:: LATTICE8 semantic presentation
begin
definition
let L be RelStr ;
attrL is finitely_typed means :Def1: :: LATTICE8:def 1
ex A being non empty set st
( ( for e being set st e in the carrier of L holds
e is Equivalence_Relation of A ) & ex o being Element of NAT st
for e1, e2 being Equivalence_Relation of A
for x, y being set st e1 in the carrier of L & e2 in the carrier of L & [x,y] in e1 "\/" e2 holds
ex F being non empty FinSequence of A st
( len F = o & x,y are_joint_by F,e1,e2 ) );
end;
:: deftheorem Def1 defines finitely_typed LATTICE8:def_1_:_
for L being RelStr holds
( L is finitely_typed iff ex A being non empty set st
( ( for e being set st e in the carrier of L holds
e is Equivalence_Relation of A ) & ex o being Element of NAT st
for e1, e2 being Equivalence_Relation of A
for x, y being set st e1 in the carrier of L & e2 in the carrier of L & [x,y] in e1 "\/" e2 holds
ex F being non empty FinSequence of A st
( len F = o & x,y are_joint_by F,e1,e2 ) ) );
definition
let L be lower-bounded LATTICE;
let n be Element of NAT ;
predL has_a_representation_of_type<= n means :Def2: :: LATTICE8:def 2
ex A being non trivial set ex f being Homomorphism of L,(EqRelLATT A) st
( f is V14() & Image f is finitely_typed & ex e being Equivalence_Relation of A st
( e in the carrier of (Image f) & e <> id A ) & type_of (Image f) <= n );
end;
:: deftheorem Def2 defines has_a_representation_of_type<= LATTICE8:def_2_:_
for L being lower-bounded LATTICE
for n being Element of NAT holds
( L has_a_representation_of_type<= n iff ex A being non trivial set ex f being Homomorphism of L,(EqRelLATT A) st
( f is V14() & Image f is finitely_typed & ex e being Equivalence_Relation of A st
( e in the carrier of (Image f) & e <> id A ) & type_of (Image f) <= n ) );
registration
cluster non empty finite reflexive transitive antisymmetric lower-bounded distributive with_suprema with_infima for RelStr ;
existence
ex b1 being LATTICE st
( b1 is lower-bounded & b1 is distributive & b1 is finite )
proof
set L = the finite distributive LATTICE;
take the finite distributive LATTICE ; ::_thesis: ( the finite distributive LATTICE is lower-bounded & the finite distributive LATTICE is distributive & the finite distributive LATTICE is finite )
thus ( the finite distributive LATTICE is lower-bounded & the finite distributive LATTICE is distributive & the finite distributive LATTICE is finite ) ; ::_thesis: verum
end;
end;
Lm1: 1 is odd
proof
(2 * 0) + 1 = 1 ;
hence 1 is odd ; ::_thesis: verum
end;
Lm2: 2 is even
proof
2 * 1 = 2 ;
hence 2 is even ; ::_thesis: verum
end;
registration
let A be non trivial set ;
cluster non empty non trivial full meet-inheriting join-inheriting finitely_typed for SubRelStr of EqRelLATT A;
existence
ex b1 being non empty Sublattice of EqRelLATT A st
( not b1 is trivial & b1 is finitely_typed & b1 is full )
proof
reconsider e1 = nabla A, e2 = id A as Element of (EqRelLATT A) by LATTICE5:4;
set a = the Element of A;
set b = the Element of A \ { the Element of A};
set Y = subrelstr {e1,e2};
A1: the carrier of (subrelstr {e1,e2}) = {e1,e2} by YELLOW_0:def_15;
e1 = [:A,A:] by EQREL_1:def_1;
then A2: e2 <= e1 by LATTICE5:6;
A3: for x, y being Element of (EqRelLATT A) st x in the carrier of (subrelstr {e1,e2}) & y in the carrier of (subrelstr {e1,e2}) & ex_inf_of {x,y}, EqRelLATT A holds
inf {x,y} in the carrier of (subrelstr {e1,e2})
proof
let x, y be Element of (EqRelLATT A); ::_thesis: ( x in the carrier of (subrelstr {e1,e2}) & y in the carrier of (subrelstr {e1,e2}) & ex_inf_of {x,y}, EqRelLATT A implies inf {x,y} in the carrier of (subrelstr {e1,e2}) )
assume that
A4: ( x in the carrier of (subrelstr {e1,e2}) & y in the carrier of (subrelstr {e1,e2}) ) and
ex_inf_of {x,y}, EqRelLATT A ; ::_thesis: inf {x,y} in the carrier of (subrelstr {e1,e2})
percases ( ( x = e1 & y = e1 ) or ( x = e1 & y = e2 ) or ( x = e2 & y = e1 ) or ( x = e2 & y = e2 ) ) by A1, A4, TARSKI:def_2;
suppose ( x = e1 & y = e1 ) ; ::_thesis: inf {x,y} in the carrier of (subrelstr {e1,e2})
then inf {x,y} = e1 "/\" e1 by YELLOW_0:40
.= e1 by YELLOW_5:2 ;
hence inf {x,y} in the carrier of (subrelstr {e1,e2}) by A1, TARSKI:def_2; ::_thesis: verum
end;
suppose ( x = e1 & y = e2 ) ; ::_thesis: inf {x,y} in the carrier of (subrelstr {e1,e2})
then inf {x,y} = e1 "/\" e2 by YELLOW_0:40
.= e2 by A2, YELLOW_5:10 ;
hence inf {x,y} in the carrier of (subrelstr {e1,e2}) by A1, TARSKI:def_2; ::_thesis: verum
end;
suppose ( x = e2 & y = e1 ) ; ::_thesis: inf {x,y} in the carrier of (subrelstr {e1,e2})
then inf {x,y} = e2 "/\" e1 by YELLOW_0:40
.= e2 by A2, YELLOW_5:10 ;
hence inf {x,y} in the carrier of (subrelstr {e1,e2}) by A1, TARSKI:def_2; ::_thesis: verum
end;
suppose ( x = e2 & y = e2 ) ; ::_thesis: inf {x,y} in the carrier of (subrelstr {e1,e2})
then inf {x,y} = e2 "/\" e2 by YELLOW_0:40
.= e2 by YELLOW_5:2 ;
hence inf {x,y} in the carrier of (subrelstr {e1,e2}) by A1, TARSKI:def_2; ::_thesis: verum
end;
end;
end;
A5: subrelstr {e1,e2} is finitely_typed
proof
take A ; :: according to LATTICE8:def_1 ::_thesis: ( ( for e being set st e in the carrier of (subrelstr {e1,e2}) holds
e is Equivalence_Relation of A ) & ex o being Element of NAT st
for e1, e2 being Equivalence_Relation of A
for x, y being set st e1 in the carrier of (subrelstr {e1,e2}) & e2 in the carrier of (subrelstr {e1,e2}) & [x,y] in e1 "\/" e2 holds
ex F being non empty FinSequence of A st
( len F = o & x,y are_joint_by F,e1,e2 ) )
thus for e being set st e in the carrier of (subrelstr {e1,e2}) holds
e is Equivalence_Relation of A by A1, TARSKI:def_2; ::_thesis: ex o being Element of NAT st
for e1, e2 being Equivalence_Relation of A
for x, y being set st e1 in the carrier of (subrelstr {e1,e2}) & e2 in the carrier of (subrelstr {e1,e2}) & [x,y] in e1 "\/" e2 holds
ex F being non empty FinSequence of A st
( len F = o & x,y are_joint_by F,e1,e2 )
take o = 3; ::_thesis: for e1, e2 being Equivalence_Relation of A
for x, y being set st e1 in the carrier of (subrelstr {e1,e2}) & e2 in the carrier of (subrelstr {e1,e2}) & [x,y] in e1 "\/" e2 holds
ex F being non empty FinSequence of A st
( len F = o & x,y are_joint_by F,e1,e2 )
thus for eq1, eq2 being Equivalence_Relation of A
for x1, y1 being set st eq1 in the carrier of (subrelstr {e1,e2}) & eq2 in the carrier of (subrelstr {e1,e2}) & [x1,y1] in eq1 "\/" eq2 holds
ex F being non empty FinSequence of A st
( len F = o & x1,y1 are_joint_by F,eq1,eq2 ) ::_thesis: verum
proof
let eq1, eq2 be Equivalence_Relation of A; ::_thesis: for x1, y1 being set st eq1 in the carrier of (subrelstr {e1,e2}) & eq2 in the carrier of (subrelstr {e1,e2}) & [x1,y1] in eq1 "\/" eq2 holds
ex F being non empty FinSequence of A st
( len F = o & x1,y1 are_joint_by F,eq1,eq2 )
let x1, y1 be set ; ::_thesis: ( eq1 in the carrier of (subrelstr {e1,e2}) & eq2 in the carrier of (subrelstr {e1,e2}) & [x1,y1] in eq1 "\/" eq2 implies ex F being non empty FinSequence of A st
( len F = o & x1,y1 are_joint_by F,eq1,eq2 ) )
assume that
A6: eq1 in the carrier of (subrelstr {e1,e2}) and
A7: eq2 in the carrier of (subrelstr {e1,e2}) and
A8: [x1,y1] in eq1 "\/" eq2 ; ::_thesis: ex F being non empty FinSequence of A st
( len F = o & x1,y1 are_joint_by F,eq1,eq2 )
( eq1 = e2 or eq1 <> e2 ) ;
then consider z being set such that
A9: ( ( eq1 = e2 & z = x1 ) or ( eq1 <> e2 & z = y1 ) ) ;
ex x2, y2 being set st
( [x1,y1] = [x2,y2] & x2 in A & y2 in A ) by A8, RELSET_1:2;
then ( x1 in A & y1 in A ) by XTUPLE_0:1;
then reconsider F = <*x1,z,y1*> as non empty FinSequence of A by A9, FINSEQ_2:14;
take F ; ::_thesis: ( len F = o & x1,y1 are_joint_by F,eq1,eq2 )
percases ( ( eq1 = e2 & z = x1 ) or ( eq1 <> e2 & z = y1 ) ) by A9;
supposeA10: ( eq1 = e2 & z = x1 ) ; ::_thesis: ( len F = o & x1,y1 are_joint_by F,eq1,eq2 )
then A11: F . 2 = x1 by FINSEQ_1:45;
A12: F . 1 = x1 by FINSEQ_1:45;
A13: for i being Element of NAT st 1 <= i & i < len F holds
( ( i is odd implies [(F . i),(F . (i + 1))] in eq1 ) & ( i is even implies [(F . i),(F . (i + 1))] in eq2 ) )
proof
let i be Element of NAT ; ::_thesis: ( 1 <= i & i < len F implies ( ( i is odd implies [(F . i),(F . (i + 1))] in eq1 ) & ( i is even implies [(F . i),(F . (i + 1))] in eq2 ) ) )
assume that
A14: 1 <= i and
A15: i < len F ; ::_thesis: ( ( i is odd implies [(F . i),(F . (i + 1))] in eq1 ) & ( i is even implies [(F . i),(F . (i + 1))] in eq2 ) )
i < 2 + 1 by A15, FINSEQ_1:45;
then i <= 2 by NAT_1:13;
then A16: ( i = 0 or i = 1 or i = 2 ) by NAT_1:26;
percases ( ( i = 1 & i is odd & eq1 = e2 & eq2 = e1 ) or ( i = 1 & i is odd & eq1 = e2 & eq2 = e2 ) or ( i = 2 & i is even & eq1 = e2 & eq2 = e1 ) or ( i = 2 & i is even & eq1 = e2 & eq2 = e2 ) ) by A1, A7, A10, A14, A16, Lm1, Lm2, TARSKI:def_2;
supposeA17: ( i = 1 & i is odd & eq1 = e2 & eq2 = e1 ) ; ::_thesis: ( ( i is odd implies [(F . i),(F . (i + 1))] in eq1 ) & ( i is even implies [(F . i),(F . (i + 1))] in eq2 ) )
ex x2, y2 being set st
( [x1,y1] = [x2,y2] & x2 in A & y2 in A ) by A8, RELSET_1:2;
then x1 in A by XTUPLE_0:1;
hence ( ( i is odd implies [(F . i),(F . (i + 1))] in eq1 ) & ( i is even implies [(F . i),(F . (i + 1))] in eq2 ) ) by A12, A11, A17, EQREL_1:5; ::_thesis: verum
end;
supposeA18: ( i = 1 & i is odd & eq1 = e2 & eq2 = e2 ) ; ::_thesis: ( ( i is odd implies [(F . i),(F . (i + 1))] in eq1 ) & ( i is even implies [(F . i),(F . (i + 1))] in eq2 ) )
ex x2, y2 being set st
( [x1,y1] = [x2,y2] & x2 in A & y2 in A ) by A8, RELSET_1:2;
then x1 in A by XTUPLE_0:1;
hence ( ( i is odd implies [(F . i),(F . (i + 1))] in eq1 ) & ( i is even implies [(F . i),(F . (i + 1))] in eq2 ) ) by A12, A11, A18, EQREL_1:5; ::_thesis: verum
end;
supposeA19: ( i = 2 & i is even & eq1 = e2 & eq2 = e1 ) ; ::_thesis: ( ( i is odd implies [(F . i),(F . (i + 1))] in eq1 ) & ( i is even implies [(F . i),(F . (i + 1))] in eq2 ) )
then eq1 "\/" eq2 = e2 "\/" e1 by LATTICE5:10
.= eq2 by A2, A19, YELLOW_5:8 ;
hence ( ( i is odd implies [(F . i),(F . (i + 1))] in eq1 ) & ( i is even implies [(F . i),(F . (i + 1))] in eq2 ) ) by A8, A11, A19, FINSEQ_1:45; ::_thesis: verum
end;
suppose ( i = 2 & i is even & eq1 = e2 & eq2 = e2 ) ; ::_thesis: ( ( i is odd implies [(F . i),(F . (i + 1))] in eq1 ) & ( i is even implies [(F . i),(F . (i + 1))] in eq2 ) )
hence ( ( i is odd implies [(F . i),(F . (i + 1))] in eq1 ) & ( i is even implies [(F . i),(F . (i + 1))] in eq2 ) ) by A8, A11, FINSEQ_1:45; ::_thesis: verum
end;
end;
end;
( len F = 3 & F . 3 = y1 ) by FINSEQ_1:45;
hence ( len F = o & x1,y1 are_joint_by F,eq1,eq2 ) by A12, A13, LATTICE5:def_3; ::_thesis: verum
end;
supposeA20: ( eq1 <> e2 & z = y1 ) ; ::_thesis: ( len F = o & x1,y1 are_joint_by F,eq1,eq2 )
then A21: F . 2 = y1 by FINSEQ_1:45;
A22: F . 3 = y1 by FINSEQ_1:45;
A23: for i being Element of NAT st 1 <= i & i < len F holds
( ( i is odd implies [(F . i),(F . (i + 1))] in eq1 ) & ( i is even implies [(F . i),(F . (i + 1))] in eq2 ) )
proof
let i be Element of NAT ; ::_thesis: ( 1 <= i & i < len F implies ( ( i is odd implies [(F . i),(F . (i + 1))] in eq1 ) & ( i is even implies [(F . i),(F . (i + 1))] in eq2 ) ) )
assume that
A24: 1 <= i and
A25: i < len F ; ::_thesis: ( ( i is odd implies [(F . i),(F . (i + 1))] in eq1 ) & ( i is even implies [(F . i),(F . (i + 1))] in eq2 ) )
i < 2 + 1 by A25, FINSEQ_1:45;
then i <= 2 by NAT_1:13;
then A26: ( i = 0 or i = 1 or i = 2 ) by NAT_1:26;
percases ( ( i = 1 & i is odd & eq1 = e1 & eq2 = e1 ) or ( i = 1 & i is odd & eq1 = e1 & eq2 = e2 ) or ( i = 2 & i is even & eq1 = e1 & eq2 = e1 ) or ( i = 2 & i is even & eq1 = e1 & eq2 = e2 ) ) by A1, A6, A7, A20, A24, A26, Lm1, Lm2, TARSKI:def_2;
suppose ( i = 1 & i is odd & eq1 = e1 & eq2 = e1 ) ; ::_thesis: ( ( i is odd implies [(F . i),(F . (i + 1))] in eq1 ) & ( i is even implies [(F . i),(F . (i + 1))] in eq2 ) )
hence ( ( i is odd implies [(F . i),(F . (i + 1))] in eq1 ) & ( i is even implies [(F . i),(F . (i + 1))] in eq2 ) ) by A8, A21, FINSEQ_1:45; ::_thesis: verum
end;
supposeA27: ( i = 1 & i is odd & eq1 = e1 & eq2 = e2 ) ; ::_thesis: ( ( i is odd implies [(F . i),(F . (i + 1))] in eq1 ) & ( i is even implies [(F . i),(F . (i + 1))] in eq2 ) )
then eq1 "\/" eq2 = e1 "\/" e2 by LATTICE5:10
.= eq1 by A2, A27, YELLOW_5:8 ;
hence ( ( i is odd implies [(F . i),(F . (i + 1))] in eq1 ) & ( i is even implies [(F . i),(F . (i + 1))] in eq2 ) ) by A8, A21, A27, FINSEQ_1:45; ::_thesis: verum
end;
supposeA28: ( i = 2 & i is even & eq1 = e1 & eq2 = e1 ) ; ::_thesis: ( ( i is odd implies [(F . i),(F . (i + 1))] in eq1 ) & ( i is even implies [(F . i),(F . (i + 1))] in eq2 ) )
ex x2, y2 being set st
( [x1,y1] = [x2,y2] & x2 in A & y2 in A ) by A8, RELSET_1:2;
then y1 in A by XTUPLE_0:1;
hence ( ( i is odd implies [(F . i),(F . (i + 1))] in eq1 ) & ( i is even implies [(F . i),(F . (i + 1))] in eq2 ) ) by A21, A22, A28, EQREL_1:5; ::_thesis: verum
end;
supposeA29: ( i = 2 & i is even & eq1 = e1 & eq2 = e2 ) ; ::_thesis: ( ( i is odd implies [(F . i),(F . (i + 1))] in eq1 ) & ( i is even implies [(F . i),(F . (i + 1))] in eq2 ) )
ex x2, y2 being set st
( [x1,y1] = [x2,y2] & x2 in A & y2 in A ) by A8, RELSET_1:2;
then y1 in A by XTUPLE_0:1;
hence ( ( i is odd implies [(F . i),(F . (i + 1))] in eq1 ) & ( i is even implies [(F . i),(F . (i + 1))] in eq2 ) ) by A21, A22, A29, EQREL_1:5; ::_thesis: verum
end;
end;
end;
( len F = 3 & F . 1 = x1 ) by FINSEQ_1:45;
hence ( len F = o & x1,y1 are_joint_by F,eq1,eq2 ) by A22, A23, LATTICE5:def_3; ::_thesis: verum
end;
end;
end;
end;
A30: for x, y being Element of (EqRelLATT A) st x in the carrier of (subrelstr {e1,e2}) & y in the carrier of (subrelstr {e1,e2}) & ex_sup_of {x,y}, EqRelLATT A holds
sup {x,y} in the carrier of (subrelstr {e1,e2})
proof
let x, y be Element of (EqRelLATT A); ::_thesis: ( x in the carrier of (subrelstr {e1,e2}) & y in the carrier of (subrelstr {e1,e2}) & ex_sup_of {x,y}, EqRelLATT A implies sup {x,y} in the carrier of (subrelstr {e1,e2}) )
assume that
A31: ( x in the carrier of (subrelstr {e1,e2}) & y in the carrier of (subrelstr {e1,e2}) ) and
ex_sup_of {x,y}, EqRelLATT A ; ::_thesis: sup {x,y} in the carrier of (subrelstr {e1,e2})
percases ( ( x = e1 & y = e1 ) or ( x = e1 & y = e2 ) or ( x = e2 & y = e1 ) or ( x = e2 & y = e2 ) ) by A1, A31, TARSKI:def_2;
suppose ( x = e1 & y = e1 ) ; ::_thesis: sup {x,y} in the carrier of (subrelstr {e1,e2})
then sup {x,y} = e1 "\/" e1 by YELLOW_0:41
.= e1 by YELLOW_5:1 ;
hence sup {x,y} in the carrier of (subrelstr {e1,e2}) by A1, TARSKI:def_2; ::_thesis: verum
end;
suppose ( x = e1 & y = e2 ) ; ::_thesis: sup {x,y} in the carrier of (subrelstr {e1,e2})
then sup {x,y} = e1 "\/" e2 by YELLOW_0:41
.= e1 by A2, YELLOW_5:8 ;
hence sup {x,y} in the carrier of (subrelstr {e1,e2}) by A1, TARSKI:def_2; ::_thesis: verum
end;
suppose ( x = e2 & y = e1 ) ; ::_thesis: sup {x,y} in the carrier of (subrelstr {e1,e2})
then sup {x,y} = e2 "\/" e1 by YELLOW_0:41
.= e1 by A2, YELLOW_5:8 ;
hence sup {x,y} in the carrier of (subrelstr {e1,e2}) by A1, TARSKI:def_2; ::_thesis: verum
end;
suppose ( x = e2 & y = e2 ) ; ::_thesis: sup {x,y} in the carrier of (subrelstr {e1,e2})
then sup {x,y} = e2 "\/" e2 by YELLOW_0:41
.= e2 by YELLOW_5:1 ;
hence sup {x,y} in the carrier of (subrelstr {e1,e2}) by A1, TARSKI:def_2; ::_thesis: verum
end;
end;
end;
A32: the Element of A <> the Element of A \ { the Element of A} by ZFMISC_1:56;
A33: not subrelstr {e1,e2} is trivial
proof
assume subrelstr {e1,e2} is trivial ; ::_thesis: contradiction
then ex s being Element of (subrelstr {e1,e2}) st the carrier of (subrelstr {e1,e2}) = {s} by TEX_1:def_1;
then nabla A = id A by A1, ZFMISC_1:5;
then [:A,A:] = id A by EQREL_1:def_1;
then [ the Element of A, the Element of A \ { the Element of A}] in id A by ZFMISC_1:def_2;
hence contradiction by A32, RELAT_1:def_10; ::_thesis: verum
end;
reconsider Y = subrelstr {e1,e2} as non empty full Sublattice of EqRelLATT A by A3, A30, YELLOW_0:def_16, YELLOW_0:def_17;
take Y ; ::_thesis: ( not Y is trivial & Y is finitely_typed & Y is full )
thus ( not Y is trivial & Y is finitely_typed & Y is full ) by A33, A5; ::_thesis: verum
end;
end;
theorem Th1: :: LATTICE8:1
for A being non empty set
for L being lower-bounded LATTICE
for d being distance_function of A,L holds succ {} c= DistEsti d
proof
let A be non empty set ; ::_thesis: for L being lower-bounded LATTICE
for d being distance_function of A,L holds succ {} c= DistEsti d
let L be lower-bounded LATTICE; ::_thesis: for d being distance_function of A,L holds succ {} c= DistEsti d
let d be distance_function of A,L; ::_thesis: succ {} c= DistEsti d
( succ {} c= DistEsti d or DistEsti d in succ {} ) by ORDINAL1:16;
then ( succ {} c= DistEsti d or DistEsti d c= {} ) by ORDINAL1:22;
hence succ {} c= DistEsti d by LATTICE5:20, XBOOLE_1:3; ::_thesis: verum
end;
theorem :: LATTICE8:2
for L being trivial Semilattice holds L is modular
proof
let L be trivial Semilattice; ::_thesis: L is modular
let a, b, c be Element of L; :: according to YELLOW11:def_3 ::_thesis: ( not a <= c or a "\/" (b "/\" c) = (a "\/" b) "/\" c )
assume a <= c ; ::_thesis: a "\/" (b "/\" c) = (a "\/" b) "/\" c
thus a "\/" (b "/\" c) = (a "\/" b) "/\" c by STRUCT_0:def_10; ::_thesis: verum
end;
theorem :: LATTICE8:3
for A being non empty set
for L being non empty Sublattice of EqRelLATT A holds
( L is trivial or ex e being Equivalence_Relation of A st
( e in the carrier of L & e <> id A ) )
proof
let A be non empty set ; ::_thesis: for L being non empty Sublattice of EqRelLATT A holds
( L is trivial or ex e being Equivalence_Relation of A st
( e in the carrier of L & e <> id A ) )
let L be non empty Sublattice of EqRelLATT A; ::_thesis: ( L is trivial or ex e being Equivalence_Relation of A st
( e in the carrier of L & e <> id A ) )
now__::_thesis:_(_(_for_e_being_Equivalence_Relation_of_A_holds_
(_not_e_in_the_carrier_of_L_or_not_e_<>_id_A_)_)_implies_L_is_trivial_)
assume A1: for e being Equivalence_Relation of A holds
( not e in the carrier of L or not e <> id A ) ; ::_thesis: L is trivial
thus L is trivial ::_thesis: verum
proof
consider x being set such that
A2: x in the carrier of L by XBOOLE_0:def_1;
the carrier of L c= the carrier of (EqRelLATT A) by YELLOW_0:def_13;
