
set w = the Element of X1;
the Element of X1 in X1 by A10;
then A147: ex y being Element of R1 st
( y = the Element of X1 & [y,x] in the InternalRel of G ) ;
set Z = {u,v,x, the Element of X1};
{u,v,x, the Element of X1} c= the carrier of G
proof
the Element of X1 in X1 by A10;
then ex y2 being Element of R1 st
( y2 = the Element of X1 & [y2,x] in the InternalRel of G ) ;
then the Element of X1 in the carrier of R1 \/ the carrier of R2 by XBOOLE_0:def_3;
then A148: the Element of X1 in the carrier of R by A2, NECKLA_2:def_2;
v in X2 by A18;
then ex y1 being 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 set ; :: according to TARSKI:def_3 ::_thesis: ( not q in {u,v,x, the Element of X1} or q in the carrier of G )
assume q in {u,v,x, the Element of X1} ; ::_thesis: q in the carrier of G
then A151: ( q = u or q = v or q = x or q = the Element of X1 ) by ENUMSET1:def_2;
the carrier of R c= the carrier of G by A19, XBOOLE_1:7;
hence q in the carrier of G by A151, A150, A149, A148; ::_thesis: verum
end;
then reconsider Z = {u,v,x, the Element of X1} as Subset of G ;
reconsider H = subrelstr Z as non empty full SubRelStr of G by YELLOW_0:def_15;
A152: the Element of X1 in X1 by A10;
reconsider w = the Element of X1 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 being 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 [x,w] in the InternalRel of G by RELAT_2:def_3; ::_thesis: verum
end;
A155: u in Y2 by A115;
reconsider u = u, v = v as Element of G by A146, ZFMISC_1:87;
A156: [v,x] in the InternalRel of G
proof
ex y1 being Element of R2 st
( v = y1 & [y1,x] in the InternalRel of G ) by A153;
hence [v,x] in the InternalRel of G ; ::_thesis: verum
end;
A157: w <> u
proof
assume A158: not w <> u ; ::_thesis: contradiction
( ex y1 being Element of R1 st
( w = y1 & [y1,x] in the InternalRel of G ) & ex y2 being Element of R2 st
( u = y2 & not [y2,x] in the InternalRel of G ) ) by A155, A152;
hence contradiction by A158; ::_thesis: verum
end;
A159: not [u,x] in the InternalRel of G
proof
ex y1 being Element of R2 st
( u = y1 & not [y1,x] in the InternalRel of G ) by A155;
hence not [u,x] in the InternalRel of G ; ::_thesis: verum
end;
A160: not [v,w] in the InternalRel of G
proof
A161: ex y2 being 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 = w as Element of R by A2, NECKLA_2:def_2;
A162: ex y1 being 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 = v as Element of R by A2, NECKLA_2:def_2;
assume [v,w] in the InternalRel of G ; ::_thesis: contradiction
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;
percases ( [v,w] in the InternalRel of R1 or [v,w] in the InternalRel of R2 ) by A163, XBOOLE_0:def_3;
suppose [v,w] in the InternalRel of R1 ; ::_thesis: contradiction
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, XBOOLE_0:def_7; ::_thesis: verum
end;
suppose [v,w] in the InternalRel of R2 ; ::_thesis: contradiction
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, XBOOLE_0:def_7; ::_thesis: verum
end;
end;
end;
A164: w <> x
proof
assume A165: not w <> x ; ::_thesis: contradiction
ex y1 being 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; ::_thesis: verum
end;
A166: not [u,w] in the InternalRel of G
proof
A167: ex y2 being 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 = w as Element of R by A2, NECKLA_2:def_2;
A168: ex y1 being 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 = u as Element of R by A2, NECKLA_2:def_2;
assume [u,w] in the InternalRel of G ; ::_thesis: contradiction
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;
percases ( [u,w] in the InternalRel of R1 or [u,w] in the InternalRel of R2 ) by A169, XBOOLE_0:def_3;
suppose [u,w] in the InternalRel of R1 ; ::_thesis: contradiction
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, XBOOLE_0:def_7; ::_thesis: verum
end;
suppose [u,w] in the InternalRel of R2 ; ::_thesis: contradiction
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, XBOOLE_0:def_7; ::_thesis: verum
end;
end;
end;
A170: x <> u
proof
assume A171: not x <> u ; ::_thesis: contradiction
ex y1 being 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; ::_thesis: verum
end;
A172: w <> v
proof
consider y1 being Element of R1 such that
A173: w = y1 and
[y1,x] in the InternalRel of G by A152;
assume A174: not w <> v ; ::_thesis: contradiction
ex y2 being 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, XBOOLE_0:def_7; ::_thesis: verum
end;
A175: v <> x
proof
assume A176: not v <> x ; ::_thesis: contradiction
ex y1 being 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; ::_thesis: verum
end;
u <> v
proof
assume A177: not u <> v ; ::_thesis: contradiction