then reconsider e = x as Equivalence_Relation of A by A2, LATTICE5:4;
the carrier of L = {x}
proof
thus the carrier of L c= {x} :: according to XBOOLE_0:def_10 ::_thesis: {x} c= the carrier of L
proof
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in the carrier of L or a in {x} )
assume A3: a in the carrier of L ; ::_thesis: a in {x}
the carrier of L c= the carrier of (EqRelLATT A) by YELLOW_0:def_13;
then reconsider B = a as Equivalence_Relation of A by A3, LATTICE5:4;
B = id A by A1, A3
.= e by A1, A2 ;
hence a in {x} by TARSKI:def_1; ::_thesis: verum
end;
let A be set ; :: according to TARSKI:def_3 ::_thesis: ( not A in {x} or A in the carrier of L )
assume A in {x} ; ::_thesis: A in the carrier of L
hence A in the carrier of L by A2, TARSKI:def_1; ::_thesis: verum
end;
hence L is trivial ; ::_thesis: verum
end;
end;
hence ( L is trivial or ex e being Equivalence_Relation of A st
( e in the carrier of L & e <> id A ) ) ; ::_thesis: verum
end;
theorem Th4: :: LATTICE8:4
for L1, L2 being lower-bounded LATTICE
for f being Function of L1,L2 st f is infs-preserving & f is sups-preserving holds
( f is meet-preserving & f is join-preserving )
proof
let L1, L2 be lower-bounded LATTICE; ::_thesis: for f being Function of L1,L2 st f is infs-preserving & f is sups-preserving holds
( f is meet-preserving & f is join-preserving )
let f be Function of L1,L2; ::_thesis: ( f is infs-preserving & f is sups-preserving implies ( f is meet-preserving & f is join-preserving ) )
assume A1: ( f is infs-preserving & f is sups-preserving ) ; ::_thesis: ( f is meet-preserving & f is join-preserving )
thus f is meet-preserving ::_thesis: f is join-preserving
proof
let x, y be Element of L1; :: according to WAYBEL_0:def_34 ::_thesis: f preserves_inf_of {x,y}
thus f preserves_inf_of {x,y} by A1, WAYBEL_0:def_32; ::_thesis: verum
end;
thus f is join-preserving ::_thesis: verum
proof
let x, y be Element of L1; :: according to WAYBEL_0:def_35 ::_thesis: f preserves_sup_of {x,y}
thus f preserves_sup_of {x,y} by A1, WAYBEL_0:def_33; ::_thesis: verum
end;
end;
theorem Th5: :: LATTICE8:5
for L1, L2 being lower-bounded LATTICE st L1,L2 are_isomorphic & L1 is modular holds
L2 is modular
proof
let L1, L2 be lower-bounded LATTICE; ::_thesis: ( L1,L2 are_isomorphic & L1 is modular implies L2 is modular )
assume that
A1: L1,L2 are_isomorphic and
A2: L1 is modular ; ::_thesis: L2 is modular
let a, b, c be Element of L2; :: according to YELLOW11:def_3 ::_thesis: ( not a <= c or a "\/" (b "/\" c) = (a "\/" b) "/\" c )
consider f being Function of L1,L2 such that
A3: f is isomorphic by A1, WAYBEL_1:def_8;
set C = (f ") . c;
set A = (f ") . a;
set B = (f ") . b;
A4: ( f is V14() & rng f = the carrier of L2 ) by A3, WAYBEL_0:66;
then A5: b = f . ((f ") . b) by FUNCT_1:35;
A6: (f ") . c in dom f by A4, FUNCT_1:32;
A7: ( (f ") . a in dom f & (f ") . b in dom f ) by A4, FUNCT_1:32;
A8: ( a = f . ((f ") . a) & c = f . ((f ") . c) ) by A4, FUNCT_1:35;
reconsider A = (f ") . a, B = (f ") . b, C = (f ") . c as Element of L1 by A7, A6;
assume a <= c ; ::_thesis: a "\/" (b "/\" c) = (a "\/" b) "/\" c
then A <= C by A3, A8, WAYBEL_0:66;
then A9: A "\/" (B "/\" C) = (A "\/" B) "/\" C by A2, YELLOW11:def_3;
( f is infs-preserving & f is sups-preserving ) by A3, WAYBEL13:20;
then A10: ( f is meet-preserving & f is join-preserving ) by Th4;
hence a "\/" (b "/\" c) = (f . A) "\/" (f . (B "/\" C)) by A5, A8, WAYBEL_6:1
.= f . ((A "\/" B) "/\" C) by A10, A9, WAYBEL_6:2
.= (f . (A "\/" B)) "/\" (f . C) by A10, WAYBEL_6:1
.= (a "\/" b) "/\" c by A10, A5, A8, WAYBEL_6:2 ;
::_thesis: verum
end;
theorem Th6: :: LATTICE8:6
for S being non empty lower-bounded Poset
for T being non empty Poset
for f being monotone Function of S,T holds Image f is lower-bounded
proof
let S be non empty lower-bounded Poset; ::_thesis: for T being non empty Poset
for f being monotone Function of S,T holds Image f is lower-bounded
let T be non empty Poset; ::_thesis: for f being monotone Function of S,T holds Image f is lower-bounded
let f be monotone Function of S,T; ::_thesis: Image f is lower-bounded
thus Image f is lower-bounded ::_thesis: verum
proof
consider x being Element of S such that
A1: x is_<=_than the carrier of S by YELLOW_0:def_4;
dom f = the carrier of S by FUNCT_2:def_1;
then f . x in rng f by FUNCT_1:def_3;
then reconsider fx = f . x as Element of (Image f) by YELLOW_0:def_15;
take fx ; :: according to YELLOW_0:def_4 ::_thesis: fx is_<=_than the carrier of (Image f)
let b be Element of (Image f); :: according to LATTICE3:def_8 ::_thesis: ( not b in the carrier of (Image f) or fx <= b )
b in the carrier of (subrelstr (rng f)) ;
then b in rng f by YELLOW_0:def_15;
then consider c being set such that
A2: c in dom f and
A3: f . c = b by FUNCT_1:def_3;
A4: the carrier of (Image f) c= the carrier of T by YELLOW_0:def_13;
assume b in the carrier of (Image f) ; ::_thesis: fx <= b
then reconsider b1 = b as Element of T by A4;
reconsider c = c as Element of S by A2;
x <= c by A1, LATTICE3:def_8;
then f . x <= b1 by A3, ORDERS_3:def_5;
hence fx <= b by YELLOW_0:60; ::_thesis: verum
end;
end;
theorem Th7: :: LATTICE8:7
for L being lower-bounded LATTICE
for x, y being Element of L
for A being non empty set
for f being Homomorphism of L,(EqRelLATT A) st f is V14() & (corestr f) . x <= (corestr f) . y holds
x <= y
proof
let L be lower-bounded LATTICE; ::_thesis: for x, y being Element of L
for A being non empty set
for f being Homomorphism of L,(EqRelLATT A) st f is V14() & (corestr f) . x <= (corestr f) . y holds
x <= y
let x, y be Element of L; ::_thesis: for A being non empty set
for f being Homomorphism of L,(EqRelLATT A) st f is V14() & (corestr f) . x <= (corestr f) . y holds
x <= y
let A be non empty set ; ::_thesis: for f being Homomorphism of L,(EqRelLATT A) st f is V14() & (corestr f) . x <= (corestr f) . y holds
x <= y
let f be Homomorphism of L,(EqRelLATT A); ::_thesis: ( f is V14() & (corestr f) . x <= (corestr f) . y implies x <= y )
assume that
A1: f is V14() and
A2: (corestr f) . x <= (corestr f) . y ; ::_thesis: x <= y
now__::_thesis:_x_<=_y
A3: corestr f = f by WAYBEL_1:30;
A4: for x, y being Element of L holds (corestr f) . (x "/\" y) = ((corestr f) . x) "/\" ((corestr f) . y)
proof
let x, y be Element of L; ::_thesis: (corestr f) . (x "/\" y) = ((corestr f) . x) "/\" ((corestr f) . y)
thus (corestr f) . (x "/\" y) = (f . x) "/\" (f . y) by A3, WAYBEL_6:1
.= ((corestr f) . x) "/\" ((corestr f) . y) by A3, YELLOW_0:69 ; ::_thesis: verum
end;
A5: corestr f is V14() by A1, WAYBEL_1:30;
((corestr f) . y) "/\" ((corestr f) . x) = (corestr f) . x by A2, YELLOW_5:10;
then (corestr f) . x = (corestr f) . (x "/\" y) by A4;
then A6: x = x "/\" y by A5, WAYBEL_1:def_1;
assume not x <= y ; ::_thesis: contradiction
hence contradiction by A6, YELLOW_0:25; ::_thesis: verum
end;
hence x <= y ; ::_thesis: verum
end;
begin
theorem Th8: :: LATTICE8:8
for A being non trivial set
for L being non empty full finitely_typed Sublattice of EqRelLATT A
for e being Equivalence_Relation of A st e in the carrier of L & e <> id A & type_of L <= 2 holds
L is modular
proof
let A be non trivial set ; ::_thesis: for L being non empty full finitely_typed Sublattice of EqRelLATT A
for e being Equivalence_Relation of A st e in the carrier of L & e <> id A & type_of L <= 2 holds
L is modular
let L be non empty full finitely_typed Sublattice of EqRelLATT A; ::_thesis: for e being Equivalence_Relation of A st e in the carrier of L & e <> id A & type_of L <= 2 holds
L is modular
let e be Equivalence_Relation of A; ::_thesis: ( e in the carrier of L & e <> id A & type_of L <= 2 implies L is modular )
assume that
A1: e in the carrier of L and
A2: e <> id A ; ::_thesis: ( not type_of L <= 2 or L is modular )
assume A3: type_of L <= 2 ; ::_thesis: L is modular
let a, b, c be Element of L; :: according to YELLOW11:def_3 ::_thesis: ( not a <= c or a "\/" (b "/\" c) = (a "\/" b) "/\" c )
A4: the carrier of L c= the carrier of (EqRelLATT A) by YELLOW_0:def_13;
A5: b in the carrier of L ;
then reconsider b9 = b as Equivalence_Relation of A by A4, LATTICE5:4;
reconsider b99 = b9 as Element of (EqRelLATT A) by A4, A5;
A6: a in the carrier of L ;
then reconsider a9 = a as Equivalence_Relation of A by A4, LATTICE5:4;
A7: c in the carrier of L ;
then reconsider c9 = c as Equivalence_Relation of A by A4, LATTICE5:4;
reconsider c99 = c9 as Element of (EqRelLATT A) by A4, A7;
reconsider a99 = a9 as Element of (EqRelLATT A) by A4, A6;
A8: (a99 "\/" b99) "/\" c99 = (a99 "\/" b99) /\ c9 by LATTICE5:8
.= (a9 "\/" b9) /\ c9 by LATTICE5:10 ;
assume A9: a <= c ; ::_thesis: a "\/" (b "/\" c) = (a "\/" b) "/\" c
then a99 <= c99 by YELLOW_0:59;
then A10: a9 c= c9 by LATTICE5:6;
A11: a99 "\/" (b99 "/\" c99) <= (a99 "\/" b99) "/\" c99 by A9, YELLOW11:8, YELLOW_0:59;
A12: b9 /\ c9 = b99 "/\" c99 by LATTICE5:8;
then a9 "\/" (b9 /\ c9) = a99 "\/" (b99 "/\" c99) by LATTICE5:10;
then A13: a9 "\/" (b9 /\ c9) c= (a9 "\/" b9) /\ c9 by A11, A8, LATTICE5:6;
consider AA being non empty set such that
A14: for e being set st e in the carrier of L holds
e is Equivalence_Relation of AA and
A15: ex i being Element of NAT st
for e1, e2 being Equivalence_Relation of AA
for x, y being set st e1 in the carrier of L & e2 in the carrier of L & [x,y] in e1 "\/" e2 holds
ex F being non empty FinSequence of AA st
( len F = i & x,y are_joint_by F,e1,e2 ) by Def1;
e is Equivalence_Relation of AA by A1, A14;
then A16: ( field e = A & field e = AA ) by EQREL_1:9;
A17: (a9 "\/" b9) /\ c9 c= a9 "\/" (b9 /\ c9)
proof
let x, y be Element of A; :: according to RELSET_1:def_1 ::_thesis: ( not [x,y] in (a9 "\/" b9) /\ c9 or [x,y] in a9 "\/" (b9 /\ c9) )
assume A18: [x,y] in (a9 "\/" b9) /\ c9 ; ::_thesis: [x,y] in a9 "\/" (b9 /\ c9)
then A19: [x,y] in a9 "\/" b9 by XBOOLE_0:def_4;
A20: [x,y] in c9 by A18, XBOOLE_0:def_4;
percases ( type_of L = 2 or type_of L = 1 or type_of L = 0 ) by A3, NAT_1:26;
suppose type_of L = 2 ; ::_thesis: [x,y] in a9 "\/" (b9 /\ c9)
then consider F being non empty FinSequence of A such that
A21: len F = 2 + 2 and
A22: x,y are_joint_by F,a9,b9 by A1, A2, A15, A16, A19, LATTICE5:def_4;
A23: F . 4 = y by A21, A22, LATTICE5:def_3;
consider l being Element of NAT such that
A24: l = 1 ;
(2 * l) + 1 = 3 by A24;
then A25: [(F . 3),(F . (3 + 1))] in a9 by A21, A22, LATTICE5:def_3;
consider k being Element of NAT such that
A26: k = 1 ;
2 * k = 2 by A26;
then A27: [(F . 2),(F . (2 + 1))] in b9 by A21, A22, LATTICE5:def_3;
A28: F . 1 = x by A22, LATTICE5:def_3;
reconsider BC = b9 /\ c9 as Element of (EqRelLATT A) by A12;
set z1 = F . 2;
set z2 = F . 3;
consider j being Element of NAT such that
A29: j = 0 ;
(2 * j) + 1 = 1 by A29;
then A30: [(F . 1),(F . (1 + 1))] in a9 by A21, A22, LATTICE5:def_3;
A31: a9 "\/" (b9 /\ c9) = a99 "\/" BC by LATTICE5:10;
BC <= BC "\/" a99 by YELLOW_0:22;
then A32: b9 /\ c9 c= a9 "\/" (b9 /\ c9) by A31, LATTICE5:6;
a99 <= a99 "\/" BC by YELLOW_0:22;
then A33: a9 c= a9 "\/" (b9 /\ c9) by A31, LATTICE5:6;
[y,x] in c9 by A20, EQREL_1:6;
then [(F . 3),x] in c9 by A10, A23, A25, EQREL_1:7;
then [(F . 3),(F . 2)] in c9 by A10, A28, A30, EQREL_1:7;
then [(F . 2),(F . 3)] in c9 by EQREL_1:6;
then [(F . 2),(F . 3)] in b9 /\ c9 by A27, XBOOLE_0:def_4;
then [x,(F . 3)] in a9 "\/" (b9 /\ c9) by A28, A30, A33, A32, EQREL_1:7;
hence [x,y] in a9 "\/" (b9 /\ c9) by A23, A25, A33, EQREL_1:7; ::_thesis: verum
end;
suppose type_of L = 1 ; ::_thesis: [x,y] in a9 "\/" (b9 /\ c9)
then consider F being non empty FinSequence of A such that
A34: len F = 1 + 2 and
A35: x,y are_joint_by F,a9,b9 by A1, A2, A15, A16, A19, LATTICE5:def_4;
set z1 = F . 2;
consider k being Element of NAT such that
A36: k = 1 ;
reconsider BC = b9 /\ c9 as Element of (EqRelLATT A) by A12;
consider j being Element of NAT such that
A37: j = 0 ;
(2 * j) + 1 = 1 by A37;
then A38: [(F . 1),(F . (1 + 1))] in a9 by A34, A35, LATTICE5:def_3;
2 * k = 2 by A36;
then A39: [(F . 2),(F . (2 + 1))] in b9 by A34, A35, LATTICE5:def_3;
A40: a9 "\/" (b9 /\ c9) = a99 "\/" BC by LATTICE5:10;
A41: [x,y] in c9 by A18, XBOOLE_0:def_4;
A42: F . 1 = x by A35, LATTICE5:def_3;
then [(F . 2),x] in c9 by A10, A38, EQREL_1:6;
then A43: [(F . 2),y] in c9 by A41, EQREL_1:7;
BC <= BC "\/" a99 by YELLOW_0:22;
then A44: b9 /\ c9 c= a9 "\/" (b9 /\ c9) by A40, LATTICE5:6;
a99 <= a99 "\/" BC by YELLOW_0:22;
then A45: a9 c= a9 "\/" (b9 /\ c9) by A40, LATTICE5:6;
F . 3 = y by A34, A35, LATTICE5:def_3;
then [(F . 2),y] in b9 /\ c9 by A39, A43, XBOOLE_0:def_4;
hence [x,y] in a9 "\/" (b9 /\ c9) by A42, A38, A45, A44, EQREL_1:7; ::_thesis: verum
end;
suppose type_of L = 0 ; ::_thesis: [x,y] in a9 "\/" (b9 /\ c9)
then consider F being non empty FinSequence of A such that
A46: ( len F = 0 + 2 & x,y are_joint_by F,a9,b9 ) by A1, A2, A15, A16, A19, LATTICE5:def_4;
reconsider BC = b9 /\ c9 as Element of (EqRelLATT A) by A12;
consider j being Element of NAT such that
A47: j = 0 ;
(2 * j) + 1 = 1 by A47;
then A48: [(F . 1),(F . (1 + 1))] in a9 by A46, LATTICE5:def_3;
( a99 <= a99 "\/" BC & a9 "\/" (b9 /\ c9) = a99 "\/" BC ) by LATTICE5:10, YELLOW_0:22;
then A49: a9 c= a9 "\/" (b9 /\ c9) by LATTICE5:6;
( F . 1 = x & F . 2 = y ) by A46, LATTICE5:def_3;
hence [x,y] in a9 "\/" (b9 /\ c9) by A48, A49; ::_thesis: verum
end;
end;
end;
a99 "\/" b99 = a "\/" b by YELLOW_0:70;
then A50: (a "\/" b) "/\" c = (a99 "\/" b99) "/\" c99 by YELLOW_0:69
.= (a99 "\/" b99) /\ c9 by LATTICE5:8
.= (a9 "\/" b9) /\ c9 by LATTICE5:10 ;
A51: b99 "/\" c99 = b "/\" c by YELLOW_0:69;
a9 "\/" (b9 /\ c9) = a99 "\/" (b99 "/\" c99) by A12, LATTICE5:10
.= a "\/" (b "/\" c) by A51, YELLOW_0:70 ;
hence a "\/" (b "/\" c) = (a "\/" b) "/\" c by A13, A17, A50, XBOOLE_0:def_10; ::_thesis: verum
end;
theorem Th9: :: LATTICE8:9
for L being lower-bounded LATTICE st L has_a_representation_of_type<= 2 holds
L is modular
proof
let L be lower-bounded LATTICE; ::_thesis: ( L has_a_representation_of_type<= 2 implies L is modular )
assume L has_a_representation_of_type<= 2 ; ::_thesis: L is modular
then consider A being non trivial set , f being Homomorphism of L,(EqRelLATT A) such that
A1: f is V14() and
A2: ( Image f is finitely_typed & ex e being Equivalence_Relation of A st
( e in the carrier of (Image f) & e <> id A ) & type_of (Image f) <= 2 ) by Def2;
A3: ( rng (corestr f) = the carrier of (Image f) & ( for x, y being Element of L st x <= y holds
(corestr f) . x <= (corestr f) . y ) ) by FUNCT_2:def_3, WAYBEL_1:def_2;
( corestr f is V14() & ( for x, y being Element of L st (corestr f) . x <= (corestr f) . y holds
x <= y ) ) by A1, Th7, WAYBEL_1:30;
then corestr f is isomorphic by A3, WAYBEL_0:66;
then A4: L, Image f are_isomorphic by WAYBEL_1:def_8;
A5: Image f is lower-bounded LATTICE by Th6;
Image f is modular by A2, Th8;
hence L is modular by A5, A4, Th5, WAYBEL_1:6; ::_thesis: verum
end;
definition
let A be set ;
func new_set2 A -> set equals :: LATTICE8:def 3
A \/ {{A},{{A}}};
correctness
coherence
A \/ {{A},{{A}}} is set ;
;
end;
:: deftheorem defines new_set2 LATTICE8:def_3_:_
for A being set holds new_set2 A = A \/ {{A},{{A}}};
registration
let A be set ;
cluster new_set2 A -> non empty ;
coherence
not new_set2 A is empty ;
end;
definition
let A be non empty set ;
let L be lower-bounded LATTICE;
let d be BiFunction of A,L;
let q be Element of [:A,A, the carrier of L, the carrier of L:];
func new_bi_fun2 (d,q) -> BiFunction of (new_set2 A),L means :Def4: :: LATTICE8:def 4
( ( for u, v being Element of A holds it . (u,v) = d . (u,v) ) & it . ({A},{A}) = Bottom L & it . ({{A}},{{A}}) = Bottom L & it . ({A},{{A}}) = ((d . ((q `1_4),(q `2_4))) "\/" (q `3_4)) "/\" (q `4_4) & it . ({{A}},{A}) = ((d . ((q `1_4),(q `2_4))) "\/" (q `3_4)) "/\" (q `4_4) & ( for u being Element of A holds
( it . (u,{A}) = (d . (u,(q `1_4))) "\/" (q `3_4) & it . ({A},u) = (d . (u,(q `1_4))) "\/" (q `3_4) & it . (u,{{A}}) = (d . (u,(q `2_4))) "\/" (q `3_4) & it . ({{A}},u) = (d . (u,(q `2_4))) "\/" (q `3_4) ) ) );
existence
ex b1 being BiFunction of (new_set2 A),L st
( ( for u, v being Element of A holds b1 . (u,v) = d . (u,v) ) & b1 . ({A},{A}) = Bottom L & b1 . ({{A}},{{A}}) = Bottom L & b1 . ({A},{{A}}) = ((d . ((q `1_4),(q `2_4))) "\/" (q `3_4)) "/\" (q `4_4) & b1 . ({{A}},{A}) = ((d . ((q `1_4),(q `2_4))) "\/" (q `3_4)) "/\" (q `4_4) & ( for u being Element of A holds
( b1 . (u,{A}) = (d . (u,(q `1_4))) "\/" (q `3_4) & b1 . ({A},u) = (d . (u,(q `1_4))) "\/" (q `3_4) & b1 . (u,{{A}}) = (d . (u,(q `2_4))) "\/" (q `3_4) & b1 . ({{A}},u) = (d . (u,(q `2_4))) "\/" (q `3_4) ) ) )
proof
reconsider a = q `3_4 , b = q `4_4 as Element of L ;
set x = q `1_4 ;
set y = q `2_4 ;
defpred S1[ Element of new_set2 A, Element of new_set2 A, set ] means ( ( $1 in A & $2 in A implies $3 = d . ($1,$2) ) & ( ( ( $1 = {A} & $2 = {{A}} ) or ( $2 = {A} & $1 = {{A}} ) ) implies $3 = ((d . ((q `1_4),(q `2_4))) "\/" a) "/\" b ) & ( ( $1 = {A} or $1 = {{A}} ) & $2 = $1 implies $3 = Bottom L ) & ( $1 in A & $2 = {A} implies ex p9 being Element of A st
( p9 = $1 & $3 = (d . (p9,(q `1_4))) "\/" a ) ) & ( $1 in A & $2 = {{A}} implies ex p9 being Element of A st
( p9 = $1 & $3 = (d . (p9,(q `2_4))) "\/" a ) ) & ( $2 in A & $1 = {A} implies ex q9 being Element of A st
( q9 = $2 & $3 = (d . (q9,(q `1_4))) "\/" a ) ) & ( $2 in A & $1 = {{A}} implies ex q9 being Element of A st
( q9 = $2 & $3 = (d . (q9,(q `2_4))) "\/" a ) ) );
{{A}} in {{A},{{A}}} by TARSKI:def_2;
then A1: {{A}} in new_set2 A by XBOOLE_0:def_3;
A2: for p, q being Element of new_set2 A ex r being Element of L st S1[p,q,r]
proof
let p, q be Element of new_set2 A; ::_thesis: ex r being Element of L st S1[p,q,r]
A3: ( p in A or p in {{A},{{A}}} ) by XBOOLE_0:def_3;
A4: ( q in A or q in {{A},{{A}}} ) by XBOOLE_0:def_3;
A5: ( ( ( p = {A} or p = {{A}} ) & p = q ) iff ( ( p = {A} & q = {A} ) or ( p = {{A}} & q = {{A}} ) ) ) ;
A6: not {A} in A by TARSKI:def_1;
A7: {A} <> {{A}}
proof
assume {A} = {{A}} ; ::_thesis: contradiction
then {A} in {A} by TARSKI:def_1;
hence contradiction ; ::_thesis: verum
end;
A8: not {{A}} in A
proof
A9: ( A in {A} & {A} in {{A}} ) by TARSKI:def_1;
assume {{A}} in A ; ::_thesis: contradiction