( ex y1 being Element of R2 st
( u = y1 & not [y1,x] in the InternalRel of G ) & ex y2 being Element of R2 st
( v = y2 & [y2,x] in the InternalRel of G ) ) by A155, A153;
hence contradiction by A177; ::_thesis: verum
end;
then u,v,x,w are_mutually_different 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 G embeds Necklace 4 ; ::_thesis: contradiction
then G is N-free by NECKLA_2:def_1;
then H is N-free by Th23;
hence contradiction by A178, NECKLA_2:def_1; ::_thesis: verum
end;
hence G embeds Necklace 4 ; ::_thesis: verum
end;
end;
end;
end;
theorem :: NECKLA_3:40
for G being non empty finite strict symmetric irreflexive RelStr st G is N-free & the carrier of G in FinSETS holds
RelStr(# the carrier of G, the InternalRel of G #) in fin_RelStr_sp
proof
let R be non empty finite strict symmetric irreflexive RelStr ; ::_thesis: ( R is N-free & the carrier of R in FinSETS implies RelStr(# the carrier of R, the InternalRel of R #) in fin_RelStr_sp )
defpred S1[ Nat] means for G being non empty finite strict symmetric irreflexive RelStr st G is N-free & card the carrier of G = $1 & the carrier of G in FinSETS holds
RelStr(# the carrier of G, the InternalRel of G #) in fin_RelStr_sp ;
A1: for n being Nat st ( for k being Nat st k < n holds
S1[k] ) holds
S1[n]
proof
let n be Nat; ::_thesis: ( ( for k being Nat st k < n holds
S1[k] ) implies S1[n] )
assume A2: for k being Nat st k < n holds
S1[k] ; ::_thesis: S1[n]
let G be non empty finite strict symmetric irreflexive RelStr ; ::_thesis: ( G is N-free & card the carrier of G = n & the carrier of G in FinSETS implies RelStr(# the carrier of G, the InternalRel of G #) in fin_RelStr_sp )
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 ; ::_thesis: RelStr(# the carrier of G, the InternalRel of G #) in fin_RelStr_sp
percases ( G is trivial or ( not G is path-connected & not G is trivial ) or ( not ComplRelStr G is path-connected & not G is trivial ) or ( not G is trivial & G is path-connected & ComplRelStr G is path-connected ) ) ;
suppose G is trivial ; ::_thesis: RelStr(# the carrier of G, the InternalRel of G #) in fin_RelStr_sp
then the carrier of G is 1 -element ;
then reconsider G = G as 1 -element RelStr by STRUCT_0:def_19;
RelStr(# the carrier of G, the InternalRel of G #) in fin_RelStr_sp by A5, NECKLA_2:def_5;
hence RelStr(# the carrier of G, the InternalRel of G #) in fin_RelStr_sp ; ::_thesis: verum
end;
suppose ( not G is path-connected & not G is trivial ) ; ::_thesis: RelStr(# the carrier of G, the InternalRel of G #) in fin_RelStr_sp
then consider G1, G2 being non empty strict symmetric irreflexive RelStr such that
A6: the carrier of G1 misses the carrier of G2 and
A7: RelStr(# the carrier of G, the InternalRel of G #) = union_of (G1,G2) by Th30;
set cG1 = the carrier of G1;
set cG2 = the carrier of G2;
set R = RelStr(# the carrier of G, the InternalRel of G #);
set cR = the carrier of RelStr(# the carrier of G, the InternalRel of G #);
A8: the carrier of RelStr(# the carrier of G, the InternalRel of G #) = the carrier of G1 \/ the carrier of G2 by A7, NECKLA_2:def_2;
then A9: card the carrier of G1 in card the carrier of RelStr(# the carrier of G, the InternalRel of G #) by A6, Lm1;
then reconsider G1 = G1 as non empty finite strict symmetric irreflexive RelStr by CARD_2:49;
reconsider cR = the carrier of RelStr(# the carrier of G, the InternalRel of G #) as finite set ;
A10: card the carrier of G2 in card cR by A6, A8, Lm1;
then reconsider G2 = G2 as non empty finite strict symmetric irreflexive RelStr by CARD_2:49;
reconsider cG2 = the carrier of G2 as finite set by A10, CARD_2:49;
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 = the carrier of G1 as finite set by A9, CARD_2:49;
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 RelStr(# the carrier of G, the InternalRel of G #) in fin_RelStr_sp by A6, A7, A13, NECKLA_2:def_5; ::_thesis: verum
end;
suppose ( not ComplRelStr G is path-connected & not G is trivial ) ; ::_thesis: RelStr(# the carrier of G, the InternalRel of G #) in fin_RelStr_sp
then consider G1, G2 being non empty strict symmetric irreflexive RelStr such that
A16: the carrier of G1 misses the carrier of G2 and
A17: RelStr(# the carrier of G, the InternalRel of G #) = sum_of (G1,G2) by Th31;
set cG1 = the carrier of G1;
set cG2 = the carrier of G2;
set R = RelStr(# the carrier of G, the InternalRel of G #);
set cR = the carrier of RelStr(# the carrier of G, the InternalRel of G #);
A18: the carrier of RelStr(# the carrier of G, the InternalRel of G #) = the carrier of G1 \/ the carrier of G2 by A17, NECKLA_2:def_3;