hence contradiction by A9, XREGULAR:7; ::_thesis: verum
end;
percases ( ( p in A & q in A ) or ( p = {A} & q = {{A}} ) or ( q = {A} & p = {{A}} ) or ( ( p = {A} or p = {{A}} ) & q = p ) or ( p in A & q = {A} ) or ( p in A & q = {{A}} ) or ( q in A & p = {A} ) or ( q in A & p = {{A}} ) ) by A3, A4, A5, TARSKI:def_2;
suppose ( p in A & q in A ) ; ::_thesis: ex r being Element of L st S1[p,q,r]
then reconsider p9 = p, q9 = q as Element of A ;
take d . (p9,q9) ; ::_thesis: S1[p,q,d . (p9,q9)]
thus S1[p,q,d . (p9,q9)] by A6, A8; ::_thesis: verum
end;
supposeA10: ( ( p = {A} & q = {{A}} ) or ( q = {A} & p = {{A}} ) ) ; ::_thesis: ex r being Element of L st S1[p,q,r]
take ((d . ((q `1_4),(q `2_4))) "\/" a) "/\" b ; ::_thesis: S1[p,q,((d . ((q `1_4),(q `2_4))) "\/" a) "/\" b]
thus S1[p,q,((d . ((q `1_4),(q `2_4))) "\/" a) "/\" b] by A7, A8, A10, TARSKI:def_1; ::_thesis: verum
end;
supposeA11: ( ( p = {A} or p = {{A}} ) & q = p ) ; ::_thesis: ex r being Element of L st S1[p,q,r]
take Bottom L ; ::_thesis: S1[p,q, Bottom L]
thus S1[p,q, Bottom L] by A7, A8, A11, TARSKI:def_1; ::_thesis: verum
end;
supposeA12: ( p in A & q = {A} ) ; ::_thesis: ex r being Element of L st S1[p,q,r]
then reconsider p9 = p as Element of A ;
take (d . (p9,(q `1_4))) "\/" a ; ::_thesis: S1[p,q,(d . (p9,(q `1_4))) "\/" a]
thus S1[p,q,(d . (p9,(q `1_4))) "\/" a] by A7, A8, A12, TARSKI:def_1; ::_thesis: verum
end;
supposeA13: ( p in A & q = {{A}} ) ; ::_thesis: ex r being Element of L st S1[p,q,r]
then reconsider p9 = p as Element of A ;
take (d . (p9,(q `2_4))) "\/" a ; ::_thesis: S1[p,q,(d . (p9,(q `2_4))) "\/" a]
thus S1[p,q,(d . (p9,(q `2_4))) "\/" a] by A7, A8, A13, TARSKI:def_1; ::_thesis: verum
end;
supposeA14: ( q in A & p = {A} ) ; ::_thesis: ex r being Element of L st S1[p,q,r]
then reconsider q9 = q as Element of A ;
take (d . (q9,(q `1_4))) "\/" a ; ::_thesis: S1[p,q,(d . (q9,(q `1_4))) "\/" a]
thus S1[p,q,(d . (q9,(q `1_4))) "\/" a] by A7, A8, A14, TARSKI:def_1; ::_thesis: verum
end;
supposeA15: ( q in A & p = {{A}} ) ; ::_thesis: ex r being Element of L st S1[p,q,r]
then reconsider q9 = q as Element of A ;
take (d . (q9,(q `2_4))) "\/" a ; ::_thesis: S1[p,q,(d . (q9,(q `2_4))) "\/" a]
thus S1[p,q,(d . (q9,(q `2_4))) "\/" a] by A7, A8, A15, TARSKI:def_1; ::_thesis: verum
end;
end;
end;
consider f being Function of [:(new_set2 A),(new_set2 A):], the carrier of L such that
A16: for p, q being Element of new_set2 A holds S1[p,q,f . (p,q)] from BINOP_1:sch_3(A2);
reconsider f = f as BiFunction of (new_set2 A),L ;
{A} in {{A},{{A}}} by TARSKI:def_2;
then A17: {A} in new_set2 A by XBOOLE_0:def_3;
A18: for u being Element of A holds
( f . ({A},u) = (d . (u,(q `1_4))) "\/" a & f . ({{A}},u) = (d . (u,(q `2_4))) "\/" a )
proof
let u be Element of A; ::_thesis: ( f . ({A},u) = (d . (u,(q `1_4))) "\/" a & f . ({{A}},u) = (d . (u,(q `2_4))) "\/" a )
reconsider u9 = u as Element of new_set2 A by XBOOLE_0:def_3;
ex u1 being Element of A st
( u1 = u9 & f . ({A},u9) = (d . (u1,(q `1_4))) "\/" a ) by A17, A16;
hence f . ({A},u) = (d . (u,(q `1_4))) "\/" a ; ::_thesis: f . ({{A}},u) = (d . (u,(q `2_4))) "\/" a
ex u2 being Element of A st
( u2 = u9 & f . ({{A}},u9) = (d . (u2,(q `2_4))) "\/" a ) by A1, A16;
hence f . ({{A}},u) = (d . (u,(q `2_4))) "\/" a ; ::_thesis: verum
end;
take f ; ::_thesis: ( ( for u, v being Element of A holds f . (u,v) = d . (u,v) ) & f . ({A},{A}) = Bottom L & f . ({{A}},{{A}}) = Bottom L & f . ({A},{{A}}) = ((d . ((q `1_4),(q `2_4))) "\/" (q `3_4)) "/\" (q `4_4) & f . ({{A}},{A}) = ((d . ((q `1_4),(q `2_4))) "\/" (q `3_4)) "/\" (q `4_4) & ( for u being Element of A holds
( f . (u,{A}) = (d . (u,(q `1_4))) "\/" (q `3_4) & f . ({A},u) = (d . (u,(q `1_4))) "\/" (q `3_4) & f . (u,{{A}}) = (d . (u,(q `2_4))) "\/" (q `3_4) & f . ({{A}},u) = (d . (u,(q `2_4))) "\/" (q `3_4) ) ) )
A19: for u, v being Element of A holds f . (u,v) = d . (u,v)
proof
let u, v be Element of A; ::_thesis: f . (u,v) = d . (u,v)
reconsider u9 = u, v9 = v as Element of new_set2 A by XBOOLE_0:def_3;
thus f . (u,v) = f . (u9,v9)
.= d . (u,v) by A16 ; ::_thesis: verum
end;
for u being Element of A holds
( f . (u,{A}) = (d . (u,(q `1_4))) "\/" a & f . (u,{{A}}) = (d . (u,(q `2_4))) "\/" a )
proof
let u be Element of A; ::_thesis: ( f . (u,{A}) = (d . (u,(q `1_4))) "\/" a & f . (u,{{A}}) = (d . (u,(q `2_4))) "\/" a )
reconsider u9 = u as Element of new_set2 A by XBOOLE_0:def_3;
ex u1 being Element of A st
( u1 = u9 & f . (u9,{A}) = (d . (u1,(q `1_4))) "\/" a ) by A17, A16;
hence f . (u,{A}) = (d . (u,(q `1_4))) "\/" a ; ::_thesis: f . (u,{{A}}) = (d . (u,(q `2_4))) "\/" a
ex u2 being Element of A st
( u2 = u9 & f . (u9,{{A}}) = (d . (u2,(q `2_4))) "\/" a ) by A1, A16;
hence f . (u,{{A}}) = (d . (u,(q `2_4))) "\/" a ; ::_thesis: verum
end;
hence ( ( for u, v being Element of A holds f . (u,v) = d . (u,v) ) & f . ({A},{A}) = Bottom L & f . ({{A}},{{A}}) = Bottom L & f . ({A},{{A}}) = ((d . ((q `1_4),(q `2_4))) "\/" (q `3_4)) "/\" (q `4_4) & f . ({{A}},{A}) = ((d . ((q `1_4),(q `2_4))) "\/" (q `3_4)) "/\" (q `4_4) & ( for u being Element of A holds
( f . (u,{A}) = (d . (u,(q `1_4))) "\/" (q `3_4) & f . ({A},u) = (d . (u,(q `1_4))) "\/" (q `3_4) & f . (u,{{A}}) = (d . (u,(q `2_4))) "\/" (q `3_4) & f . ({{A}},u) = (d . (u,(q `2_4))) "\/" (q `3_4) ) ) ) by A17, A1, A16, A19, A18; ::_thesis: verum
end;
uniqueness
for b1, b2 being BiFunction of (new_set2 A),L st ( for u, v being Element of A holds b1 . (u,v) = d . (u,v) ) & b1 . ({A},{A}) = Bottom L & b1 . ({{A}},{{A}}) = Bottom L & b1 . ({A},{{A}}) = ((d . ((q `1_4),(q `2_4))) "\/" (q `3_4)) "/\" (q `4_4) & b1 . ({{A}},{A}) = ((d . ((q `1_4),(q `2_4))) "\/" (q `3_4)) "/\" (q `4_4) & ( for u being Element of A holds
( b1 . (u,{A}) = (d . (u,(q `1_4))) "\/" (q `3_4) & b1 . ({A},u) = (d . (u,(q `1_4))) "\/" (q `3_4) & b1 . (u,{{A}}) = (d . (u,(q `2_4))) "\/" (q `3_4) & b1 . ({{A}},u) = (d . (u,(q `2_4))) "\/" (q `3_4) ) ) & ( for u, v being Element of A holds b2 . (u,v) = d . (u,v) ) & b2 . ({A},{A}) = Bottom L & b2 . ({{A}},{{A}}) = Bottom L & b2 . ({A},{{A}}) = ((d . ((q `1_4),(q `2_4))) "\/" (q `3_4)) "/\" (q `4_4) & b2 . ({{A}},{A}) = ((d . ((q `1_4),(q `2_4))) "\/" (q `3_4)) "/\" (q `4_4) & ( for u being Element of A holds
( b2 . (u,{A}) = (d . (u,(q `1_4))) "\/" (q `3_4) & b2 . ({A},u) = (d . (u,(q `1_4))) "\/" (q `3_4) & b2 . (u,{{A}}) = (d . (u,(q `2_4))) "\/" (q `3_4) & b2 . ({{A}},u) = (d . (u,(q `2_4))) "\/" (q `3_4) ) ) holds
b1 = b2
proof
set x = q `1_4 ;
set y = q `2_4 ;
set a = q `3_4 ;
let f1, f2 be BiFunction of (new_set2 A),L; ::_thesis: ( ( for u, v being Element of A holds f1 . (u,v) = d . (u,v) ) & f1 . ({A},{A}) = Bottom L & f1 . ({{A}},{{A}}) = Bottom L & f1 . ({A},{{A}}) = ((d . ((q `1_4),(q `2_4))) "\/" (q `3_4)) "/\" (q `4_4) & f1 . ({{A}},{A}) = ((d . ((q `1_4),(q `2_4))) "\/" (q `3_4)) "/\" (q `4_4) & ( for u being Element of A holds
( f1 . (u,{A}) = (d . (u,(q `1_4))) "\/" (q `3_4) & f1 . ({A},u) = (d . (u,(q `1_4))) "\/" (q `3_4) & f1 . (u,{{A}}) = (d . (u,(q `2_4))) "\/" (q `3_4) & f1 . ({{A}},u) = (d . (u,(q `2_4))) "\/" (q `3_4) ) ) & ( for u, v being Element of A holds f2 . (u,v) = d . (u,v) ) & f2 . ({A},{A}) = Bottom L & f2 . ({{A}},{{A}}) = Bottom L & f2 . ({A},{{A}}) = ((d . ((q `1_4),(q `2_4))) "\/" (q `3_4)) "/\" (q `4_4) & f2 . ({{A}},{A}) = ((d . ((q `1_4),(q `2_4))) "\/" (q `3_4)) "/\" (q `4_4) & ( for u being Element of A holds
( f2 . (u,{A}) = (d . (u,(q `1_4))) "\/" (q `3_4) & f2 . ({A},u) = (d . (u,(q `1_4))) "\/" (q `3_4) & f2 . (u,{{A}}) = (d . (u,(q `2_4))) "\/" (q `3_4) & f2 . ({{A}},u) = (d . (u,(q `2_4))) "\/" (q `3_4) ) ) implies f1 = f2 )
assume that
A20: for u, v being Element of A holds f1 . (u,v) = d . (u,v) and
A21: f1 . ({A},{A}) = Bottom L and
A22: f1 . ({{A}},{{A}}) = Bottom L and
A23: f1 . ({A},{{A}}) = ((d . ((q `1_4),(q `2_4))) "\/" (q `3_4)) "/\" (q `4_4) and
A24: f1 . ({{A}},{A}) = ((d . ((q `1_4),(q `2_4))) "\/" (q `3_4)) "/\" (q `4_4) and
A25: for u being Element of A holds
( f1 . (u,{A}) = (d . (u,(q `1_4))) "\/" (q `3_4) & f1 . ({A},u) = (d . (u,(q `1_4))) "\/" (q `3_4) & f1 . (u,{{A}}) = (d . (u,(q `2_4))) "\/" (q `3_4) & f1 . ({{A}},u) = (d . (u,(q `2_4))) "\/" (q `3_4) ) and
A26: for u, v being Element of A holds f2 . (u,v) = d . (u,v) and
A27: f2 . ({A},{A}) = Bottom L and
A28: f2 . ({{A}},{{A}}) = Bottom L and
A29: f2 . ({A},{{A}}) = ((d . ((q `1_4),(q `2_4))) "\/" (q `3_4)) "/\" (q `4_4) and
A30: f2 . ({{A}},{A}) = ((d . ((q `1_4),(q `2_4))) "\/" (q `3_4)) "/\" (q `4_4) and
A31: for u being Element of A holds
( f2 . (u,{A}) = (d . (u,(q `1_4))) "\/" (q `3_4) & f2 . ({A},u) = (d . (u,(q `1_4))) "\/" (q `3_4) & f2 . (u,{{A}}) = (d . (u,(q `2_4))) "\/" (q `3_4) & f2 . ({{A}},u) = (d . (u,(q `2_4))) "\/" (q `3_4) ) ; ::_thesis: f1 = f2
now__::_thesis:_for_p,_q_being_Element_of_new_set2_A_holds_f1_._(p,q)_=_f2_._(p,q)
let p, q be Element of new_set2 A; ::_thesis: f1 . (b1,b2) = f2 . (b1,b2)
A32: ( p in A or p in {{A},{{A}}} ) by XBOOLE_0:def_3;
A33: ( q in A or q in {{A},{{A}}} ) by XBOOLE_0:def_3;
percases ( ( p in A & q in A ) or ( p in A & q = {A} ) or ( p in A & q = {{A}} ) or ( p = {A} & q in A ) or ( p = {A} & q = {A} ) or ( p = {A} & q = {{A}} ) or ( p = {{A}} & q in A ) or ( p = {{A}} & q = {A} ) or ( p = {{A}} & q = {{A}} ) ) by A32, A33, TARSKI:def_2;
supposeA34: ( p in A & q in A ) ; ::_thesis: f1 . (b1,b2) = f2 . (b1,b2)
hence f1 . (p,q) = d . (p,q) by A20
.= f2 . (p,q) by A26, A34 ;
::_thesis: verum
end;
supposeA35: ( p in A & q = {A} ) ; ::_thesis: f1 . (b1,b2) = f2 . (b1,b2)
then reconsider p9 = p as Element of A ;
thus f1 . (p,q) = (d . (p9,(q `1_4))) "\/" (q `3_4) by A25, A35
.= f2 . (p,q) by A31, A35 ; ::_thesis: verum
end;
supposeA36: ( p in A & q = {{A}} ) ; ::_thesis: f1 . (b1,b2) = f2 . (b1,b2)
then reconsider p9 = p as Element of A ;
thus f1 . (p,q) = (d . (p9,(q `2_4))) "\/" (q `3_4) by A25, A36
.= f2 . (p,q) by A31, A36 ; ::_thesis: verum
end;
supposeA37: ( p = {A} & q in A ) ; ::_thesis: f1 . (b1,b2) = f2 . (b1,b2)
then reconsider q9 = q as Element of A ;
thus f1 . (p,q) = (d . (q9,(q `1_4))) "\/" (q `3_4) by A25, A37
.= f2 . (p,q) by A31, A37 ; ::_thesis: verum
end;
suppose ( p = {A} & q = {A} ) ; ::_thesis: f1 . (b1,b2) = f2 . (b1,b2)
hence f1 . (p,q) = f2 . (p,q) by A21, A27; ::_thesis: verum
end;
suppose ( p = {A} & q = {{A}} ) ; ::_thesis: f1 . (b1,b2) = f2 . (b1,b2)
hence f1 . (p,q) = f2 . (p,q) by A23, A29; ::_thesis: verum
end;
supposeA38: ( p = {{A}} & q in A ) ; ::_thesis: f1 . (b1,b2) = f2 . (b1,b2)
then reconsider q9 = q as Element of A ;
thus f1 . (p,q) = (d . (q9,(q `2_4))) "\/" (q `3_4) by A25, A38
.= f2 . (p,q) by A31, A38 ; ::_thesis: verum
end;
suppose ( p = {{A}} & q = {A} ) ; ::_thesis: f1 . (b1,b2) = f2 . (b1,b2)
hence f1 . (p,q) = f2 . (p,q) by A24, A30; ::_thesis: verum
end;
suppose ( p = {{A}} & q = {{A}} ) ; ::_thesis: f1 . (b1,b2) = f2 . (b1,b2)
hence f1 . (p,q) = f2 . (p,q) by A22, A28; ::_thesis: verum
end;
end;
end;
hence f1 = f2 by BINOP_1:2; ::_thesis: verum
end;
end;
:: deftheorem Def4 defines new_bi_fun2 LATTICE8:def_4_:_
for A being non empty set
for L being lower-bounded LATTICE
for d being BiFunction of A,L
for q being Element of [:A,A, the carrier of L, the carrier of L:]
for b5 being BiFunction of (new_set2 A),L holds
( b5 = new_bi_fun2 (d,q) iff ( ( for u, v being Element of A holds b5 . (u,v) = d . (u,v) ) & b5 . ({A},{A}) = Bottom L & b5 . ({{A}},{{A}}) = Bottom L & b5 . ({A},{{A}}) = ((d . ((q `1_4),(q `2_4))) "\/" (q `3_4)) "/\" (q `4_4) & b5 . ({{A}},{A}) = ((d . ((q `1_4),(q `2_4))) "\/" (q `3_4)) "/\" (q `4_4) & ( for u being Element of A holds
( b5 . (u,{A}) = (d . (u,(q `1_4))) "\/" (q `3_4) & b5 . ({A},u) = (d . (u,(q `1_4))) "\/" (q `3_4) & b5 . (u,{{A}}) = (d . (u,(q `2_4))) "\/" (q `3_4) & b5 . ({{A}},u) = (d . (u,(q `2_4))) "\/" (q `3_4) ) ) ) );
theorem Th10: :: LATTICE8:10
for A being non empty set
for L being lower-bounded LATTICE
for d being BiFunction of A,L st d is zeroed holds
for q being Element of [:A,A, the carrier of L, the carrier of L:] holds new_bi_fun2 (d,q) is zeroed
proof
let A be non empty set ; ::_thesis: for L being lower-bounded LATTICE
for d being BiFunction of A,L st d is zeroed holds
for q being Element of [:A,A, the carrier of L, the carrier of L:] holds new_bi_fun2 (d,q) is zeroed
let L be lower-bounded LATTICE; ::_thesis: for d being BiFunction of A,L st d is zeroed holds
for q being Element of [:A,A, the carrier of L, the carrier of L:] holds new_bi_fun2 (d,q) is zeroed
let d be BiFunction of A,L; ::_thesis: ( d is zeroed implies for q being Element of [:A,A, the carrier of L, the carrier of L:] holds new_bi_fun2 (d,q) is zeroed )
assume A1: d is zeroed ; ::_thesis: for q being Element of [:A,A, the carrier of L, the carrier of L:] holds new_bi_fun2 (d,q) is zeroed
let q be Element of [:A,A, the carrier of L, the carrier of L:]; ::_thesis: new_bi_fun2 (d,q) is zeroed
set f = new_bi_fun2 (d,q);
for u being Element of new_set2 A holds (new_bi_fun2 (d,q)) . (u,u) = Bottom L
proof
let u be Element of new_set2 A; ::_thesis: (new_bi_fun2 (d,q)) . (u,u) = Bottom L
A2: ( u in A or u in {{A},{{A}}} ) by XBOOLE_0:def_3;
percases ( u in A or u = {A} or u = {{A}} ) by A2, TARSKI:def_2;
suppose u in A ; ::_thesis: (new_bi_fun2 (d,q)) . (u,u) = Bottom L
then reconsider u9 = u as Element of A ;
thus (new_bi_fun2 (d,q)) . (u,u) = d . (u9,u9) by Def4
.= Bottom L by A1, LATTICE5:def_6 ; ::_thesis: verum
end;
suppose ( u = {A} or u = {{A}} ) ; ::_thesis: (new_bi_fun2 (d,q)) . (u,u) = Bottom L
hence (new_bi_fun2 (d,q)) . (u,u) = Bottom L by Def4; ::_thesis: verum
end;
end;
end;
hence new_bi_fun2 (d,q) is zeroed by LATTICE5:def_6; ::_thesis: verum
end;
theorem Th11: :: LATTICE8:11
for A being non empty set
for L being lower-bounded LATTICE
for d being BiFunction of A,L st d is symmetric holds
for q being Element of [:A,A, the carrier of L, the carrier of L:] holds new_bi_fun2 (d,q) is symmetric
proof
let A be non empty set ; ::_thesis: for L being lower-bounded LATTICE
for d being BiFunction of A,L st d is symmetric holds
for q being Element of [:A,A, the carrier of L, the carrier of L:] holds new_bi_fun2 (d,q) is symmetric
let L be lower-bounded LATTICE; ::_thesis: for d being BiFunction of A,L st d is symmetric holds
for q being Element of [:A,A, the carrier of L, the carrier of L:] holds new_bi_fun2 (d,q) is symmetric
let d be BiFunction of A,L; ::_thesis: ( d is symmetric implies for q being Element of [:A,A, the carrier of L, the carrier of L:] holds new_bi_fun2 (d,q) is symmetric )
assume A1: d is symmetric ; ::_thesis: for q being Element of [:A,A, the carrier of L, the carrier of L:] holds new_bi_fun2 (d,q) is symmetric
let q be Element of [:A,A, the carrier of L, the carrier of L:]; ::_thesis: new_bi_fun2 (d,q) is symmetric
set f = new_bi_fun2 (d,q);
set x = q `1_4 ;
set y = q `2_4 ;
set a = q `3_4 ;
set b = q `4_4 ;
let p, q be Element of new_set2 A; :: according to LATTICE5:def_5 ::_thesis: (new_bi_fun2 (d,q)) . (p,q) = (new_bi_fun2 (d,q)) . (q,p)
A2: ( p in A or p in {{A},{{A}}} ) by XBOOLE_0:def_3;
A3: ( q in A or q in {{A},{{A}}} ) by XBOOLE_0:def_3;
percases ( ( p in A & q in A ) or ( p in A & q = {A} ) or ( p in A & q = {{A}} ) or ( p = {A} & q in A ) or ( p = {A} & q = {A} ) or ( p = {A} & q = {{A}} ) or ( p = {{A}} & q in A ) or ( p = {{A}} & q = {A} ) or ( p = {{A}} & q = {{A}} ) ) by A2, A3, TARSKI:def_2;
suppose ( p in A & q in A ) ; ::_thesis: (new_bi_fun2 (d,q)) . (p,q) = (new_bi_fun2 (d,q)) . (q,p)
then reconsider p9 = p, q9 = q as Element of A ;
thus (new_bi_fun2 (d,q)) . (p,q) = d . (p9,q9) by Def4
.= d . (q9,p9) by A1, LATTICE5:def_5
.= (new_bi_fun2 (d,q)) . (q,p) by Def4 ; ::_thesis: verum
end;
supposeA4: ( p in A & q = {A} ) ; ::_thesis: (new_bi_fun2 (d,q)) . (p,q) = (new_bi_fun2 (d,q)) . (q,p)
then reconsider p9 = p as Element of A ;
thus (new_bi_fun2 (d,q)) . (p,q) = (d . (p9,(q `1_4))) "\/" (q `3_4) by A4, Def4
.= (new_bi_fun2 (d,q)) . (q,p) by A4, Def4 ; ::_thesis: verum
end;
supposeA5: ( p in A & q = {{A}} ) ; ::_thesis: (new_bi_fun2 (d,q)) . (p,q) = (new_bi_fun2 (d,q)) . (q,p)
then reconsider p9 = p as Element of A ;
thus (new_bi_fun2 (d,q)) . (p,q) = (d . (p9,(q `2_4))) "\/" (q `3_4) by A5, Def4
.= (new_bi_fun2 (d,q)) . (q,p) by A5, Def4 ; ::_thesis: verum
end;
supposeA6: ( p = {A} & q in A ) ; ::_thesis: (new_bi_fun2 (d,q)) . (p,q) = (new_bi_fun2 (d,q)) . (q,p)
then reconsider q9 = q as Element of A ;
thus (new_bi_fun2 (d,q)) . (p,q) = (d . (q9,(q `1_4))) "\/" (q `3_4) by A6, Def4
.= (new_bi_fun2 (d,q)) . (q,p) by A6, Def4 ; ::_thesis: verum
end;
suppose ( p = {A} & q = {A} ) ; ::_thesis: (new_bi_fun2 (d,q)) . (p,q) = (new_bi_fun2 (d,q)) . (q,p)
hence (new_bi_fun2 (d,q)) . (p,q) = (new_bi_fun2 (d,q)) . (q,p) ; ::_thesis: verum
end;
supposeA7: ( p = {A} & q = {{A}} ) ; ::_thesis: (new_bi_fun2 (d,q)) . (p,q) = (new_bi_fun2 (d,q)) . (q,p)
hence (new_bi_fun2 (d,q)) . (p,q) = ((d . ((q `1_4),(q `2_4))) "\/" (q `3_4)) "/\" (q `4_4) by Def4
.= (new_bi_fun2 (d,q)) . (q,p) by A7, Def4 ;
::_thesis: verum
end;
supposeA8: ( p = {{A}} & q in A ) ; ::_thesis: (new_bi_fun2 (d,q)) . (p,q) = (new_bi_fun2 (d,q)) . (q,p)
then reconsider q9 = q as Element of A ;
thus (new_bi_fun2 (d,q)) . (p,q) = (d . (q9,(q `2_4))) "\/" (q `3_4) by A8, Def4
.= (new_bi_fun2 (d,q)) . (q,p) by A8, Def4 ; ::_thesis: verum
end;
supposeA9: ( p = {{A}} & q = {A} ) ; ::_thesis: (new_bi_fun2 (d,q)) . (p,q) = (new_bi_fun2 (d,q)) . (q,p)
hence (new_bi_fun2 (d,q)) . (p,q) = ((d . ((q `1_4),(q `2_4))) "\/" (q `3_4)) "/\" (q `4_4) by Def4
.= (new_bi_fun2 (d,q)) . (q,p) by A9, Def4 ;
::_thesis: verum
end;
suppose ( p = {{A}} & q = {{A}} ) ; ::_thesis: (new_bi_fun2 (d,q)) . (p,q) = (new_bi_fun2 (d,q)) . (q,p)
hence (new_bi_fun2 (d,q)) . (p,q) = (new_bi_fun2 (d,q)) . (q,p) ; ::_thesis: verum
end;
end;
end;
theorem Th12: :: LATTICE8:12
for A being non empty set
for L being lower-bounded LATTICE st L is modular holds
for d being BiFunction of A,L st d is symmetric & d is u.t.i. holds
for q being Element of [:A,A, the carrier of L, the carrier of L:] st d . ((q `1_4),(q `2_4)) <= (q `3_4) "\/" (q `4_4) holds
new_bi_fun2 (d,q) is u.t.i.