then A19: card the carrier of G1 in card the carrier of RelStr(# the carrier of G, the InternalRel of G #) by A16, Lm1;
then reconsider G1 = G1 as non empty finite strict symmetric irreflexive RelStr by CARD_2:49;
reconsider cR = the carrier of RelStr(# the carrier of G, the InternalRel of G #) as finite set ;
A20: card the carrier of G2 in card cR by A16, A18, Lm1;
then reconsider G2 = G2 as non empty finite strict symmetric irreflexive RelStr by CARD_2:49;
reconsider cG2 = the carrier of G2 as finite set by A20, CARD_2:49;
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 = the carrier of G1 as finite set by A19, CARD_2:49;
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 RelStr(# the carrier of G, the InternalRel of G #) in fin_RelStr_sp by A16, A17, A23, NECKLA_2:def_5; ::_thesis: verum
end;
supposeA26: ( not G is trivial & G is path-connected & ComplRelStr G is path-connected ) ; ::_thesis: RelStr(# the carrier of G, the InternalRel of G #) in fin_RelStr_sp
consider x being set such that
A27: x in the carrier of G by XBOOLE_0:def_1;
reconsider x = x as Element of G by A27;
set A = the carrier of G \ {x};
A28: the carrier of G \ {x} c= the carrier of G ;
reconsider A = the carrier of G \ {x} as Subset of G ;
set R = subrelstr A;
reconsider R = subrelstr A as non empty finite symmetric irreflexive 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 RelStr(# the carrier of G, the InternalRel of G #) in fin_RelStr_sp ::_thesis: verum
proof
percases ( R is trivial RelStr or ex R1, R2 being 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) ) ) ) by A32, NECKLA_2:6;
supposeA33: R is trivial RelStr ; ::_thesis: RelStr(# the carrier of G, the InternalRel of G #) in fin_RelStr_sp
not the carrier of R is empty ;
then A34: not A is empty by YELLOW_0:def_15;
A is trivial by A33, YELLOW_0:def_15;
then consider a being set 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 :: according to XBOOLE_0:def_10 ::_thesis: the carrier of G c= the carrier of G \/ {x}
proof
let c be set ; :: according to TARSKI:def_3 ::_thesis: ( not c in the carrier of G \/ {x} or c in the carrier of G )
assume c in the carrier of G \/ {x} ; ::_thesis: c in the carrier of G
then ( c in the carrier of G or c in {x} ) by XBOOLE_0:def_3;
hence c in the carrier of G ; ::_thesis: verum
end;
let c be set ; :: according to TARSKI:def_3 ::_thesis: ( not c in the carrier of G or c in the carrier of G \/ {x} )
assume c in the carrier of G ; ::_thesis: c in the carrier of G \/ {x}
hence c in the carrier of G \/ {x} by XBOOLE_0:def_3; ::_thesis: verum
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 RelStr(# the carrier of G, the InternalRel of G #) in fin_RelStr_sp by A5, Th33; ::_thesis: verum
end;
suppose ex R1, R2 being 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) ) ) ; ::_thesis: RelStr(# the carrier of G, the InternalRel of G #) in fin_RelStr_sp
then consider R1, R2 being 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 RelStr(# the carrier of G, the InternalRel of G #) in fin_RelStr_sp ::_thesis: verum
proof
percases ( R = union_of (R1,R2) or R = sum_of (R1,R2) ) by A40;
supposeA41: R = union_of (R1,R2) ; ::_thesis: RelStr(# the carrier of G, the InternalRel of G #) in fin_RelStr_sp
R2 is SubRelStr of R by A37, A40, Th10;
then reconsider R2 = 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 = 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 RelStr(# the carrier of G, the InternalRel of G #) in fin_RelStr_sp by A3, NECKLA_2:def_1; ::_thesis: verum
end;
supposeA42: R = sum_of (R1,R2) ; ::_thesis: RelStr(# the carrier of G, the InternalRel of G #) in fin_RelStr_sp
not ComplRelStr R2 is empty
proof
assume ComplRelStr R2 is empty ; ::_thesis: contradiction
then R2 is empty ;
hence contradiction by A39, NECKLA_2:4; ::_thesis: verum
end;
then reconsider R22 = ComplRelStr R2 as non empty RelStr ;
not ComplRelStr R1 is empty
proof
assume ComplRelStr R1 is empty ; ::_thesis: contradiction
then R1 is empty ;
hence contradiction by A38, NECKLA_2:4; ::_thesis: verum
end;
then reconsider R11 = ComplRelStr R1 as non empty RelStr ;
reconsider G9 = ComplRelStr G as non empty symmetric irreflexive 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 & not G9 is 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 RelStr(# the carrier of G, the InternalRel of G #) in fin_RelStr_sp by A45, NECKLA_2:def_1; ::_thesis: verum
end;
end;
end;
end;
end;
end;
end;
end;
end;
A47: for k being Nat holds S1[k] from NAT_1:sch_4(A1);
card the carrier of R is Nat ;
hence ( R is N-free & the carrier of R in FinSETS implies RelStr(# the carrier of R, the InternalRel of R #) in fin_RelStr_sp ) by A47; ::_thesis: verum
end;