proof
let A be non empty set ; ::_thesis: for L being lower-bounded LATTICE st L is modular holds
for d being BiFunction of A,L st d is symmetric & d is u.t.i. holds
for q being Element of [:A,A, the carrier of L, the carrier of L:] st d . ((q `1_4),(q `2_4)) <= (q `3_4) "\/" (q `4_4) holds
new_bi_fun2 (d,q) is u.t.i.
let L be lower-bounded LATTICE; ::_thesis: ( L is modular implies for d being BiFunction of A,L st d is symmetric & d is u.t.i. holds
for q being Element of [:A,A, the carrier of L, the carrier of L:] st d . ((q `1_4),(q `2_4)) <= (q `3_4) "\/" (q `4_4) holds
new_bi_fun2 (d,q) is u.t.i. )
assume A1: L is modular ; ::_thesis: for d being BiFunction of A,L st d is symmetric & d is u.t.i. holds
for q being Element of [:A,A, the carrier of L, the carrier of L:] st d . ((q `1_4),(q `2_4)) <= (q `3_4) "\/" (q `4_4) holds
new_bi_fun2 (d,q) is u.t.i.
reconsider B = {{A},{{A}}} as non empty set ;
let d be BiFunction of A,L; ::_thesis: ( d is symmetric & d is u.t.i. implies for q being Element of [:A,A, the carrier of L, the carrier of L:] st d . ((q `1_4),(q `2_4)) <= (q `3_4) "\/" (q `4_4) holds
new_bi_fun2 (d,q) is u.t.i. )
assume that
A2: d is symmetric and
A3: d is u.t.i. ; ::_thesis: for q being Element of [:A,A, the carrier of L, the carrier of L:] st d . ((q `1_4),(q `2_4)) <= (q `3_4) "\/" (q `4_4) holds
new_bi_fun2 (d,q) is u.t.i.
let q be Element of [:A,A, the carrier of L, the carrier of L:]; ::_thesis: ( d . ((q `1_4),(q `2_4)) <= (q `3_4) "\/" (q `4_4) implies new_bi_fun2 (d,q) is u.t.i. )
set x = q `1_4 ;
set y = q `2_4 ;
set a = q `3_4 ;
set b = q `4_4 ;
set f = new_bi_fun2 (d,q);
reconsider a = q `3_4 , b = q `4_4 as Element of L ;
A4: for p, q, u being Element of new_set2 A st p in A & q in A & u in B holds
(new_bi_fun2 (d,q)) . (p,u) <= ((new_bi_fun2 (d,q)) . (p,q)) "\/" ((new_bi_fun2 (d,q)) . (q,u))
proof
let p, q, u be Element of new_set2 A; ::_thesis: ( p in A & q in A & u in B implies (new_bi_fun2 (d,q)) . (p,u) <= ((new_bi_fun2 (d,q)) . (p,q)) "\/" ((new_bi_fun2 (d,q)) . (q,u)) )
assume A5: ( p in A & q in A & u in B ) ; ::_thesis: (new_bi_fun2 (d,q)) . (p,u) <= ((new_bi_fun2 (d,q)) . (p,q)) "\/" ((new_bi_fun2 (d,q)) . (q,u))
percases ( ( p in A & q in A & u = {A} ) or ( p in A & q in A & u = {{A}} ) ) by A5, TARSKI:def_2;
supposeA6: ( p in A & q in A & u = {A} ) ; ::_thesis: (new_bi_fun2 (d,q)) . (p,u) <= ((new_bi_fun2 (d,q)) . (p,q)) "\/" ((new_bi_fun2 (d,q)) . (q,u))
then reconsider p9 = p, q9 = q as Element of A ;
A7: (new_bi_fun2 (d,q)) . (p,q) = d . (p9,q9) by Def4;
d . (p9,(q `1_4)) <= (d . (p9,q9)) "\/" (d . (q9,(q `1_4))) by A3, LATTICE5:def_7;
then A8: (d . (p9,(q `1_4))) "\/" a <= ((d . (p9,q9)) "\/" (d . (q9,(q `1_4)))) "\/" a by WAYBEL_1:2;
( (new_bi_fun2 (d,q)) . (p,u) = (d . (p9,(q `1_4))) "\/" a & (new_bi_fun2 (d,q)) . (q,u) = (d . (q9,(q `1_4))) "\/" a ) by A6, Def4;
hence (new_bi_fun2 (d,q)) . (p,u) <= ((new_bi_fun2 (d,q)) . (p,q)) "\/" ((new_bi_fun2 (d,q)) . (q,u)) by A7, A8, LATTICE3:14; ::_thesis: verum
end;
supposeA9: ( p in A & q in A & u = {{A}} ) ; ::_thesis: (new_bi_fun2 (d,q)) . (p,u) <= ((new_bi_fun2 (d,q)) . (p,q)) "\/" ((new_bi_fun2 (d,q)) . (q,u))
then reconsider p9 = p, q9 = q as Element of A ;
A10: (new_bi_fun2 (d,q)) . (p,q) = d . (p9,q9) by Def4;
d . (p9,(q `2_4)) <= (d . (p9,q9)) "\/" (d . (q9,(q `2_4))) by A3, LATTICE5:def_7;
then A11: (d . (p9,(q `2_4))) "\/" a <= ((d . (p9,q9)) "\/" (d . (q9,(q `2_4)))) "\/" a by WAYBEL_1:2;
( (new_bi_fun2 (d,q)) . (p,u) = (d . (p9,(q `2_4))) "\/" a & (new_bi_fun2 (d,q)) . (q,u) = (d . (q9,(q `2_4))) "\/" a ) by A9, Def4;
hence (new_bi_fun2 (d,q)) . (p,u) <= ((new_bi_fun2 (d,q)) . (p,q)) "\/" ((new_bi_fun2 (d,q)) . (q,u)) by A10, A11, LATTICE3:14; ::_thesis: verum
end;
end;
end;
assume A12: d . ((q `1_4),(q `2_4)) <= (q `3_4) "\/" (q `4_4) ; ::_thesis: new_bi_fun2 (d,q) is u.t.i.
A13: for p, q, u being Element of new_set2 A st p in A & q in B & u in B holds
(new_bi_fun2 (d,q)) . (p,u) <= ((new_bi_fun2 (d,q)) . (p,q)) "\/" ((new_bi_fun2 (d,q)) . (q,u))
proof
let p, q, u be Element of new_set2 A; ::_thesis: ( p in A & q in B & u in B implies (new_bi_fun2 (d,q)) . (p,u) <= ((new_bi_fun2 (d,q)) . (p,q)) "\/" ((new_bi_fun2 (d,q)) . (q,u)) )
assume A14: ( p in A & q in B & u in B ) ; ::_thesis: (new_bi_fun2 (d,q)) . (p,u) <= ((new_bi_fun2 (d,q)) . (p,q)) "\/" ((new_bi_fun2 (d,q)) . (q,u))
percases ( ( p in A & q = {A} & u = {A} ) or ( p in A & q = {A} & u = {{A}} ) or ( p in A & q = {{A}} & u = {A} ) or ( p in A & q = {{A}} & u = {{A}} ) ) by A14, TARSKI:def_2;
supposeA15: ( p in A & q = {A} & u = {A} ) ; ::_thesis: (new_bi_fun2 (d,q)) . (p,u) <= ((new_bi_fun2 (d,q)) . (p,q)) "\/" ((new_bi_fun2 (d,q)) . (q,u))
then ((new_bi_fun2 (d,q)) . (p,q)) "\/" ((new_bi_fun2 (d,q)) . (q,u)) = (Bottom L) "\/" ((new_bi_fun2 (d,q)) . (p,q)) by Def4
.= (new_bi_fun2 (d,q)) . (p,q) by WAYBEL_1:3 ;
hence (new_bi_fun2 (d,q)) . (p,u) <= ((new_bi_fun2 (d,q)) . (p,q)) "\/" ((new_bi_fun2 (d,q)) . (q,u)) by A15; ::_thesis: verum
end;
supposeA16: ( p in A & q = {A} & u = {{A}} ) ; ::_thesis: (new_bi_fun2 (d,q)) . (p,u) <= ((new_bi_fun2 (d,q)) . (p,q)) "\/" ((new_bi_fun2 (d,q)) . (q,u))
then reconsider p9 = p as Element of A ;
a <= a "\/" (d . ((q `1_4),(q `2_4))) by YELLOW_0:22;
then A17: a "\/" (((d . ((q `1_4),(q `2_4))) "\/" a) "/\" b) = (a "\/" b) "/\" (a "\/" (d . ((q `1_4),(q `2_4)))) by A1, YELLOW11:def_3;
d . (p9,(q `2_4)) <= (d . (p9,(q `1_4))) "\/" (d . ((q `1_4),(q `2_4))) by A3, LATTICE5:def_7;
then A18: (d . (p9,(q `2_4))) "\/" a <= ((d . (p9,(q `1_4))) "\/" (d . ((q `1_4),(q `2_4)))) "\/" a by YELLOW_5:7;
a <= a ;
then (d . ((q `1_4),(q `2_4))) "\/" a <= (a "\/" b) "\/" a by A12, YELLOW_3:3;
then ((d . ((q `1_4),(q `2_4))) "\/" a) "/\" ((d . ((q `1_4),(q `2_4))) "\/" a) <= ((d . ((q `1_4),(q `2_4))) "\/" a) "/\" ((a "\/" b) "\/" a) by YELLOW_5:6;
then A19: ( (d . ((q `1_4),(q `2_4))) "\/" a = ((d . ((q `1_4),(q `2_4))) "\/" a) "/\" ((d . ((q `1_4),(q `2_4))) "\/" a) & (d . (p9,(q `1_4))) "\/" (((d . ((q `1_4),(q `2_4))) "\/" a) "/\" ((d . ((q `1_4),(q `2_4))) "\/" a)) <= (d . (p9,(q `1_4))) "\/" (((d . ((q `1_4),(q `2_4))) "\/" a) "/\" ((a "\/" b) "\/" a)) ) by WAYBEL_1:2, YELLOW_5:2;
( (new_bi_fun2 (d,q)) . (p,q) = (d . (p9,(q `1_4))) "\/" a & (new_bi_fun2 (d,q)) . (q,u) = ((d . ((q `1_4),(q `2_4))) "\/" a) "/\" b ) by A16, Def4;
then ((new_bi_fun2 (d,q)) . (p,q)) "\/" ((new_bi_fun2 (d,q)) . (q,u)) = (d . (p9,(q `1_4))) "\/" ((a "\/" b) "/\" (a "\/" (d . ((q `1_4),(q `2_4))))) by A17, LATTICE3:14
.= (d . (p9,(q `1_4))) "\/" (((d . ((q `1_4),(q `2_4))) "\/" a) "/\" ((a "\/" a) "\/" b)) by YELLOW_5:1
.= (d . (p9,(q `1_4))) "\/" (((d . ((q `1_4),(q `2_4))) "\/" a) "/\" ((a "\/" b) "\/" a)) by LATTICE3:14 ;
then ((d . (p9,(q `1_4))) "\/" (d . ((q `1_4),(q `2_4)))) "\/" a <= ((new_bi_fun2 (d,q)) . (p,q)) "\/" ((new_bi_fun2 (d,q)) . (q,u)) by A19, LATTICE3:14;
then (d . (p9,(q `2_4))) "\/" a <= ((new_bi_fun2 (d,q)) . (p,q)) "\/" ((new_bi_fun2 (d,q)) . (q,u)) by A18, ORDERS_2:3;
hence (new_bi_fun2 (d,q)) . (p,u) <= ((new_bi_fun2 (d,q)) . (p,q)) "\/" ((new_bi_fun2 (d,q)) . (q,u)) by A16, Def4; ::_thesis: verum
end;
supposeA20: ( p in A & q = {{A}} & u = {A} ) ; ::_thesis: (new_bi_fun2 (d,q)) . (p,u) <= ((new_bi_fun2 (d,q)) . (p,q)) "\/" ((new_bi_fun2 (d,q)) . (q,u))
then reconsider p9 = p as Element of A ;
a <= a "\/" (d . ((q `1_4),(q `2_4))) by YELLOW_0:22;
then A21: a "\/" (((d . ((q `1_4),(q `2_4))) "\/" a) "/\" b) = (a "\/" b) "/\" (a "\/" (d . ((q `1_4),(q `2_4)))) by A1, YELLOW11:def_3;
a <= a ;
then (d . ((q `1_4),(q `2_4))) "\/" a <= (a "\/" b) "\/" a by A12, YELLOW_3:3;
then ((d . ((q `1_4),(q `2_4))) "\/" a) "/\" ((d . ((q `1_4),(q `2_4))) "\/" a) <= ((d . ((q `1_4),(q `2_4))) "\/" a) "/\" ((a "\/" b) "\/" a) by YELLOW_5:6;
then A22: ( (d . ((q `1_4),(q `2_4))) "\/" a = ((d . ((q `1_4),(q `2_4))) "\/" a) "/\" ((d . ((q `1_4),(q `2_4))) "\/" a) & (d . (p9,(q `2_4))) "\/" (((d . ((q `1_4),(q `2_4))) "\/" a) "/\" ((d . ((q `1_4),(q `2_4))) "\/" a)) <= (d . (p9,(q `2_4))) "\/" (((d . ((q `1_4),(q `2_4))) "\/" a) "/\" ((a "\/" b) "\/" a)) ) by WAYBEL_1:2, YELLOW_5:2;
d . ((q `2_4),(q `1_4)) = d . ((q `1_4),(q `2_4)) by A2, LATTICE5:def_5;
then d . (p9,(q `1_4)) <= (d . (p9,(q `2_4))) "\/" (d . ((q `1_4),(q `2_4))) by A3, LATTICE5:def_7;
then A23: (d . (p9,(q `1_4))) "\/" a <= ((d . (p9,(q `2_4))) "\/" (d . ((q `1_4),(q `2_4)))) "\/" a by YELLOW_5:7;
( (new_bi_fun2 (d,q)) . (p,q) = (d . (p9,(q `2_4))) "\/" a & (new_bi_fun2 (d,q)) . (q,u) = ((d . ((q `1_4),(q `2_4))) "\/" a) "/\" b ) by A20, Def4;
then ((new_bi_fun2 (d,q)) . (p,q)) "\/" ((new_bi_fun2 (d,q)) . (q,u)) = (d . (p9,(q `2_4))) "\/" ((a "\/" b) "/\" (a "\/" (d . ((q `1_4),(q `2_4))))) by A21, LATTICE3:14
.= (d . (p9,(q `2_4))) "\/" (((d . ((q `1_4),(q `2_4))) "\/" a) "/\" ((a "\/" a) "\/" b)) by YELLOW_5:1
.= (d . (p9,(q `2_4))) "\/" (((d . ((q `1_4),(q `2_4))) "\/" a) "/\" ((a "\/" b) "\/" a)) by LATTICE3:14 ;
then ((d . (p9,(q `2_4))) "\/" (d . ((q `1_4),(q `2_4)))) "\/" a <= ((new_bi_fun2 (d,q)) . (p,q)) "\/" ((new_bi_fun2 (d,q)) . (q,u)) by A22, LATTICE3:14;
then (d . (p9,(q `1_4))) "\/" a <= ((new_bi_fun2 (d,q)) . (p,q)) "\/" ((new_bi_fun2 (d,q)) . (q,u)) by A23, ORDERS_2:3;
hence (new_bi_fun2 (d,q)) . (p,u) <= ((new_bi_fun2 (d,q)) . (p,q)) "\/" ((new_bi_fun2 (d,q)) . (q,u)) by A20, Def4; ::_thesis: verum
end;
supposeA24: ( p in A & q = {{A}} & u = {{A}} ) ; ::_thesis: (new_bi_fun2 (d,q)) . (p,u) <= ((new_bi_fun2 (d,q)) . (p,q)) "\/" ((new_bi_fun2 (d,q)) . (q,u))
then ((new_bi_fun2 (d,q)) . (p,q)) "\/" ((new_bi_fun2 (d,q)) . (q,u)) = (Bottom L) "\/" ((new_bi_fun2 (d,q)) . (p,q)) by Def4
.= (new_bi_fun2 (d,q)) . (p,q) by WAYBEL_1:3 ;
hence (new_bi_fun2 (d,q)) . (p,u) <= ((new_bi_fun2 (d,q)) . (p,q)) "\/" ((new_bi_fun2 (d,q)) . (q,u)) by A24; ::_thesis: verum
end;
end;
end;
A25: for p, q, u being Element of new_set2 A st p in B & q in A & u in B holds
(new_bi_fun2 (d,q)) . (p,u) <= ((new_bi_fun2 (d,q)) . (p,q)) "\/" ((new_bi_fun2 (d,q)) . (q,u))
proof
let p, q, u be Element of new_set2 A; ::_thesis: ( p in B & q in A & u in B implies (new_bi_fun2 (d,q)) . (p,u) <= ((new_bi_fun2 (d,q)) . (p,q)) "\/" ((new_bi_fun2 (d,q)) . (q,u)) )
assume A26: ( p in B & q in A & u in B ) ; ::_thesis: (new_bi_fun2 (d,q)) . (p,u) <= ((new_bi_fun2 (d,q)) . (p,q)) "\/" ((new_bi_fun2 (d,q)) . (q,u))
percases ( ( q in A & p = {A} & u = {A} ) or ( q in A & p = {A} & u = {{A}} ) or ( q in A & p = {{A}} & u = {A} ) or ( q in A & p = {{A}} & u = {{A}} ) ) by A26, TARSKI:def_2;
suppose ( q in A & p = {A} & u = {A} ) ; ::_thesis: (new_bi_fun2 (d,q)) . (p,u) <= ((new_bi_fun2 (d,q)) . (p,q)) "\/" ((new_bi_fun2 (d,q)) . (q,u))
then (new_bi_fun2 (d,q)) . (p,u) = Bottom L by Def4;
hence (new_bi_fun2 (d,q)) . (p,u) <= ((new_bi_fun2 (d,q)) . (p,q)) "\/" ((new_bi_fun2 (d,q)) . (q,u)) by YELLOW_0:44; ::_thesis: verum
end;
supposeA27: ( q in A & p = {A} & u = {{A}} ) ; ::_thesis: (new_bi_fun2 (d,q)) . (p,u) <= ((new_bi_fun2 (d,q)) . (p,q)) "\/" ((new_bi_fun2 (d,q)) . (q,u))
then reconsider q9 = q as Element of A ;
d . (q9,(q `1_4)) = d . ((q `1_4),q9) by A2, LATTICE5:def_5;
then A28: d . ((q `1_4),(q `2_4)) <= (d . (q9,(q `1_4))) "\/" (d . (q9,(q `2_4))) by A3, LATTICE5:def_7;
(new_bi_fun2 (d,q)) . (p,q) = (d . (q9,(q `1_4))) "\/" a by A27, Def4;
then ((new_bi_fun2 (d,q)) . (p,q)) "\/" ((new_bi_fun2 (d,q)) . (q,u)) = ((d . (q9,(q `1_4))) "\/" a) "\/" ((d . (q9,(q `2_4))) "\/" a) by A27, Def4
.= ((d . (q9,(q `1_4))) "\/" ((d . (q9,(q `2_4))) "\/" a)) "\/" a by LATTICE3:14
.= (((d . (q9,(q `1_4))) "\/" (d . (q9,(q `2_4)))) "\/" a) "\/" a by LATTICE3:14
.= ((d . (q9,(q `1_4))) "\/" (d . (q9,(q `2_4)))) "\/" (a "\/" a) by LATTICE3:14
.= ((d . (q9,(q `1_4))) "\/" (d . (q9,(q `2_4)))) "\/" a by YELLOW_5:1 ;
then A29: (d . ((q `1_4),(q `2_4))) "\/" a <= ((new_bi_fun2 (d,q)) . (p,q)) "\/" ((new_bi_fun2 (d,q)) . (q,u)) by A28, YELLOW_5:7;
A30: ((d . ((q `1_4),(q `2_4))) "\/" a) "/\" b <= (d . ((q `1_4),(q `2_4))) "\/" a by YELLOW_0:23;
(new_bi_fun2 (d,q)) . (p,u) = ((d . ((q `1_4),(q `2_4))) "\/" a) "/\" b by A27, Def4;
hence (new_bi_fun2 (d,q)) . (p,u) <= ((new_bi_fun2 (d,q)) . (p,q)) "\/" ((new_bi_fun2 (d,q)) . (q,u)) by A29, A30, ORDERS_2:3; ::_thesis: verum
end;
supposeA31: ( q in A & p = {{A}} & u = {A} ) ; ::_thesis: (new_bi_fun2 (d,q)) . (p,u) <= ((new_bi_fun2 (d,q)) . (p,q)) "\/" ((new_bi_fun2 (d,q)) . (q,u))
then reconsider q9 = q as Element of A ;
d . (q9,(q `1_4)) = d . ((q `1_4),q9) by A2, LATTICE5:def_5;
then A32: d . ((q `1_4),(q `2_4)) <= (d . (q9,(q `1_4))) "\/" (d . (q9,(q `2_4))) by A3, LATTICE5:def_7;
(new_bi_fun2 (d,q)) . (p,q) = (d . (q9,(q `2_4))) "\/" a by A31, Def4;
then ((new_bi_fun2 (d,q)) . (p,q)) "\/" ((new_bi_fun2 (d,q)) . (q,u)) = ((d . (q9,(q `1_4))) "\/" a) "\/" ((d . (q9,(q `2_4))) "\/" a) by A31, Def4
.= ((d . (q9,(q `1_4))) "\/" ((d . (q9,(q `2_4))) "\/" a)) "\/" a by LATTICE3:14
.= (((d . (q9,(q `1_4))) "\/" (d . (q9,(q `2_4)))) "\/" a) "\/" a by LATTICE3:14
.= ((d . (q9,(q `1_4))) "\/" (d . (q9,(q `2_4)))) "\/" (a "\/" a) by LATTICE3:14
.= ((d . (q9,(q `1_4))) "\/" (d . (q9,(q `2_4)))) "\/" a by YELLOW_5:1 ;
then A33: (d . ((q `1_4),(q `2_4))) "\/" a <= ((new_bi_fun2 (d,q)) . (p,q)) "\/" ((new_bi_fun2 (d,q)) . (q,u)) by A32, YELLOW_5:7;
A34: ((d . ((q `1_4),(q `2_4))) "\/" a) "/\" b <= (d . ((q `1_4),(q `2_4))) "\/" a by YELLOW_0:23;
(new_bi_fun2 (d,q)) . (p,u) = ((d . ((q `1_4),(q `2_4))) "\/" a) "/\" b by A31, Def4;
hence (new_bi_fun2 (d,q)) . (p,u) <= ((new_bi_fun2 (d,q)) . (p,q)) "\/" ((new_bi_fun2 (d,q)) . (q,u)) by A33, A34, ORDERS_2:3; ::_thesis: verum
end;
suppose ( q in A & p = {{A}} & u = {{A}} ) ; ::_thesis: (new_bi_fun2 (d,q)) . (p,u) <= ((new_bi_fun2 (d,q)) . (p,q)) "\/" ((new_bi_fun2 (d,q)) . (q,u))
then (new_bi_fun2 (d,q)) . (p,u) = Bottom L by Def4;
hence (new_bi_fun2 (d,q)) . (p,u) <= ((new_bi_fun2 (d,q)) . (p,q)) "\/" ((new_bi_fun2 (d,q)) . (q,u)) by YELLOW_0:44; ::_thesis: verum
end;
end;
end;
A35: for p, q, u being Element of new_set2 A st p in B & q in B & u in B holds
(new_bi_fun2 (d,q)) . (p,u) <= ((new_bi_fun2 (d,q)) . (p,q)) "\/" ((new_bi_fun2 (d,q)) . (q,u))
proof
let p, q, u be Element of new_set2 A; ::_thesis: ( p in B & q in B & u in B implies (new_bi_fun2 (d,q)) . (p,u) <= ((new_bi_fun2 (d,q)) . (p,q)) "\/" ((new_bi_fun2 (d,q)) . (q,u)) )
assume A36: ( p in B & q in B & u in B ) ; ::_thesis: (new_bi_fun2 (d,q)) . (p,u) <= ((new_bi_fun2 (d,q)) . (p,q)) "\/" ((new_bi_fun2 (d,q)) . (q,u))
percases ( ( p = {A} & q = {A} & u = {A} ) or ( p = {A} & q = {A} & u = {{A}} ) or ( p = {A} & q = {{A}} & u = {A} ) or ( p = {A} & q = {{A}} & u = {{A}} ) or ( p = {{A}} & q = {A} & u = {A} ) or ( p = {{A}} & q = {A} & u = {{A}} ) or ( p = {{A}} & q = {{A}} & u = {A} ) or ( p = {{A}} & q = {{A}} & u = {{A}} ) ) by A36, TARSKI:def_2;
supposeA37: ( p = {A} & q = {A} & u = {A} ) ; ::_thesis: (new_bi_fun2 (d,q)) . (p,u) <= ((new_bi_fun2 (d,q)) . (p,q)) "\/" ((new_bi_fun2 (d,q)) . (q,u))
Bottom L <= ((new_bi_fun2 (d,q)) . (p,q)) "\/" ((new_bi_fun2 (d,q)) . (q,u)) by YELLOW_0:44;
hence (new_bi_fun2 (d,q)) . (p,u) <= ((new_bi_fun2 (d,q)) . (p,q)) "\/" ((new_bi_fun2 (d,q)) . (q,u)) by A37, Def4; ::_thesis: verum
end;
supposeA38: ( p = {A} & q = {A} & u = {{A}} ) ; ::_thesis: (new_bi_fun2 (d,q)) . (p,u) <= ((new_bi_fun2 (d,q)) . (p,q)) "\/" ((new_bi_fun2 (d,q)) . (q,u))
then ((new_bi_fun2 (d,q)) . (p,q)) "\/" ((new_bi_fun2 (d,q)) . (q,u)) = (Bottom L) "\/" ((new_bi_fun2 (d,q)) . (p,u)) by Def4
.= (Bottom L) "\/" (((d . ((q `1_4),(q `2_4))) "\/" a) "/\" b) by A38, Def4
.= ((d . ((q `1_4),(q `2_4))) "\/" a) "/\" b by WAYBEL_1:3 ;
hence (new_bi_fun2 (d,q)) . (p,u) <= ((new_bi_fun2 (d,q)) . (p,q)) "\/" ((new_bi_fun2 (d,q)) . (q,u)) by A38, Def4; ::_thesis: verum
end;
supposeA39: ( p = {A} & q = {{A}} & u = {A} ) ; ::_thesis: (new_bi_fun2 (d,q)) . (p,u) <= ((new_bi_fun2 (d,q)) . (p,q)) "\/" ((new_bi_fun2 (d,q)) . (q,u))
Bottom L <= ((new_bi_fun2 (d,q)) . (p,q)) "\/" ((new_bi_fun2 (d,q)) . (q,u)) by YELLOW_0:44;
hence (new_bi_fun2 (d,q)) . (p,u) <= ((new_bi_fun2 (d,q)) . (p,q)) "\/" ((new_bi_fun2 (d,q)) . (q,u)) by A39, Def4; ::_thesis: verum
end;
supposeA40: ( p = {A} & q = {{A}} & u = {{A}} ) ; ::_thesis: (new_bi_fun2 (d,q)) . (p,u) <= ((new_bi_fun2 (d,q)) . (p,q)) "\/" ((new_bi_fun2 (d,q)) . (q,u))
then ((new_bi_fun2 (d,q)) . (p,q)) "\/" ((new_bi_fun2 (d,q)) . (q,u)) = (((d . ((q `1_4),(q `2_4))) "\/" a) "/\" b) "\/" ((new_bi_fun2 (d,q)) . (q,u)) by Def4
.= (Bottom L) "\/" (((d . ((q `1_4),(q `2_4))) "\/" a) "/\" b) by A40, Def4
.= ((d . ((q `1_4),(q `2_4))) "\/" a) "/\" b by WAYBEL_1:3 ;
hence (new_bi_fun2 (d,q)) . (p,u) <= ((new_bi_fun2 (d,q)) . (p,q)) "\/" ((new_bi_fun2 (d,q)) . (q,u)) by A40, Def4; ::_thesis: verum
end;
supposeA41: ( p = {{A}} & q = {A} & u = {A} ) ; ::_thesis: (new_bi_fun2 (d,q)) . (p,u) <= ((new_bi_fun2 (d,q)) . (p,q)) "\/" ((new_bi_fun2 (d,q)) . (q,u))
then ((new_bi_fun2 (d,q)) . (p,q)) "\/" ((new_bi_fun2 (d,q)) . (q,u)) = (((d . ((q `1_4),(q `2_4))) "\/" a) "/\" b) "\/" ((new_bi_fun2 (d,q)) . (q,u)) by Def4
.= (Bottom L) "\/" (((d . ((q `1_4),(q `2_4))) "\/" a) "/\" b) by A41, Def4
.= ((d . ((q `1_4),(q `2_4))) "\/" a) "/\" b by WAYBEL_1:3
.= (new_bi_fun2 (d,q)) . (p,q) by A41, Def4 ;
hence (new_bi_fun2 (d,q)) . (p,u) <= ((new_bi_fun2 (d,q)) . (p,q)) "\/" ((new_bi_fun2 (d,q)) . (q,u)) by A41; ::_thesis: verum
end;
supposeA42: ( p = {{A}} & q = {A} & u = {{A}} ) ; ::_thesis: (new_bi_fun2 (d,q)) . (p,u) <= ((new_bi_fun2 (d,q)) . (p,q)) "\/" ((new_bi_fun2 (d,q)) . (q,u))
Bottom L <= ((new_bi_fun2 (d,q)) . (p,q)) "\/" ((new_bi_fun2 (d,q)) . (q,u)) by YELLOW_0:44;
hence (new_bi_fun2 (d,q)) . (p,u) <= ((new_bi_fun2 (d,q)) . (p,q)) "\/" ((new_bi_fun2 (d,q)) . (q,u)) by A42, Def4; ::_thesis: verum
end;
supposeA43: ( p = {{A}} & q = {{A}} & u = {A} ) ; ::_thesis: (new_bi_fun2 (d,q)) . (p,u) <= ((new_bi_fun2 (d,q)) . (p,q)) "\/" ((new_bi_fun2 (d,q)) . (q,u))
then ((new_bi_fun2 (d,q)) . (p,q)) "\/" ((new_bi_fun2 (d,q)) . (q,u)) = (Bottom L) "\/" ((new_bi_fun2 (d,q)) . (p,u)) by Def4
.= (Bottom L) "\/" (((d . ((q `1_4),(q `2_4))) "\/" a) "/\" b) by A43, Def4
.= ((d . ((q `1_4),(q `2_4))) "\/" a) "/\" b by WAYBEL_1:3 ;
hence (new_bi_fun2 (d,q)) . (p,u) <= ((new_bi_fun2 (d,q)) . (p,q)) "\/" ((new_bi_fun2 (d,q)) . (q,u)) by A43, Def4; ::_thesis: verum
end;
supposeA44: ( p = {{A}} & q = {{A}} & u = {{A}} ) ; ::_thesis: (new_bi_fun2 (d,q)) . (p,u) <= ((new_bi_fun2 (d,q)) . (p,q)) "\/" ((new_bi_fun2 (d,q)) . (q,u))
Bottom L <= ((new_bi_fun2 (d,q)) . (p,q)) "\/" ((new_bi_fun2 (d,q)) . (q,u)) by YELLOW_0:44;
hence (new_bi_fun2 (d,q)) . (p,u) <= ((new_bi_fun2 (d,q)) . (p,q)) "\/" ((new_bi_fun2 (d,q)) . (q,u)) by A44, Def4; ::_thesis: verum
end;
end;
end;
A45: for p, q, u being Element of new_set2 A st p in B & q in B & u in A holds
(new_bi_fun2 (d,q)) . (p,u) <= ((new_bi_fun2 (d,q)) . (p,q)) "\/" ((new_bi_fun2 (d,q)) . (q,u))
proof
let p, q, u be Element of new_set2 A; ::_thesis: ( p in B & q in B & u in A implies (new_bi_fun2 (d,q)) . (p,u) <= ((new_bi_fun2 (d,q)) . (p,q)) "\/" ((new_bi_fun2 (d,q)) . (q,u)) )
assume A46: ( p in B & q in B & u in A ) ; ::_thesis: (new_bi_fun2 (d,q)) . (p,u) <= ((new_bi_fun2 (d,q)) . (p,q)) "\/" ((new_bi_fun2 (d,q)) . (q,u))
percases ( ( u in A & q = {A} & p = {A} ) or ( u in A & q = {A} & p = {{A}} ) or ( u in A & q = {{A}} & p = {A} ) or ( u in A & q = {{A}} & p = {{A}} ) ) by A46, TARSKI:def_2;
supposeA47: ( u in A & q = {A} & p = {A} ) ; ::_thesis: (new_bi_fun2 (d,q)) . (p,u) <= ((new_bi_fun2 (d,q)) . (p,q)) "\/" ((new_bi_fun2 (d,q)) . (q,u))
then ((new_bi_fun2 (d,q)) . (p,q)) "\/" ((new_bi_fun2 (d,q)) . (q,u)) = (Bottom L) "\/" ((new_bi_fun2 (d,q)) . (q,u)) by Def4
.= (new_bi_fun2 (d,q)) . (p,u) by A47, WAYBEL_1:3 ;
hence (new_bi_fun2 (d,q)) . (p,u) <= ((new_bi_fun2 (d,q)) . (p,q)) "\/" ((new_bi_fun2 (d,q)) . (q,u)) ; ::_thesis: verum
end;
supposeA48: ( u in A & q = {A} & p = {{A}} ) ; ::_thesis: (new_bi_fun2 (d,q)) . (p,u) <= ((new_bi_fun2 (d,q)) . (p,q)) "\/" ((new_bi_fun2 (d,q)) . (q,u))
then reconsider u9 = u as Element of A ;
a <= a "\/" (d . ((q `1_4),(q `2_4))) by YELLOW_0:22;
then A49: a "\/" (((d . ((q `1_4),(q `2_4))) "\/" a) "/\" b) = (a "\/" b) "/\" (a "\/" (d . ((q `1_4),(q `2_4)))) by A1, YELLOW11:def_3;
d . (u9,(q `2_4)) <= (d . (u9,(q `1_4))) "\/" (d . ((q `1_4),(q `2_4))) by A3, LATTICE5:def_7;
then A50: (d . (u9,(q `2_4))) "\/" a <= ((d . (u9,(q `1_4))) "\/" (d . ((q `1_4),(q `2_4)))) "\/" a by YELLOW_5:7;
a <= a ;
then (d . ((q `1_4),(q `2_4))) "\/" a <= (a "\/" b) "\/" a by A12, YELLOW_3:3;
then ((d . ((q `1_4),(q `2_4))) "\/" a) "/\" ((d . ((q `1_4),(q `2_4))) "\/" a) <= ((d . ((q `1_4),(q `2_4))) "\/" a) "/\" ((a "\/" b) "\/" a) by YELLOW_5:6;
then A51: ( (d . ((q `1_4),(q `2_4))) "\/" a = ((d . ((q `1_4),(q `2_4))) "\/" a) "/\" ((d . ((q `1_4),(q `2_4))) "\/" a) & (d . (u9,(q `1_4))) "\/" (((d . ((q `1_4),(q `2_4))) "\/" a) "/\" ((d . ((q `1_4),(q `2_4))) "\/" a)) <= (d . (u9,(q `1_4))) "\/" (((d . ((q `1_4),(q `2_4))) "\/" a) "/\" ((a "\/" b) "\/" a)) ) by WAYBEL_1:2, YELLOW_5:2;
( (new_bi_fun2 (d,q)) . (p,q) = ((d . ((q `1_4),(q `2_4))) "\/" a) "/\" b & (new_bi_fun2 (d,q)) . (q,u) = (d . (u9,(q `1_4))) "\/" a ) by A48, Def4;
then ((new_bi_fun2 (d,q)) . (p,q)) "\/" ((new_bi_fun2 (d,q)) . (q,u)) = (d . (u9,(q `1_4))) "\/" ((a "\/" b) "/\" (a "\/" (d . ((q `1_4),(q `2_4))))) by A49, LATTICE3:14
.= (d . (u9,(q `1_4))) "\/" (((d . ((q `1_4),(q `2_4))) "\/" a) "/\" ((a "\/" a) "\/" b)) by YELLOW_5:1
.= (d . (u9,(q `1_4))) "\/" (((d . ((q `1_4),(q `2_4))) "\/" a) "/\" ((a "\/" b) "\/" a)) by LATTICE3:14 ;
then ((d . (u9,(q `1_4))) "\/" (d . ((q `1_4),(q `2_4)))) "\/" a <= ((new_bi_fun2 (d,q)) . (p,q)) "\/" ((new_bi_fun2 (d,q)) . (q,u)) by A51, LATTICE3:14;
then (d . (u9,(q `2_4))) "\/" a <= ((new_bi_fun2 (d,q)) . (p,q)) "\/" ((new_bi_fun2 (d,q)) . (q,u)) by A50, ORDERS_2:3;
hence (new_bi_fun2 (d,q)) . (p,u) <= ((new_bi_fun2 (d,q)) . (p,q)) "\/" ((new_bi_fun2 (d,q)) . (q,u)) by A48, Def4; ::_thesis: verum
end;
supposeA52: ( u in A & q = {{A}} & p = {A} ) ; ::_thesis: (new_bi_fun2 (d,q)) . (p,u) <= ((new_bi_fun2 (d,q)) . (p,q)) "\/" ((new_bi_fun2 (d,q)) . (q,u))
then reconsider u9 = u as Element of A ;
a <= a "\/" (d . ((q `1_4),(q `2_4))) by YELLOW_0:22;
then A53: a "\/" (((d . ((q `1_4),(q `2_4))) "\/" a) "/\" b) = (a "\/" b) "/\" (a "\/" (d . ((q `1_4),(q `2_4)))) by A1, YELLOW11:def_3;
a <= a ;
then (d . ((q `1_4),(q `2_4))) "\/" a <= (a "\/" b) "\/" a by A12, YELLOW_3:3;
then ((d . ((q `1_4),(q `2_4))) "\/" a) "/\" ((d . ((q `1_4),(q `2_4))) "\/" a) <= ((d . ((q `1_4),(q `2_4))) "\/" a) "/\" ((a "\/" b) "\/" a) by YELLOW_5:6;
then A54: ( (d . ((q `1_4),(q `2_4))) "\/" a = ((d . ((q `1_4),(q `2_4))) "\/" a) "/\" ((d . ((q `1_4),(q `2_4))) "\/" a) & (d . (u9,(q `2_4))) "\/" (((d . ((q `1_4),(q `2_4))) "\/" a) "/\" ((d . ((q `1_4),(q `2_4))) "\/" a)) <= (d . (u9,(q `2_4))) "\/" (((d . ((q `1_4),(q `2_4))) "\/" a) "/\" ((a "\/" b) "\/" a)) ) by WAYBEL_1:2, YELLOW_5:2;
d . ((q `2_4),(q `1_4)) = d . ((q `1_4),(q `2_4)) by A2, LATTICE5:def_5;
then d . (u9,(q `1_4)) <= (d . (u9,(q `2_4))) "\/" (d . ((q `1_4),(q `2_4))) by A3, LATTICE5:def_7;
then A55: (d . (u9,(q `1_4))) "\/" a <= ((d . (u9,(q `2_4))) "\/" (d . ((q `1_4),(q `2_4)))) "\/" a by YELLOW_5:7;
( (new_bi_fun2 (d,q)) . (p,q) = ((d . ((q `1_4),(q `2_4))) "\/" a) "/\" b & (new_bi_fun2 (d,q)) . (q,u) = (d . (u9,(q `2_4))) "\/" a ) by A52, Def4;
then ((new_bi_fun2 (d,q)) . (p,q)) "\/" ((new_bi_fun2 (d,q)) . (q,u)) = (d . (u9,(q `2_4))) "\/" ((a "\/" b) "/\" (a "\/" (d . ((q `1_4),(q `2_4))))) by A53, LATTICE3:14
.= (d . (u9,(q `2_4))) "\/" (((d . ((q `1_4),(q `2_4))) "\/" a) "/\" ((a "\/" a) "\/" b)) by YELLOW_5:1
.= (d . (u9,(q `2_4))) "\/" (((d . ((q `1_4),(q `2_4))) "\/" a) "/\" ((a "\/" b) "\/" a)) by LATTICE3:14 ;
then ((d . (u9,(q `2_4))) "\/" (d . ((q `1_4),(q `2_4)))) "\/" a <= ((new_bi_fun2 (d,q)) . (p,q)) "\/" ((new_bi_fun2 (d,q)) . (q,u)) by A54, LATTICE3:14;
then (d . (u9,(q `1_4))) "\/" a <= ((new_bi_fun2 (d,q)) . (p,q)) "\/" ((new_bi_fun2 (d,q)) . (q,u)) by A55, ORDERS_2:3;
hence (new_bi_fun2 (d,q)) . (p,u) <= ((new_bi_fun2 (d,q)) . (p,q)) "\/" ((new_bi_fun2 (d,q)) . (q,u)) by A52, Def4; ::_thesis: verum
end;
supposeA56: ( u in A & q = {{A}} & p = {{A}} ) ; ::_thesis: (new_bi_fun2 (d,q)) . (p,u) <= ((new_bi_fun2 (d,q)) . (p,q)) "\/" ((new_bi_fun2 (d,q)) . (q,u))
then ((new_bi_fun2 (d,q)) . (p,q)) "\/" ((new_bi_fun2 (d,q)) . (q,u)) = (Bottom L) "\/" ((new_bi_fun2 (d,q)) . (q,u)) by Def4
.= (new_bi_fun2 (d,q)) . (p,u) by A56, WAYBEL_1:3 ;
hence (new_bi_fun2 (d,q)) . (p,u) <= ((new_bi_fun2 (d,q)) . (p,q)) "\/" ((new_bi_fun2 (d,q)) . (q,u)) ; ::_thesis: verum
end;
end;
end;
A57: for p, q, u being Element of new_set2 A st p in B & q in A & u in A holds
(new_bi_fun2 (d,q)) . (p,u) <= ((new_bi_fun2 (d,q)) . (p,q)) "\/" ((new_bi_fun2 (d,q)) . (q,u))
proof
let p, q, u be Element of new_set2 A; ::_thesis: ( p in B & q in A & u in A implies (new_bi_fun2 (d,q)) . (p,u) <= ((new_bi_fun2 (d,q)) . (p,q)) "\/" ((new_bi_fun2 (d,q)) . (q,u)) )
assume that
A58: p in B and
A59: ( q in A & u in A ) ; ::_thesis: (new_bi_fun2 (d,q)) . (p,u) <= ((new_bi_fun2 (d,q)) . (p,q)) "\/" ((new_bi_fun2 (d,q)) . (q,u))
reconsider q9 = q, u9 = u as Element of A by A59;
percases ( ( p = {A} & q in A & u in A ) or ( p = {{A}} & q in A & u in A ) ) by A58, A59, TARSKI:def_2;
supposeA60: ( p = {A} & q in A & u in A ) ; ::_thesis: (new_bi_fun2 (d,q)) . (p,u) <= ((new_bi_fun2 (d,q)) . (p,q)) "\/" ((new_bi_fun2 (d,q)) . (q,u))
d . (u9,(q `1_4)) <= (d . (u9,q9)) "\/" (d . (q9,(q `1_4))) by A3, LATTICE5:def_7;
then d . (u9,(q `1_4)) <= (d . (q9,u9)) "\/" (d . (q9,(q `1_4))) by A2, LATTICE5:def_5;
then (d . (u9,(q `1_4))) "\/" a <= ((d . (q9,(q `1_4))) "\/" (d . (q9,u9))) "\/" a by WAYBEL_1:2;
then A61: (d . (u9,(q `1_4))) "\/" a <= ((d . (q9,(q `1_4))) "\/" a) "\/" (d . (q9,u9)) by LATTICE3:14;
A62: (new_bi_fun2 (d,q)) . (q,u) = d . (q9,u9) by Def4;
(new_bi_fun2 (d,q)) . (p,q) = (d . (q9,(q `1_4))) "\/" a by A60, Def4;
hence (new_bi_fun2 (d,q)) . (p,u) <= ((new_bi_fun2 (d,q)) . (p,q)) "\/" ((new_bi_fun2 (d,q)) . (q,u)) by A60, A62, A61, Def4; ::_thesis: verum
end;
supposeA63: ( p = {{A}} & q in A & u in A ) ; ::_thesis: (new_bi_fun2 (d,q)) . (p,u) <= ((new_bi_fun2 (d,q)) . (p,q)) "\/" ((new_bi_fun2 (d,q)) . (q,u))
d . (u9,(q `2_4)) <= (d . (u9,q9)) "\/" (d . (q9,(q `2_4))) by A3, LATTICE5:def_7;
then d . (u9,(q `2_4)) <= (d . (q9,u9)) "\/" (d . (q9,(q `2_4))) by A2, LATTICE5:def_5;
then (d . (u9,(q `2_4))) "\/" a <= ((d . (q9,(q `2_4))) "\/" (d . (q9,u9))) "\/" a by WAYBEL_1:2;
then A64: (d . (u9,(q `2_4))) "\/" a <= ((d . (q9,(q `2_4))) "\/" a) "\/" (d . (q9,u9)) by LATTICE3:14;
A65: (new_bi_fun2 (d,q)) . (q,u) = d . (q9,u9) by Def4;
(new_bi_fun2 (d,q)) . (p,q) = (d . (q9,(q `2_4))) "\/" a by A63, Def4;
hence (new_bi_fun2 (d,q)) . (p,u) <= ((new_bi_fun2 (d,q)) . (p,q)) "\/" ((new_bi_fun2 (d,q)) . (q,u)) by A63, A65, A64, Def4; ::_thesis: verum
end;
end;
end;
A66: for p, q, u being Element of new_set2 A st p in A & q in B & u in A holds
(new_bi_fun2 (d,q)) . (p,u) <= ((new_bi_fun2 (d,q)) . (p,q)) "\/" ((new_bi_fun2 (d,q)) . (q,u))
proof
let p, q, u be Element of new_set2 A; ::_thesis: ( p in A & q in B & u in A implies (new_bi_fun2 (d,q)) . (p,u) <= ((new_bi_fun2 (d,q)) . (p,q)) "\/" ((new_bi_fun2 (d,q)) . (q,u)) )
assume A67: ( p in A & q in B & u in A ) ; ::_thesis: (new_bi_fun2 (d,q)) . (p,u) <= ((new_bi_fun2 (d,q)) . (p,q)) "\/" ((new_bi_fun2 (d,q)) . (q,u))
percases ( ( p in A & u in A & q = {A} ) or ( p in A & u in A & q = {{A}} ) ) by A67, TARSKI:def_2;
supposeA68: ( p in A & u in A & q = {A} ) ; ::_thesis: (new_bi_fun2 (d,q)) . (p,u) <= ((new_bi_fun2 (d,q)) . (p,q)) "\/" ((new_bi_fun2 (d,q)) . (q,u))
then reconsider p9 = p, u9 = u as Element of A ;
d . (p9,u9) <= (d . (p9,(q `1_4))) "\/" (d . ((q `1_4),u9)) by A3, LATTICE5:def_7;
then A69: ( (d . (p9,(q `1_4))) "\/" (d . (u9,(q `1_4))) <= ((d . (p9,(q `1_4))) "\/" (d . (u9,(q `1_4)))) "\/" a & d . (p9,u9) <= (d . (p9,(q `1_4))) "\/" (d . (u9,(q `1_4))) ) by A2, LATTICE5:def_5, YELLOW_0:22;
((d . (p9,(q `1_4))) "\/" (d . (u9,(q `1_4)))) "\/" a = (d . (p9,(q `1_4))) "\/" ((d . (u9,(q `1_4))) "\/" a) by LATTICE3:14
.= (d . (p9,(q `1_4))) "\/" ((d . (u9,(q `1_4))) "\/" (a "\/" a)) by YELLOW_5:1
.= (d . (p9,(q `1_4))) "\/" (((d . (u9,(q `1_4))) "\/" a) "\/" a) by LATTICE3:14
.= ((d . (p9,(q `1_4))) "\/" a) "\/" ((d . (u9,(q `1_4))) "\/" a) by LATTICE3:14 ;
then A70: d . (p9,u9) <= ((d . (p9,(q `1_4))) "\/" a) "\/" ((d . (u9,(q `1_4))) "\/" a) by A69, ORDERS_2:3;
( (new_bi_fun2 (d,q)) . (p,q) = (d . (p9,(q `1_4))) "\/" a & (new_bi_fun2 (d,q)) . (q,u) = (d . (u9,(q `1_4))) "\/" a ) by A68, Def4;
hence (new_bi_fun2 (d,q)) . (p,u) <= ((new_bi_fun2 (d,q)) . (p,q)) "\/" ((new_bi_fun2 (d,q)) . (q,u)) by A70, Def4; ::_thesis: verum
end;
supposeA71: ( p in A & u in A & q = {{A}} ) ; ::_thesis: (new_bi_fun2 (d,q)) . (p,u) <= ((new_bi_fun2 (d,q)) . (p,q)) "\/" ((new_bi_fun2 (d,q)) . (q,u))
then reconsider p9 = p, u9 = u as Element of A ;
d . (p9,u9) <= (d . (p9,(q `2_4))) "\/" (d . ((q `2_4),u9)) by A3, LATTICE5:def_7;
then A72: ( (d . (p9,(q `2_4))) "\/" (d . (u9,(q `2_4))) <= ((d . (p9,(q `2_4))) "\/" (d . (u9,(q `2_4)))) "\/" a & d . (p9,u9) <= (d . (p9,(q `2_4))) "\/" (d . (u9,(q `2_4))) ) by A2, LATTICE5:def_5, YELLOW_0:22;
((d . (p9,(q `2_4))) "\/" (d . (u9,(q `2_4)))) "\/" a = (d . (p9,(q `2_4))) "\/" ((d . (u9,(q `2_4))) "\/" a) by LATTICE3:14
.= (d . (p9,(q `2_4))) "\/" ((d . (u9,(q `2_4))) "\/" (a "\/" a)) by YELLOW_5:1
.= (d . (p9,(q `2_4))) "\/" (((d . (u9,(q `2_4))) "\/" a) "\/" a) by LATTICE3:14
.= ((d . (p9,(q `2_4))) "\/" a) "\/" ((d . (u9,(q `2_4))) "\/" a) by LATTICE3:14 ;
then A73: d . (p9,u9) <= ((d . (p9,(q `2_4))) "\/" a) "\/" ((d . (u9,(q `2_4))) "\/" a) by A72, ORDERS_2:3;
( (new_bi_fun2 (d,q)) . (p,q) = (d . (p9,(q `2_4))) "\/" a & (new_bi_fun2 (d,q)) . (q,u) = (d . (u9,(q `2_4))) "\/" a ) by A71, Def4;
hence (new_bi_fun2 (d,q)) . (p,u) <= ((new_bi_fun2 (d,q)) . (p,q)) "\/" ((new_bi_fun2 (d,q)) . (q,u)) by A73, Def4; ::_thesis: verum
end;
end;
end;
A74: for p, q, u being Element of new_set2 A st p in A & q in A & u in A holds
(new_bi_fun2 (d,q)) . (p,u) <= ((new_bi_fun2 (d,q)) . (p,q)) "\/" ((new_bi_fun2 (d,q)) . (q,u))
proof
let p, q, u be Element of new_set2 A; ::_thesis: ( p in A & q in A & u in A implies (new_bi_fun2 (d,q)) . (p,u) <= ((new_bi_fun2 (d,q)) . (p,q)) "\/" ((new_bi_fun2 (d,q)) . (q,u)) )
assume ( p in A & q in A & u in A ) ; ::_thesis: (new_bi_fun2 (d,q)) . (p,u) <= ((new_bi_fun2 (d,q)) . (p,q)) "\/" ((new_bi_fun2 (d,q)) . (q,u))
then reconsider p9 = p, q9 = q, u9 = u as Element of A ;
A75: (new_bi_fun2 (d,q)) . (q,u) = d . (q9,u9) by Def4;
( (new_bi_fun2 (d,q)) . (p,u) = d . (p9,u9) & (new_bi_fun2 (d,q)) . (p,q) = d . (p9,q9) ) by Def4;
hence (new_bi_fun2 (d,q)) . (p,u) <= ((new_bi_fun2 (d,q)) . (p,q)) "\/" ((new_bi_fun2 (d,q)) . (q,u)) by A3, A75, LATTICE5:def_7; ::_thesis: verum
end;
for p, q, u being Element of new_set2 A holds (new_bi_fun2 (d,q)) . (p,u) <= ((new_bi_fun2 (d,q)) . (p,q)) "\/" ((new_bi_fun2 (d,q)) . (q,u))
proof
let p, q, u be Element of new_set2 A; ::_thesis: (new_bi_fun2 (d,q)) . (p,u) <= ((new_bi_fun2 (d,q)) . (p,q)) "\/" ((new_bi_fun2 (d,q)) . (q,u))
percases ( ( p in A & q in A & u in A ) or ( p in A & q in A & u in B ) or ( p in A & q in B & u in A ) or ( p in A & q in B & u in B ) or ( p in B & q in A & u in A ) or ( p in B & q in A & u in B ) or ( p in B & q in B & u in A ) or ( p in B & q in B & u in B ) ) by XBOOLE_0:def_3;
suppose ( p in A & q in A & u in A ) ; ::_thesis: (new_bi_fun2 (d,q)) . (p,u) <= ((new_bi_fun2 (d,q)) . (p,q)) "\/" ((new_bi_fun2 (d,q)) . (q,u))
hence (new_bi_fun2 (d,q)) . (p,u) <= ((new_bi_fun2 (d,q)) . (p,q)) "\/" ((new_bi_fun2 (d,q)) . (q,u)) by A74; ::_thesis: verum
end;
suppose ( p in A & q in A & u in B ) ; ::_thesis: (new_bi_fun2 (d,q)) . (p,u) <= ((new_bi_fun2 (d,q)) . (p,q)) "\/" ((new_bi_fun2 (d,q)) . (q,u))
hence (new_bi_fun2 (d,q)) . (p,u) <= ((new_bi_fun2 (d,q)) . (p,q)) "\/" ((new_bi_fun2 (d,q)) . (q,u)) by A4; ::_thesis: verum
end;
suppose ( p in A & q in B & u in A ) ; ::_thesis: (new_bi_fun2 (d,q)) . (p,u) <= ((new_bi_fun2 (d,q)) . (p,q)) "\/" ((new_bi_fun2 (d,q)) . (q,u))
hence (new_bi_fun2 (d,q)) . (p,u) <= ((new_bi_fun2 (d,q)) . (p,q)) "\/" ((new_bi_fun2 (d,q)) . (q,u)) by A66; ::_thesis: verum
end;
suppose ( p in A & q in B & u in B ) ; ::_thesis: (new_bi_fun2 (d,q)) . (p,u) <= ((new_bi_fun2 (d,q)) . (p,q)) "\/" ((new_bi_fun2 (d,q)) . (q,u))
hence (new_bi_fun2 (d,q)) . (p,u) <= ((new_bi_fun2 (d,q)) . (p,q)) "\/" ((new_bi_fun2 (d,q)) . (q,u)) by A13; ::_thesis: verum
end;
suppose ( p in B & q in A & u in A ) ; ::_thesis: (new_bi_fun2 (d,q)) . (p,u) <= ((new_bi_fun2 (d,q)) . (p,q)) "\/" ((new_bi_fun2 (d,q)) . (q,u))
hence (new_bi_fun2 (d,q)) . (p,u) <= ((new_bi_fun2 (d,q)) . (p,q)) "\/" ((new_bi_fun2 (d,q)) . (q,u)) by A57; ::_thesis: verum
end;
suppose ( p in B & q in A & u in B ) ; ::_thesis: (new_bi_fun2 (d,q)) . (p,u) <= ((new_bi_fun2 (d,q)) . (p,q)) "\/" ((new_bi_fun2 (d,q)) . (q,u))
hence (new_bi_fun2 (d,q)) . (p,u) <= ((new_bi_fun2 (d,q)) . (p,q)) "\/" ((new_bi_fun2 (d,q)) . (q,u)) by A25; ::_thesis: verum
end;
suppose ( p in B & q in B & u in A ) ; ::_thesis: (new_bi_fun2 (d,q)) . (p,u) <= ((new_bi_fun2 (d,q)) . (p,q)) "\/" ((new_bi_fun2 (d,q)) . (q,u))
hence (new_bi_fun2 (d,q)) . (p,u) <= ((new_bi_fun2 (d,q)) . (p,q)) "\/" ((new_bi_fun2 (d,q)) . (q,u)) by A45; ::_thesis: verum
end;
suppose ( p in B & q in B & u in B ) ; ::_thesis: (new_bi_fun2 (d,q)) . (p,u) <= ((new_bi_fun2 (d,q)) . (p,q)) "\/" ((new_bi_fun2 (d,q)) . (q,u))
hence (new_bi_fun2 (d,q)) . (p,u) <= ((new_bi_fun2 (d,q)) . (p,q)) "\/" ((new_bi_fun2 (d,q)) . (q,u)) by A35; ::_thesis: verum
end;
end;
end;
hence new_bi_fun2 (d,q) is u.t.i. by LATTICE5:def_7; ::_thesis: verum
end;
theorem Th13: :: LATTICE8:13
for A being non empty set
for L being lower-bounded LATTICE
for d being BiFunction of A,L
for q being Element of [:A,A, the carrier of L, the carrier of L:] holds d c= new_bi_fun2 (d,q)
proof
let A be non empty set ; ::_thesis: for L being lower-bounded LATTICE
for d being BiFunction of A,L
for q being Element of [:A,A, the carrier of L, the carrier of L:] holds d c= new_bi_fun2 (d,q)
let L be lower-bounded LATTICE; ::_thesis: for d being BiFunction of A,L
for q being Element of [:A,A, the carrier of L, the carrier of L:] holds d c= new_bi_fun2 (d,q)
let d be BiFunction of A,L; ::_thesis: for q being Element of [:A,A, the carrier of L, the carrier of L:] holds d c= new_bi_fun2 (d,q)
let q be Element of [:A,A, the carrier of L, the carrier of L:]; ::_thesis: d c= new_bi_fun2 (d,q)
set g = new_bi_fun2 (d,q);
A1: A c= new_set2 A by XBOOLE_1:7;
A2: for z being set st z in dom d holds
d . z = (new_bi_fun2 (d,q)) . z
proof
let z be set ; ::_thesis: ( z in dom d implies d . z = (new_bi_fun2 (d,q)) . z )
assume A3: z in dom d ; ::_thesis: d . z = (new_bi_fun2 (d,q)) . z
then consider x, y being set such that
A4: [x,y] = z by RELAT_1:def_1;
reconsider x9 = x, y9 = y as Element of A by A3, A4, ZFMISC_1:87;
d . [x,y] = d . (x9,y9)
.= (new_bi_fun2 (d,q)) . (x9,y9) by Def4
.= (new_bi_fun2 (d,q)) . [x,y] ;
hence d . z = (new_bi_fun2 (d,q)) . z by A4; ::_thesis: verum
end;
( dom d = [:A,A:] & dom (new_bi_fun2 (d,q)) = [:(new_set2 A),(new_set2 A):] ) by FUNCT_2:def_1;
then dom d c= dom (new_bi_fun2 (d,q)) by A1, ZFMISC_1:96;
hence d c= new_bi_fun2 (d,q) by A2, GRFUNC_1:2; ::_thesis: verum
end;
definition
let A be non empty set ;
let O be Ordinal;
func ConsecutiveSet2 (A,O) -> set means :Def5: :: LATTICE8:def 5
ex L0 being T-Sequence st
( it = last L0 & dom L0 = succ O & L0 . {} = A & ( for C being Ordinal st succ C in succ O holds
L0 . (succ C) = new_set2 (L0 . C) ) & ( for C being Ordinal st C in succ O & C <> {} & C is limit_ordinal holds
L0 . C = union (rng (L0 | C)) ) );
correctness
existence
ex b1 being set ex L0 being T-Sequence st
( b1 = last L0 & dom L0 = succ O & L0 . {} = A & ( for C being Ordinal st succ C in succ O holds
L0 . (succ C) = new_set2 (L0 . C) ) & ( for C being Ordinal st C in succ O & C <> {} & C is limit_ordinal holds
L0 . C = union (rng (L0 | C)) ) );
uniqueness
for b1, b2 being set st ex L0 being T-Sequence st
( b1 = last L0 & dom L0 = succ O & L0 . {} = A & ( for C being Ordinal st succ C in succ O holds
L0 . (succ C) = new_set2 (L0 . C) ) & ( for C being Ordinal st C in succ O & C <> {} & C is limit_ordinal holds
L0 . C = union (rng (L0 | C)) ) ) & ex L0 being T-Sequence st
( b2 = last L0 & dom L0 = succ O & L0 . {} = A & ( for C being Ordinal st succ C in succ O holds
L0 . (succ C) = new_set2 (L0 . C) ) & ( for C being Ordinal st C in succ O & C <> {} & C is limit_ordinal holds
L0 . C = union (rng (L0 | C)) ) ) holds
b1 = b2;
proof
deffunc H1( set , T-Sequence) -> set = union (rng $2);
deffunc H2( Ordinal, set ) -> set = new_set2 $2;
thus ( ex x being set ex L0 being T-Sequence st
( x = last L0 & dom L0 = succ O & L0 . {} = A & ( for C being Ordinal st succ C in succ O holds
L0 . (succ C) = H2(C,L0 . C) ) & ( for C being Ordinal st C in succ O & C <> {} & C is limit_ordinal holds
L0 . C = H1(C,L0 | C) ) ) & ( for x1, x2 being set st ex L0 being T-Sequence st
( x1 = last L0 & dom L0 = succ O & L0 . {} = A & ( for C being Ordinal st succ C in succ O holds
L0 . (succ C) = H2(C,L0 . C) ) & ( for C being Ordinal st C in succ O & C <> {} & C is limit_ordinal holds
L0 . C = H1(C,L0 | C) ) ) & ex L0 being T-Sequence st
( x2 = last L0 & dom L0 = succ O & L0 . {} = A & ( for C being Ordinal st succ C in succ O holds
L0 . (succ C) = H2(C,L0 . C) ) & ( for C being Ordinal st C in succ O & C <> {} & C is limit_ordinal holds
L0 . C = H1(C,L0 | C) ) ) holds
x1 = x2 ) ) from ORDINAL2:sch_7(); ::_thesis: verum
end;
end;
:: deftheorem Def5 defines ConsecutiveSet2 LATTICE8:def_5_:_
for A being non empty set
for O being Ordinal
for b3 being set holds
( b3 = ConsecutiveSet2 (A,O) iff ex L0 being T-Sequence st
( b3 = last L0 & dom L0 = succ O & L0 . {} = A & ( for C being Ordinal st succ C in succ O holds
L0 . (succ C) = new_set2 (L0 . C) ) & ( for C being Ordinal st C in succ O & C <> {} & C is limit_ordinal holds
L0 . C = union (rng (L0 | C)) ) ) );
theorem Th14: :: LATTICE8:14
for A being non empty set holds ConsecutiveSet2 (A,{}) = A
proof
deffunc H1( set , T-Sequence) -> set = union (rng $2);
deffunc H2( Ordinal, set ) -> set = new_set2 $2;
let A be non empty set ; ::_thesis: ConsecutiveSet2 (A,{}) = A
deffunc H3( Ordinal) -> set = ConsecutiveSet2 (A,$1);
A1: for O being Ordinal
for It being set holds
( It = H3(O) iff ex L0 being T-Sequence st
( It = last L0 & dom L0 = succ O & L0 . {} = A & ( for C being Ordinal st succ C in succ O holds
L0 . (succ C) = H2(C,L0 . C) ) & ( for C being Ordinal st C in succ O & C <> {} & C is limit_ordinal holds
L0 . C = H1(C,L0 | C) ) ) ) by Def5;
thus H3( {} ) = A from ORDINAL2:sch_8(A1); ::_thesis: verum
end;
theorem Th15: :: LATTICE8:15
for A being non empty set
for O being Ordinal holds ConsecutiveSet2 (A,(succ O)) = new_set2 (ConsecutiveSet2 (A,O))
proof
deffunc H1( set , T-Sequence) -> set = union (rng $2);
deffunc H2( Ordinal, set ) -> set = new_set2 $2;
let A be non empty set ; ::_thesis: for O being Ordinal holds ConsecutiveSet2 (A,(succ O)) = new_set2 (ConsecutiveSet2 (A,O))
let O be Ordinal; ::_thesis: ConsecutiveSet2 (A,(succ O)) = new_set2 (ConsecutiveSet2 (A,O))
deffunc H3( Ordinal) -> set = ConsecutiveSet2 (A,$1);
A1: for O being Ordinal
for It being set holds
( It = H3(O) iff ex L0 being T-Sequence st
( It = last L0 & dom L0 = succ O & L0 . {} = A & ( for C being Ordinal st succ C in succ O holds
L0 . (succ C) = H2(C,L0 . C) ) & ( for C being Ordinal st C in succ O & C <> {} & C is limit_ordinal holds
L0 . C = H1(C,L0 | C) ) ) ) by Def5;
for O being Ordinal holds H3( succ O) = H2(O,H3(O)) from ORDINAL2:sch_9(A1);
hence ConsecutiveSet2 (A,(succ O)) = new_set2 (ConsecutiveSet2 (A,O)) ; ::_thesis: verum
end;
theorem Th16: :: LATTICE8:16
for A being non empty set
for O being Ordinal
for T being T-Sequence st O <> {} & O is limit_ordinal & dom T = O & ( for O1 being Ordinal st O1 in O holds
T . O1 = ConsecutiveSet2 (A,O1) ) holds
ConsecutiveSet2 (A,O) = union (rng T)
proof
deffunc H1( set , T-Sequence) -> set = union (rng $2);
deffunc H2( Ordinal, set ) -> set = new_set2 $2;
let A be non empty set ; ::_thesis: for O being Ordinal
for T being T-Sequence st O <> {} & O is limit_ordinal & dom T = O & ( for O1 being Ordinal st O1 in O holds
T . O1 = ConsecutiveSet2 (A,O1) ) holds
ConsecutiveSet2 (A,O) = union (rng T)
let O be Ordinal; ::_thesis: for T being T-Sequence st O <> {} & O is limit_ordinal & dom T = O & ( for O1 being Ordinal st O1 in O holds
T . O1 = ConsecutiveSet2 (A,O1) ) holds
ConsecutiveSet2 (A,O) = union (rng T)
let T be T-Sequence; ::_thesis: ( O <> {} & O is limit_ordinal & dom T = O & ( for O1 being Ordinal st O1 in O holds
T . O1 = ConsecutiveSet2 (A,O1) ) implies ConsecutiveSet2 (A,O) = union (rng T) )
deffunc H3( Ordinal) -> set = ConsecutiveSet2 (A,$1);
assume that
A1: ( O <> {} & O is limit_ordinal ) and
A2: dom T = O and
A3: for O1 being Ordinal st O1 in O holds
T . O1 = H3(O1) ; ::_thesis: ConsecutiveSet2 (A,O) = union (rng T)
A4: for O being Ordinal
for It being set holds
( It = H3(O) iff ex L0 being T-Sequence st
( It = last L0 & dom L0 = succ O & L0 . {} = A & ( for C being Ordinal st succ C in succ O holds
L0 . (succ C) = H2(C,L0 . C) ) & ( for C being Ordinal st C in succ O & C <> {} & C is limit_ordinal holds
L0 . C = H1(C,L0 | C) ) ) ) by Def5;
thus H3(O) = H1(O,T) from ORDINAL2:sch_10(A4, A1, A2, A3); ::_thesis: verum
end;
registration
let A be non empty set ;
let O be Ordinal;
cluster ConsecutiveSet2 (A,O) -> non empty ;
coherence
not ConsecutiveSet2 (A,O) is empty
proof
defpred S1[ Ordinal] means not ConsecutiveSet2 (A,A) is empty ;
A1: for O1 being Ordinal st S1[O1] holds
S1[ succ O1]
proof
let O1 be Ordinal; ::_thesis: ( S1[O1] implies S1[ succ O1] )
assume not ConsecutiveSet2 (A,O1) is empty ; ::_thesis: S1[ succ O1]
ConsecutiveSet2 (A,(succ O1)) = new_set2 (ConsecutiveSet2 (A,O1)) by Th15;
hence S1[ succ O1] ; ::_thesis: verum
end;
A2: for O1 being Ordinal st O1 <> {} & O1 is limit_ordinal & ( for O2 being Ordinal st O2 in O1 holds
S1[O2] ) holds
S1[O1]
proof
deffunc H1( Ordinal) -> set = ConsecutiveSet2 (A,A);
let O1 be Ordinal; ::_thesis: ( O1 <> {} & O1 is limit_ordinal & ( for O2 being Ordinal st O2 in O1 holds
S1[O2] ) implies S1[O1] )
assume that
A3: O1 <> {} and
A4: O1 is limit_ordinal and
for O2 being Ordinal st O2 in O1 holds
not ConsecutiveSet2 (A,O2) is empty ; ::_thesis: S1[O1]
A5: {} in O1 by A3, ORDINAL3:8;
consider Ls being T-Sequence such that
A6: ( dom Ls = O1 & ( for O2 being Ordinal st O2 in O1 holds
Ls . O2 = H1(O2) ) ) from ORDINAL2:sch_2();
Ls . {} = ConsecutiveSet2 (A,{}) by A3, A6, ORDINAL3:8
.= A by Th14 ;
then A7: A in rng Ls by A6, A5, FUNCT_1:def_3;
ConsecutiveSet2 (A,O1) = union (rng Ls) by A3, A4, A6, Th16;
then A c= ConsecutiveSet2 (A,O1) by A7, ZFMISC_1:74;
hence S1[O1] ; ::_thesis: verum
end;
A8: S1[ {} ] by Th14;
for O being Ordinal holds S1[O] from ORDINAL2:sch_1(A8, A1, A2);
hence not ConsecutiveSet2 (A,O) is empty ; ::_thesis: verum
end;
end;
theorem Th17: :: LATTICE8:17
for A being non empty set
for O being Ordinal holds A c= ConsecutiveSet2 (A,O)
proof
let A be non empty set ; ::_thesis: for O being Ordinal holds A c= ConsecutiveSet2 (A,O)
let O be Ordinal; ::_thesis: A c= ConsecutiveSet2 (A,O)
defpred S1[ Ordinal] means A c= ConsecutiveSet2 (A,$1);
A1: for O1 being Ordinal st S1[O1] holds
S1[ succ O1]
proof
let O1 be Ordinal; ::_thesis: ( S1[O1] implies S1[ succ O1] )
ConsecutiveSet2 (A,(succ O1)) = new_set2 (ConsecutiveSet2 (A,O1)) by Th15;
then A2: ConsecutiveSet2 (A,O1) c= ConsecutiveSet2 (A,(succ O1)) by XBOOLE_1:7;
assume A c= ConsecutiveSet2 (A,O1) ; ::_thesis: S1[ succ O1]
hence S1[ succ O1] by A2, XBOOLE_1:1; ::_thesis: verum
end;
A3: for O1 being Ordinal st O1 <> {} & O1 is limit_ordinal & ( for O2 being Ordinal st O2 in O1 holds
S1[O2] ) holds
S1[O1]
proof
deffunc H1( Ordinal) -> set = ConsecutiveSet2 (A,$1);
let O2 be Ordinal; ::_thesis: ( O2 <> {} & O2 is limit_ordinal & ( for O2 being Ordinal st O2 in O2 holds
S1[O2] ) implies S1[O2] )
assume that
A4: O2 <> {} and
A5: O2 is limit_ordinal and
for O1 being Ordinal st O1 in O2 holds
A c= ConsecutiveSet2 (A,O1) ; ::_thesis: S1[O2]
A6: {} in O2 by A4, ORDINAL3:8;
consider Ls being T-Sequence such that
A7: ( dom Ls = O2 & ( for O1 being Ordinal st O1 in O2 holds
Ls . O1 = H1(O1) ) ) from ORDINAL2:sch_2();
Ls . {} = ConsecutiveSet2 (A,{}) by A4, A7, ORDINAL3:8
.= A by Th14 ;
then A8: A in rng Ls by A7, A6, FUNCT_1:def_3;
ConsecutiveSet2 (A,O2) = union (rng Ls) by A4, A5, A7, Th16;
hence S1[O2] by A8, ZFMISC_1:74; ::_thesis: verum
end;
A9: S1[ {} ] by Th14;
for O being Ordinal holds S1[O] from ORDINAL2:sch_1(A9, A1, A3);
hence A c= ConsecutiveSet2 (A,O) ; ::_thesis: verum
end;
definition
let A be non empty set ;
let L be lower-bounded LATTICE;
let d be BiFunction of A,L;
let q be QuadrSeq of d;
let O be Ordinal;
assume A1: O in dom q ;
func Quadr2 (q,O) -> Element of [:(ConsecutiveSet2 (A,O)),(ConsecutiveSet2 (A,O)), the carrier of L, the carrier of L:] equals :Def6: :: LATTICE8:def 6
q . O;
correctness
coherence
q . O is Element of [:(ConsecutiveSet2 (A,O)),(ConsecutiveSet2 (A,O)), the carrier of L, the carrier of L:];
proof
q . O in rng q by A1, FUNCT_1:def_3;
then q . O in { [x,y,a,b] where x, y is Element of A, a, b is Element of L : d . (x,y) <= a "\/" b } by LATTICE5:def_13;
then consider x, y being Element of A, a, b being Element of L such that
A2: q . O = [x,y,a,b] and
d . (x,y) <= a "\/" b ;
reconsider a = a, b = b as Element of L ;
A3: ( x in A & y in A ) ;
A c= ConsecutiveSet2 (A,O) by Th17;
then reconsider x = x, y = y as Element of ConsecutiveSet2 (A,O) by A3;
reconsider z = [x,y,a,b] as Element of [:(ConsecutiveSet2 (A,O)),(ConsecutiveSet2 (A,O)), the carrier of L, the carrier of L:] ;
z = q . O by A2;
hence q . O is Element of [:(ConsecutiveSet2 (A,O)),(ConsecutiveSet2 (A,O)), the carrier of L, the carrier of L:] ; ::_thesis: verum
end;
end;
:: deftheorem Def6 defines Quadr2 LATTICE8:def_6_:_
for A being non empty set
for L being lower-bounded LATTICE
for d being BiFunction of A,L
for q being QuadrSeq of d
for O being Ordinal st O in dom q holds
Quadr2 (q,O) = q . O;
definition
let A be non empty set ;
let L be lower-bounded LATTICE;
let d be BiFunction of A,L;
let q be QuadrSeq of d;
let O be Ordinal;
func ConsecutiveDelta2 (q,O) -> set means :Def7: :: LATTICE8:def 7
ex L0 being T-Sequence st
( it = last L0 & dom L0 = succ O & L0 . {} = d & ( for C being Ordinal st succ C in succ O holds
L0 . (succ C) = new_bi_fun2 ((BiFun ((L0 . C),(ConsecutiveSet2 (A,C)),L)),(Quadr2 (q,C))) ) & ( for C being Ordinal st C in succ O & C <> {} & C is limit_ordinal holds
L0 . C = union (rng (L0 | C)) ) );
correctness
existence
ex b1 being set ex L0 being T-Sequence st
( b1 = last L0 & dom L0 = succ O & L0 . {} = d & ( for C being Ordinal st succ C in succ O holds
L0 . (succ C) = new_bi_fun2 ((BiFun ((L0 . C),(ConsecutiveSet2 (A,C)),L)),(Quadr2 (q,C))) ) & ( for C being Ordinal st C in succ O & C <> {} & C is limit_ordinal holds
L0 . C = union (rng (L0 | C)) ) );
uniqueness
for b1, b2 being set st ex L0 being T-Sequence st
( b1 = last L0 & dom L0 = succ O & L0 . {} = d & ( for C being Ordinal st succ C in succ O holds
L0 . (succ C) = new_bi_fun2 ((BiFun ((L0 . C),(ConsecutiveSet2 (A,C)),L)),(Quadr2 (q,C))) ) & ( for C being Ordinal st C in succ O & C <> {} & C is limit_ordinal holds
L0 . C = union (rng (L0 | C)) ) ) & ex L0 being T-Sequence st
( b2 = last L0 & dom L0 = succ O & L0 . {} = d & ( for C being Ordinal st succ C in succ O holds
L0 . (succ C) = new_bi_fun2 ((BiFun ((L0 . C),(ConsecutiveSet2 (A,C)),L)),(Quadr2 (q,C))) ) & ( for C being Ordinal st C in succ O & C <> {} & C is limit_ordinal holds
L0 . C = union (rng (L0 | C)) ) ) holds
b1 = b2;
proof
deffunc H1( set , T-Sequence) -> set = union (rng $2);
deffunc H2( Ordinal, set ) -> BiFunction of (new_set2 (ConsecutiveSet2 (A,$1))),L = new_bi_fun2 ((BiFun ($2,(ConsecutiveSet2 (A,$1)),L)),(Quadr2 (q,$1)));
thus ( ex x being set ex L0 being T-Sequence st
( x = last L0 & dom L0 = succ O & L0 . {} = d & ( for C being Ordinal st succ C in succ O holds
L0 . (succ C) = H2(C,L0 . C) ) & ( for C being Ordinal st C in succ O & C <> {} & C is limit_ordinal holds
L0 . C = H1(C,L0 | C) ) ) & ( for x1, x2 being set st ex L0 being T-Sequence st
( x1 = last L0 & dom L0 = succ O & L0 . {} = d & ( for C being Ordinal st succ C in succ O holds
L0 . (succ C) = H2(C,L0 . C) ) & ( for C being Ordinal st C in succ O & C <> {} & C is limit_ordinal holds
L0 . C = H1(C,L0 | C) ) ) & ex L0 being T-Sequence st
( x2 = last L0 & dom L0 = succ O & L0 . {} = d & ( for C being Ordinal st succ C in succ O holds
L0 . (succ C) = H2(C,L0 . C) ) & ( for C being Ordinal st C in succ O & C <> {} & C is limit_ordinal holds
L0 . C = H1(C,L0 | C) ) ) holds
x1 = x2 ) ) from ORDINAL2:sch_7(); ::_thesis: verum
end;
end;
:: deftheorem Def7 defines ConsecutiveDelta2 LATTICE8:def_7_:_
for A being non empty set
for L being lower-bounded LATTICE
for d being BiFunction of A,L
for q being QuadrSeq of d
for O being Ordinal
for b6 being set holds
( b6 = ConsecutiveDelta2 (q,O) iff ex L0 being T-Sequence st
( b6 = last L0 & dom L0 = succ O & L0 . {} = d & ( for C being Ordinal st succ C in succ O holds
L0 . (succ C) = new_bi_fun2 ((BiFun ((L0 . C),(ConsecutiveSet2 (A,C)),L)),(Quadr2 (q,C))) ) & ( for C being Ordinal st C in succ O & C <> {} & C is limit_ordinal holds
L0 . C = union (rng (L0 | C)) ) ) );
theorem Th18: :: LATTICE8:18
for A being non empty set
for L being lower-bounded LATTICE
for d being BiFunction of A,L
for q being QuadrSeq of d holds ConsecutiveDelta2 (q,{}) = d
proof
deffunc H1( set , T-Sequence) -> set = union (rng $2);
let A be non empty set ; ::_thesis: for L being lower-bounded LATTICE
for d being BiFunction of A,L
for q being QuadrSeq of d holds ConsecutiveDelta2 (q,{}) = d
let L be lower-bounded LATTICE; ::_thesis: for d being BiFunction of A,L
for q being QuadrSeq of d holds ConsecutiveDelta2 (q,{}) = d
let d be BiFunction of A,L; ::_thesis: for q being QuadrSeq of d holds ConsecutiveDelta2 (q,{}) = d
let q be QuadrSeq of d; ::_thesis: ConsecutiveDelta2 (q,{}) = d
deffunc H2( Ordinal, set ) -> BiFunction of (new_set2 (ConsecutiveSet2 (A,$1))),L = new_bi_fun2 ((BiFun ($2,(ConsecutiveSet2 (A,$1)),L)),(Quadr2 (q,$1)));
deffunc H3( Ordinal) -> set = ConsecutiveDelta2 (q,$1);
A1: for O being Ordinal
for It being set holds
( It = H3(O) iff ex L0 being T-Sequence st
( It = last L0 & dom L0 = succ O & L0 . {} = d & ( for C being Ordinal st succ C in succ O holds
L0 . (succ C) = H2(C,L0 . C) ) & ( for C being Ordinal st C in succ O & C <> {} & C is limit_ordinal holds
L0 . C = H1(C,L0 | C) ) ) ) by Def7;
thus H3( {} ) = d from ORDINAL2:sch_8(A1); ::_thesis: verum
end;
theorem Th19: :: LATTICE8:19
for A being non empty set
for L being lower-bounded LATTICE
for d being BiFunction of A,L
for q being QuadrSeq of d
for O being Ordinal holds ConsecutiveDelta2 (q,(succ O)) = new_bi_fun2 ((BiFun ((ConsecutiveDelta2 (q,O)),(ConsecutiveSet2 (A,O)),L)),(Quadr2 (q,O)))
proof
deffunc H1( set , T-Sequence) -> set = union (rng $2);
let A be non empty set ; ::_thesis: for L being lower-bounded LATTICE
for d being BiFunction of A,L
for q being QuadrSeq of d
for O being Ordinal holds ConsecutiveDelta2 (q,(succ O)) = new_bi_fun2 ((BiFun ((ConsecutiveDelta2 (q,O)),(ConsecutiveSet2 (A,O)),L)),(Quadr2 (q,O)))
let L be lower-bounded LATTICE; ::_thesis: for d being BiFunction of A,L
for q being QuadrSeq of d
for O being Ordinal holds ConsecutiveDelta2 (q,(succ O)) = new_bi_fun2 ((BiFun ((ConsecutiveDelta2 (q,O)),(ConsecutiveSet2 (A,O)),L)),(Quadr2 (q,O)))
let d be BiFunction of A,L; ::_thesis: for q being QuadrSeq of d
for O being Ordinal holds ConsecutiveDelta2 (q,(succ O)) = new_bi_fun2 ((BiFun ((ConsecutiveDelta2 (q,O)),(ConsecutiveSet2 (A,O)),L)),(Quadr2 (q,O)))
let q be QuadrSeq of d; ::_thesis: for O being Ordinal holds ConsecutiveDelta2 (q,(succ O)) = new_bi_fun2 ((BiFun ((ConsecutiveDelta2 (q,O)),(ConsecutiveSet2 (A,O)),L)),(Quadr2 (q,O)))
let O be Ordinal; ::_thesis: ConsecutiveDelta2 (q,(succ O)) = new_bi_fun2 ((BiFun ((ConsecutiveDelta2 (q,O)),(ConsecutiveSet2 (A,O)),L)),(Quadr2 (q,O)))
deffunc H2( Ordinal, set ) -> BiFunction of (new_set2 (ConsecutiveSet2 (A,$1))),L = new_bi_fun2 ((BiFun ($2,(ConsecutiveSet2 (A,$1)),L)),(Quadr2 (q,$1)));
deffunc H3( Ordinal) -> set = ConsecutiveDelta2 (q,$1);
A1: for O being Ordinal
for It being set holds
( It = H3(O) iff ex L0 being T-Sequence st
( It = last L0 & dom L0 = succ O & L0 . {} = d & ( for C being Ordinal st succ C in succ O holds
L0 . (succ C) = H2(C,L0 . C) ) & ( for C being Ordinal st C in succ O & C <> {} & C is limit_ordinal holds
L0 . C = H1(C,L0 | C) ) ) ) by Def7;
for O being Ordinal holds H3( succ O) = H2(O,H3(O)) from ORDINAL2:sch_9(A1);
hence ConsecutiveDelta2 (q,(succ O)) = new_bi_fun2 ((BiFun ((ConsecutiveDelta2 (q,O)),(ConsecutiveSet2 (A,O)),L)),(Quadr2 (q,O))) ; ::_thesis: verum
end;
theorem Th20: :: LATTICE8:20
for A being non empty set
for L being lower-bounded LATTICE
for d being BiFunction of A,L
for q being QuadrSeq of d
for T being T-Sequence
for O being Ordinal st O <> {} & O is limit_ordinal & dom T = O & ( for O1 being Ordinal st O1 in O holds
T . O1 = ConsecutiveDelta2 (q,O1) ) holds
ConsecutiveDelta2 (q,O) = union (rng T)
proof
deffunc H1( set , T-Sequence) -> set = union (rng $2);
let A be non empty set ; ::_thesis: for L being lower-bounded LATTICE
for d being BiFunction of A,L
for q being QuadrSeq of d
for T being T-Sequence
for O being Ordinal st O <> {} & O is limit_ordinal & dom T = O & ( for O1 being Ordinal st O1 in O holds
T . O1 = ConsecutiveDelta2 (q,O1) ) holds
ConsecutiveDelta2 (q,O) = union (rng T)
let L be lower-bounded LATTICE; ::_thesis: for d being BiFunction of A,L
for q being QuadrSeq of d
for T being T-Sequence
for O being Ordinal st O <> {} & O is limit_ordinal & dom T = O & ( for O1 being Ordinal st O1 in O holds
T . O1 = ConsecutiveDelta2 (q,O1) ) holds
ConsecutiveDelta2 (q,O) = union (rng T)
let d be BiFunction of A,L; ::_thesis: for q being QuadrSeq of d
for T being T-Sequence
for O being Ordinal st O <> {} & O is limit_ordinal & dom T = O & ( for O1 being Ordinal st O1 in O holds
T . O1 = ConsecutiveDelta2 (q,O1) ) holds
ConsecutiveDelta2 (q,O) = union (rng T)
let q be QuadrSeq of d; ::_thesis: for T being T-Sequence
for O being Ordinal st O <> {} & O is limit_ordinal & dom T = O & ( for O1 being Ordinal st O1 in O holds
T . O1 = ConsecutiveDelta2 (q,O1) ) holds
ConsecutiveDelta2 (q,O) = union (rng T)
let T be T-Sequence; ::_thesis: for O being Ordinal st O <> {} & O is limit_ordinal & dom T = O & ( for O1 being Ordinal st O1 in O holds
T . O1 = ConsecutiveDelta2 (q,O1) ) holds
ConsecutiveDelta2 (q,O) = union (rng T)
let O be Ordinal; ::_thesis: ( O <> {} & O is limit_ordinal & dom T = O & ( for O1 being Ordinal st O1 in O holds
T . O1 = ConsecutiveDelta2 (q,O1) ) implies ConsecutiveDelta2 (q,O) = union (rng T) )
deffunc H2( Ordinal, set ) -> BiFunction of (new_set2 (ConsecutiveSet2 (A,$1))),L = new_bi_fun2 ((BiFun ($2,(ConsecutiveSet2 (A,$1)),L)),(Quadr2 (q,$1)));
deffunc H3( Ordinal) -> set = ConsecutiveDelta2 (q,$1);
assume that
A1: ( O <> {} & O is limit_ordinal ) and
A2: dom T = O and
A3: for O1 being Ordinal st O1 in O holds
T . O1 = H3(O1) ; ::_thesis: ConsecutiveDelta2 (q,O) = union (rng T)
A4: for O being Ordinal
for It being set holds
( It = H3(O) iff ex L0 being T-Sequence st
( It = last L0 & dom L0 = succ O & L0 . {} = d & ( for C being Ordinal st succ C in succ O holds
L0 . (succ C) = H2(C,L0 . C) ) & ( for C being Ordinal st C in succ O & C <> {} & C is limit_ordinal holds
L0 . C = H1(C,L0 | C) ) ) ) by Def7;
thus H3(O) = H1(O,T) from ORDINAL2:sch_10(A4, A1, A2, A3); ::_thesis: verum
end;
theorem Th21: :: LATTICE8:21
for A being non empty set
for O, O1, O2 being Ordinal st O1 c= O2 holds
ConsecutiveSet2 (A,O1) c= ConsecutiveSet2 (A,O2)
proof
let A be non empty set ; ::_thesis: for O, O1, O2 being Ordinal st O1 c= O2 holds
ConsecutiveSet2 (A,O1) c= ConsecutiveSet2 (A,O2)
let O, O1, 02 be Ordinal; ::_thesis: ( O1 c= 02 implies ConsecutiveSet2 (A,O1) c= ConsecutiveSet2 (A,02) )
defpred S1[ Ordinal] means ( O1 c= $1 implies ConsecutiveSet2 (A,O1) c= ConsecutiveSet2 (A,$1) );
A1: for O1 being Ordinal st S1[O1] holds
S1[ succ O1]
proof
let O2 be Ordinal; ::_thesis: ( S1[O2] implies S1[ succ O2] )
assume A2: ( O1 c= O2 implies ConsecutiveSet2 (A,O1) c= ConsecutiveSet2 (A,O2) ) ; ::_thesis: S1[ succ O2]
assume A3: O1 c= succ O2 ; ::_thesis: ConsecutiveSet2 (A,O1) c= ConsecutiveSet2 (A,(succ O2))
percases ( O1 = succ O2 or O1 <> succ O2 ) ;
suppose O1 = succ O2 ; ::_thesis: ConsecutiveSet2 (A,O1) c= ConsecutiveSet2 (A,(succ O2))
hence ConsecutiveSet2 (A,O1) c= ConsecutiveSet2 (A,(succ O2)) ; ::_thesis: verum
end;
suppose O1 <> succ O2 ; ::_thesis: ConsecutiveSet2 (A,O1) c= ConsecutiveSet2 (A,(succ O2))
then O1 c< succ O2 by A3, XBOOLE_0:def_8;
then A4: O1 in succ O2 by ORDINAL1:11;
ConsecutiveSet2 (A,O2) c= new_set2 (ConsecutiveSet2 (A,O2)) by XBOOLE_1:7;
then ConsecutiveSet2 (A,O1) c= new_set2 (ConsecutiveSet2 (A,O2)) by A2, A4, ORDINAL1:22, XBOOLE_1:1;
hence ConsecutiveSet2 (A,O1) c= ConsecutiveSet2 (A,(succ O2)) by Th15; ::_thesis: verum
end;
end;
end;
A5: for O1 being Ordinal st O1 <> {} & O1 is limit_ordinal & ( for O2 being Ordinal st O2 in O1 holds
S1[O2] ) holds
S1[O1]
proof
deffunc H1( Ordinal) -> set = ConsecutiveSet2 (A,$1);
let O2 be Ordinal; ::_thesis: ( O2 <> {} & O2 is limit_ordinal & ( for O2 being Ordinal st O2 in O2 holds
S1[O2] ) implies S1[O2] )
assume that
A6: ( O2 <> {} & O2 is limit_ordinal ) and
for O3 being Ordinal st O3 in O2 & O1 c= O3 holds
ConsecutiveSet2 (A,O1) c= ConsecutiveSet2 (A,O3) ; ::_thesis: S1[O2]
consider L being T-Sequence such that
A7: ( dom L = O2 & ( for O3 being Ordinal st O3 in O2 holds
L . O3 = H1(O3) ) ) from ORDINAL2:sch_2();
A8: ConsecutiveSet2 (A,O2) = union (rng L) by A6, A7, Th16;
assume A9: O1 c= O2 ; ::_thesis: ConsecutiveSet2 (A,O1) c= ConsecutiveSet2 (A,O2)
percases ( O1 = O2 or O1 <> O2 ) ;
suppose O1 = O2 ; ::_thesis: ConsecutiveSet2 (A,O1) c= ConsecutiveSet2 (A,O2)
hence ConsecutiveSet2 (A,O1) c= ConsecutiveSet2 (A,O2) ; ::_thesis: verum
end;
suppose O1 <> O2 ; ::_thesis: ConsecutiveSet2 (A,O1) c= ConsecutiveSet2 (A,O2)
then A10: O1 c< O2 by A9, XBOOLE_0:def_8;
then O1 in O2 by ORDINAL1:11;
then A11: L . O1 in rng L by A7, FUNCT_1:def_3;
L . O1 = ConsecutiveSet2 (A,O1) by A7, A10, ORDINAL1:11;
hence ConsecutiveSet2 (A,O1) c= ConsecutiveSet2 (A,O2) by A8, A11, ZFMISC_1:74; ::_thesis: verum
end;
end;
end;
A12: S1[ {} ] ;
for O being Ordinal holds S1[O] from ORDINAL2:sch_1(A12, A1, A5);
hence ( O1 c= 02 implies ConsecutiveSet2 (A,O1) c= ConsecutiveSet2 (A,02) ) ; ::_thesis: verum
end;
theorem Th22: :: LATTICE8:22
for A being non empty set
for L being lower-bounded LATTICE
for d being BiFunction of A,L
for q being QuadrSeq of d
for O being Ordinal holds ConsecutiveDelta2 (q,O) is BiFunction of (ConsecutiveSet2 (A,O)),L
proof
let A be non empty set ; ::_thesis: for L being lower-bounded LATTICE
for d being BiFunction of A,L
for q being QuadrSeq of d
for O being Ordinal holds ConsecutiveDelta2 (q,O) is BiFunction of (ConsecutiveSet2 (A,O)),L
let L be lower-bounded LATTICE; ::_thesis: for d being BiFunction of A,L
for q being QuadrSeq of d
for O being Ordinal holds ConsecutiveDelta2 (q,O) is BiFunction of (ConsecutiveSet2 (A,O)),L
let d be BiFunction of A,L; ::_thesis: for q being QuadrSeq of d
for O being Ordinal holds ConsecutiveDelta2 (q,O) is BiFunction of (ConsecutiveSet2 (A,O)),L
let q be QuadrSeq of d; ::_thesis: for O being Ordinal holds ConsecutiveDelta2 (q,O) is BiFunction of (ConsecutiveSet2 (A,O)),L
let O be Ordinal; ::_thesis: ConsecutiveDelta2 (q,O) is BiFunction of (ConsecutiveSet2 (A,O)),L
defpred S1[ Ordinal] means ConsecutiveDelta2 (q,$1) is BiFunction of (ConsecutiveSet2 (A,$1)),L;
A1: for O1 being Ordinal st S1[O1] holds
S1[ succ O1]
proof
let O1 be Ordinal; ::_thesis: ( S1[O1] implies S1[ succ O1] )
assume ConsecutiveDelta2 (q,O1) is BiFunction of (ConsecutiveSet2 (A,O1)),L ; ::_thesis: S1[ succ O1]
then reconsider CD = ConsecutiveDelta2 (q,O1) as BiFunction of (ConsecutiveSet2 (A,O1)),L ;
A2: ConsecutiveDelta2 (q,(succ O1)) = new_bi_fun2 ((BiFun ((ConsecutiveDelta2 (q,O1)),(ConsecutiveSet2 (A,O1)),L)),(Quadr2 (q,O1))) by Th19
.= new_bi_fun2 (CD,(Quadr2 (q,O1))) by LATTICE5:def_15 ;
ConsecutiveSet2 (A,(succ O1)) = new_set2 (ConsecutiveSet2 (A,O1)) by Th15;
hence S1[ succ O1] by A2; ::_thesis: verum
end;
A3: for O1 being Ordinal st O1 <> {} & O1 is limit_ordinal & ( for O2 being Ordinal st O2 in O1 holds
S1[O2] ) holds
S1[O1]
proof
deffunc H1( Ordinal) -> set = ConsecutiveDelta2 (q,$1);
let O1 be Ordinal; ::_thesis: ( O1 <> {} & O1 is limit_ordinal & ( for O2 being Ordinal st O2 in O1 holds
S1[O2] ) implies S1[O1] )
assume that
A4: O1 <> {} and
A5: O1 is limit_ordinal and
A6: for O2 being Ordinal st O2 in O1 holds
ConsecutiveDelta2 (q,O2) is BiFunction of (ConsecutiveSet2 (A,O2)),L ; ::_thesis: S1[O1]
reconsider o1 = O1 as non empty Ordinal by A4;
set YY = { [:(ConsecutiveSet2 (A,O2)),(ConsecutiveSet2 (A,O2)):] where O2 is Element of o1 : verum } ;
consider Ls being T-Sequence such that
A7: ( dom Ls = O1 & ( for O2 being Ordinal st O2 in O1 holds
Ls . O2 = H1(O2) ) ) from ORDINAL2:sch_2();
A8: for O, O2 being Ordinal st O c= O2 & O2 in dom Ls holds
Ls . O c= Ls . O2
proof
let O be Ordinal; ::_thesis: for O2 being Ordinal st O c= O2 & O2 in dom Ls holds
Ls . O c= Ls . O2
defpred S2[ Ordinal] means ( O c= $1 & $1 in dom Ls implies Ls . O c= Ls . $1 );
A9: for O1 being Ordinal st O1 <> {} & O1 is limit_ordinal & ( for O2 being Ordinal st O2 in O1 holds
S2[O2] ) holds
S2[O1]
proof
deffunc H2( Ordinal) -> set = ConsecutiveDelta2 (q,$1);
let O2 be Ordinal; ::_thesis: ( O2 <> {} & O2 is limit_ordinal & ( for O2 being Ordinal st O2 in O2 holds
S2[O2] ) implies S2[O2] )
assume that
A10: ( O2 <> {} & O2 is limit_ordinal ) and
for O3 being Ordinal st O3 in O2 & O c= O3 & O3 in dom Ls holds
Ls . O c= Ls . O3 ; ::_thesis: S2[O2]
assume that
A11: O c= O2 and
A12: O2 in dom Ls ; ::_thesis: Ls . O c= Ls . O2
consider Lt being T-Sequence such that
A13: ( dom Lt = O2 & ( for O3 being Ordinal st O3 in O2 holds
Lt . O3 = H2(O3) ) ) from ORDINAL2:sch_2();
A14: Ls . O2 = ConsecutiveDelta2 (q,O2) by A7, A12
.= union (rng Lt) by A10, A13, Th20 ;
percases ( O = O2 or O <> O2 ) ;
suppose O = O2 ; ::_thesis: Ls . O c= Ls . O2
hence Ls . O c= Ls . O2 ; ::_thesis: verum
end;
suppose O <> O2 ; ::_thesis: Ls . O c= Ls . O2
then A15: O c< O2 by A11, XBOOLE_0:def_8;
then A16: O in O2 by ORDINAL1:11;
then Ls . O = ConsecutiveDelta2 (q,O) by A7, A12, ORDINAL1:10
.= Lt . O by A13, A15, ORDINAL1:11 ;
then Ls . O in rng Lt by A13, A16, FUNCT_1:def_3;
hence Ls . O c= Ls . O2 by A14, ZFMISC_1:74; ::_thesis: verum
end;
end;
end;
A17: for O1 being Ordinal st S2[O1] holds
S2[ succ O1]
proof
let O2 be Ordinal; ::_thesis: ( S2[O2] implies S2[ succ O2] )
assume A18: ( O c= O2 & O2 in dom Ls implies Ls . O c= Ls . O2 ) ; ::_thesis: S2[ succ O2]
assume that
A19: O c= succ O2 and
A20: succ O2 in dom Ls ; ::_thesis: Ls . O c= Ls . (succ O2)
percases ( O = succ O2 or O <> succ O2 ) ;
suppose O = succ O2 ; ::_thesis: Ls . O c= Ls . (succ O2)
hence Ls . O c= Ls . (succ O2) ; ::_thesis: verum
end;
suppose O <> succ O2 ; ::_thesis: Ls . O c= Ls . (succ O2)
then O c< succ O2 by A19, XBOOLE_0:def_8;
then A21: O in succ O2 by ORDINAL1:11;
A22: O2 in succ O2 by ORDINAL1:6;
then O2 in dom Ls by A20, ORDINAL1:10;
then reconsider Def8 = ConsecutiveDelta2 (q,O2) as BiFunction of (ConsecutiveSet2 (A,O2)),L by A6, A7;
Ls . (succ O2) = ConsecutiveDelta2 (q,(succ O2)) by A7, A20
.= new_bi_fun2 ((BiFun ((ConsecutiveDelta2 (q,O2)),(ConsecutiveSet2 (A,O2)),L)),(Quadr2 (q,O2))) by Th19
.= new_bi_fun2 (Def8,(Quadr2 (q,O2))) by LATTICE5:def_15 ;
then ConsecutiveDelta2 (q,O2) c= Ls . (succ O2) by Th13;
then Ls . O2 c= Ls . (succ O2) by A7, A20, A22, ORDINAL1:10;
hence Ls . O c= Ls . (succ O2) by A18, A20, A21, A22, ORDINAL1:10, ORDINAL1:22, XBOOLE_1:1; ::_thesis: verum
end;
end;
end;
A23: S2[ {} ] ;
thus for O being Ordinal holds S2[O] from ORDINAL2:sch_1(A23, A17, A9); ::_thesis: verum
end;
for x, y being set st x in rng Ls & y in rng Ls holds
x,y are_c=-comparable
proof
let x, y be set ; ::_thesis: ( x in rng Ls & y in rng Ls implies x,y are_c=-comparable )
assume that
A24: x in rng Ls and
A25: y in rng Ls ; ::_thesis: x,y are_c=-comparable
consider o1 being set such that
A26: o1 in dom Ls and
A27: Ls . o1 = x by A24, FUNCT_1:def_3;
consider o2 being set such that
A28: o2 in dom Ls and
A29: Ls . o2 = y by A25, FUNCT_1:def_3;
reconsider o19 = o1, o29 = o2 as Ordinal by A26, A28;
( o19 c= o29 or o29 c= o19 ) ;
then ( x c= y or y c= x ) by A8, A26, A27, A28, A29;
hence x,y are_c=-comparable by XBOOLE_0:def_9; ::_thesis: verum
end;
then A30: rng Ls is c=-linear by ORDINAL1:def_8;
set Y = the carrier of L;
set X = [:(ConsecutiveSet2 (A,O1)),(ConsecutiveSet2 (A,O1)):];
set f = union (rng Ls);
rng Ls c= PFuncs ([:(ConsecutiveSet2 (A,O1)),(ConsecutiveSet2 (A,O1)):], the carrier of L)
proof
let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in rng Ls or z in PFuncs ([:(ConsecutiveSet2 (A,O1)),(ConsecutiveSet2 (A,O1)):], the carrier of L) )
assume z in rng Ls ; ::_thesis: z in PFuncs ([:(ConsecutiveSet2 (A,O1)),(ConsecutiveSet2 (A,O1)):], the carrier of L)
then consider o being set such that
A31: o in dom Ls and
A32: z = Ls . o by FUNCT_1:def_3;
reconsider o = o as Ordinal by A31;
Ls . o = ConsecutiveDelta2 (q,o) by A7, A31;
then reconsider h = Ls . o as BiFunction of (ConsecutiveSet2 (A,o)),L by A6, A7, A31;
o c= O1 by A7, A31, ORDINAL1:def_2;
then ( dom h = [:(ConsecutiveSet2 (A,o)),(ConsecutiveSet2 (A,o)):] & ConsecutiveSet2 (A,o) c= ConsecutiveSet2 (A,O1) ) by Th21, FUNCT_2:def_1;
then ( rng h c= the carrier of L & dom h c= [:(ConsecutiveSet2 (A,O1)),(ConsecutiveSet2 (A,O1)):] ) by ZFMISC_1:96;
hence z in PFuncs ([:(ConsecutiveSet2 (A,O1)),(ConsecutiveSet2 (A,O1)):], the carrier of L) by A32, PARTFUN1:def_3; ::_thesis: verum
end;
then union (rng Ls) in PFuncs ([:(ConsecutiveSet2 (A,O1)),(ConsecutiveSet2 (A,O1)):], the carrier of L) by A30, TREES_2:40;
then A33: ex g being Function st
( union (rng Ls) = g & dom g c= [:(ConsecutiveSet2 (A,O1)),(ConsecutiveSet2 (A,O1)):] & rng g c= the carrier of L ) by PARTFUN1:def_3;
Ls is Function-yielding
proof
let x be set ; :: according to FUNCOP_1:def_6 ::_thesis: ( not x in proj1 Ls or Ls . x is set )
assume A34: x in dom Ls ; ::_thesis: Ls . x is set
then reconsider o = x as Ordinal ;
Ls . o = ConsecutiveDelta2 (q,o) by A7, A34;
hence Ls . x is set by A6, A7, A34; ::_thesis: verum
end;
then reconsider LsF = Ls as Function-yielding Function ;
A35: rng (doms Ls) = { [:(ConsecutiveSet2 (A,O2)),(ConsecutiveSet2 (A,O2)):] where O2 is Element of o1 : verum }
proof
thus rng (doms Ls) c= { [:(ConsecutiveSet2 (A,O2)),(ConsecutiveSet2 (A,O2)):] where O2 is Element of o1 : verum } :: according to XBOOLE_0:def_10 ::_thesis: { [:(ConsecutiveSet2 (A,O2)),(ConsecutiveSet2 (A,O2)):] where O2 is Element of o1 : verum } c= rng (doms Ls)
proof
let Z be set ; :: according to TARSKI:def_3 ::_thesis: ( not Z in rng (doms Ls) or Z in { [:(ConsecutiveSet2 (A,O2)),(ConsecutiveSet2 (A,O2)):] where O2 is Element of o1 : verum } )
assume Z in rng (doms Ls) ; ::_thesis: Z in { [:(ConsecutiveSet2 (A,O2)),(ConsecutiveSet2 (A,O2)):] where O2 is Element of o1 : verum }
then consider o being set such that
A36: o in dom (doms Ls) and
A37: Z = (doms Ls) . o by FUNCT_1:def_3;
A38: o in dom LsF by A36, FUNCT_6:59;
then reconsider o9 = o as Element of o1 by A7;
Ls . o9 = ConsecutiveDelta2 (q,o9) by A7;
then reconsider ls = Ls . o9 as BiFunction of (ConsecutiveSet2 (A,o9)),L by A6;
Z = dom ls by A37, A38, FUNCT_6:22
.= [:(ConsecutiveSet2 (A,o9)),(ConsecutiveSet2 (A,o9)):] by FUNCT_2:def_1 ;
hence Z in { [:(ConsecutiveSet2 (A,O2)),(ConsecutiveSet2 (A,O2)):] where O2 is Element of o1 : verum } ; ::_thesis: verum
end;
let Z be set ; :: according to TARSKI:def_3 ::_thesis: ( not Z in { [:(ConsecutiveSet2 (A,O2)),(ConsecutiveSet2 (A,O2)):] where O2 is Element of o1 : verum } or Z in rng (doms Ls) )
assume Z in { [:(ConsecutiveSet2 (A,O2)),(ConsecutiveSet2 (A,O2)):] where O2 is Element of o1 : verum } ; ::_thesis: Z in rng (doms Ls)
then consider o being Element of o1 such that
A39: Z = [:(ConsecutiveSet2 (A,o)),(ConsecutiveSet2 (A,o)):] ;
Ls . o = ConsecutiveDelta2 (q,o) by A7;
then reconsider ls = Ls . o as BiFunction of (ConsecutiveSet2 (A,o)),L by A6;
o in dom LsF by A7;
then A40: o in dom (doms LsF) by FUNCT_6:59;
Z = dom ls by A39, FUNCT_2:def_1
.= (doms Ls) . o by A7, FUNCT_6:22 ;
hence Z in rng (doms Ls) by A40, FUNCT_1:def_3; ::_thesis: verum
end;
A41: ConsecutiveDelta2 (q,O1) = union (rng Ls) by A4, A5, A7, Th20;
reconsider f = union (rng Ls) as Function by A33;
deffunc H2( Ordinal) -> set = ConsecutiveSet2 (A,$1);
consider Ts being T-Sequence such that
A42: ( dom Ts = O1 & ( for O2 being Ordinal st O2 in O1 holds
Ts . O2 = H2(O2) ) ) from ORDINAL2:sch_2();
{} in O1 by A4, ORDINAL3:8;
then reconsider RTs = rng Ts as non empty set by A42, FUNCT_1:3;
A43: { [:(ConsecutiveSet2 (A,O2)),(ConsecutiveSet2 (A,O2)):] where O2 is Element of o1 : verum } = { [:a,a:] where a is Element of RTs : a in RTs }
proof
set XX = { [:a,a:] where a is Element of RTs : a in RTs } ;
thus { [:(ConsecutiveSet2 (A,O2)),(ConsecutiveSet2 (A,O2)):] where O2 is Element of o1 : verum } c= { [:a,a:] where a is Element of RTs : a in RTs } :: according to XBOOLE_0:def_10 ::_thesis: { [:a,a:] where a is Element of RTs : a in RTs } c= { [:(ConsecutiveSet2 (A,O2)),(ConsecutiveSet2 (A,O2)):] where O2 is Element of o1 : verum }
proof
let Z be set ; :: according to TARSKI:def_3 ::_thesis: ( not Z in { [:(ConsecutiveSet2 (A,O2)),(ConsecutiveSet2 (A,O2)):] where O2 is Element of o1 : verum } or Z in { [:a,a:] where a is Element of RTs : a in RTs } )
assume Z in { [:(ConsecutiveSet2 (A,O2)),(ConsecutiveSet2 (A,O2)):] where O2 is Element of o1 : verum } ; ::_thesis: Z in { [:a,a:] where a is Element of RTs : a in RTs }
then consider o being Element of o1 such that
A44: Z = [:(ConsecutiveSet2 (A,o)),(ConsecutiveSet2 (A,o)):] ;
Ts . o = ConsecutiveSet2 (A,o) by A42;
then reconsider CoS = ConsecutiveSet2 (A,o) as Element of RTs by A42, FUNCT_1:def_3;
Z = [:CoS,CoS:] by A44;
hence Z in { [:a,a:] where a is Element of RTs : a in RTs } ; ::_thesis: verum
end;
let Z be set ; :: according to TARSKI:def_3 ::_thesis: ( not Z in { [:a,a:] where a is Element of RTs : a in RTs } or Z in { [:(ConsecutiveSet2 (A,O2)),(ConsecutiveSet2 (A,O2)):] where O2 is Element of o1 : verum } )
assume Z in { [:a,a:] where a is Element of RTs : a in RTs } ; ::_thesis: Z in { [:(ConsecutiveSet2 (A,O2)),(ConsecutiveSet2 (A,O2)):] where O2 is Element of o1 : verum }
then consider a being Element of RTs such that
A45: Z = [:a,a:] and
a in RTs ;
consider o being set such that
A46: o in dom Ts and
A47: a = Ts . o by FUNCT_1:def_3;
reconsider o9 = o as Ordinal by A46;
a = ConsecutiveSet2 (A,o9) by A42, A46, A47;
hence Z in { [:(ConsecutiveSet2 (A,O2)),(ConsecutiveSet2 (A,O2)):] where O2 is Element of o1 : verum } by A42, A45, A46; ::_thesis: verum
end;
for x, y being set st x in RTs & y in RTs holds
x,y are_c=-comparable
proof
let x, y be set ; ::_thesis: ( x in RTs & y in RTs implies x,y are_c=-comparable )
assume that
A48: x in RTs and
A49: y in RTs ; ::_thesis: x,y are_c=-comparable
consider o1 being set such that
A50: o1 in dom Ts and
A51: Ts . o1 = x by A48, FUNCT_1:def_3;
consider o2 being set such that
A52: o2 in dom Ts and
A53: Ts . o2 = y by A49, FUNCT_1:def_3;
reconsider o19 = o1, o29 = o2 as Ordinal by A50, A52;
A54: Ts . o29 = ConsecutiveSet2 (A,o29) by A42, A52;
A55: ( o19 c= o29 or o29 c= o19 ) ;
Ts . o19 = ConsecutiveSet2 (A,o19) by A42, A50;
then ( x c= y or y c= x ) by A51, A53, A54, A55, Th21;
hence x,y are_c=-comparable by XBOOLE_0:def_9; ::_thesis: verum
end;
then A56: ( dom f = union (rng (doms LsF)) & RTs is c=-linear ) by LATTICE5:1, ORDINAL1:def_8;
[:(ConsecutiveSet2 (A,O1)),(ConsecutiveSet2 (A,O1)):] = [:(union (rng Ts)),(ConsecutiveSet2 (A,O1)):] by A4, A5, A42, Th16
.= [:(union RTs),(union RTs):] by A4, A5, A42, Th16
.= dom f by A35, A56, A43, LATTICE5:3 ;
hence S1[O1] by A41, A33, FUNCT_2:def_1, RELSET_1:4; ::_thesis: verum
end;
ConsecutiveSet2 (A,{}) = A by Th14;
then A57: S1[ {} ] by Th18;
for O being Ordinal holds S1[O] from ORDINAL2:sch_1(A57, A1, A3);
hence ConsecutiveDelta2 (q,O) is BiFunction of (ConsecutiveSet2 (A,O)),L ; ::_thesis: verum
end;
definition
let A be non empty set ;
let L be lower-bounded LATTICE;
let d be BiFunction of A,L;
let q be QuadrSeq of d;
let O be Ordinal;
:: original: ConsecutiveDelta2
redefine func ConsecutiveDelta2 (q,O) -> BiFunction of (ConsecutiveSet2 (A,O)),L;
coherence
ConsecutiveDelta2 (q,O) is BiFunction of (ConsecutiveSet2 (A,O)),L by Th22;
end;
theorem Th23: :: LATTICE8:23
for A being non empty set
for L being lower-bounded LATTICE
for d being BiFunction of A,L
for q being QuadrSeq of d
for O being Ordinal holds d c= ConsecutiveDelta2 (q,O)
proof
let A be non empty set ; ::_thesis: for L being lower-bounded LATTICE
for d being BiFunction of A,L
for q being QuadrSeq of d
for O being Ordinal holds d c= ConsecutiveDelta2 (q,O)
let L be lower-bounded LATTICE; ::_thesis: for d being BiFunction of A,L
for q being QuadrSeq of d
for O being Ordinal holds d c= ConsecutiveDelta2 (q,O)
let d be BiFunction of A,L; ::_thesis: for q being QuadrSeq of d
for O being Ordinal holds d c= ConsecutiveDelta2 (q,O)
let q be QuadrSeq of d; ::_thesis: for O being Ordinal holds d c= ConsecutiveDelta2 (q,O)
let O be Ordinal; ::_thesis: d c= ConsecutiveDelta2 (q,O)
defpred S1[ Ordinal] means d c= ConsecutiveDelta2 (q,$1);
A1: for O1 being Ordinal st S1[O1] holds
S1[ succ O1]
proof
let O1 be Ordinal; ::_thesis: ( S1[O1] implies S1[ succ O1] )
ConsecutiveDelta2 (q,(succ O1)) = new_bi_fun2 ((BiFun ((ConsecutiveDelta2 (q,O1)),(ConsecutiveSet2 (A,O1)),L)),(Quadr2 (q,O1))) by Th19
.= new_bi_fun2 ((ConsecutiveDelta2 (q,O1)),(Quadr2 (q,O1))) by LATTICE5:def_15 ;
then A2: ConsecutiveDelta2 (q,O1) c= ConsecutiveDelta2 (q,(succ O1)) by Th13;
assume d c= ConsecutiveDelta2 (q,O1) ; ::_thesis: S1[ succ O1]
hence S1[ succ O1] by A2, XBOOLE_1:1; ::_thesis: verum
end;
A3: for O1 being Ordinal st O1 <> {} & O1 is limit_ordinal & ( for O2 being Ordinal st O2 in O1 holds
S1[O2] ) holds
S1[O1]
proof
deffunc H1( Ordinal) -> BiFunction of (ConsecutiveSet2 (A,$1)),L = ConsecutiveDelta2 (q,$1);
let O2 be Ordinal; ::_thesis: ( O2 <> {} & O2 is limit_ordinal & ( for O2 being Ordinal st O2 in O2 holds
S1[O2] ) implies S1[O2] )
assume that
A4: O2 <> {} and
A5: O2 is limit_ordinal and
for O1 being Ordinal st O1 in O2 holds
d c= ConsecutiveDelta2 (q,O1) ; ::_thesis: S1[O2]
A6: {} in O2 by A4, ORDINAL3:8;
consider Ls being T-Sequence such that
A7: ( dom Ls = O2 & ( for O1 being Ordinal st O1 in O2 holds
Ls . O1 = H1(O1) ) ) from ORDINAL2:sch_2();
Ls . {} = ConsecutiveDelta2 (q,{}) by A4, A7, ORDINAL3:8
.= d by Th18 ;
then A8: d in rng Ls by A7, A6, FUNCT_1:def_3;
ConsecutiveDelta2 (q,O2) = union (rng Ls) by A4, A5, A7, Th20;
hence S1[O2] by A8, ZFMISC_1:74; ::_thesis: verum
end;
A9: S1[ {} ] by Th18;
for O being Ordinal holds S1[O] from ORDINAL2:sch_1(A9, A1, A3);
hence d c= ConsecutiveDelta2 (q,O) ; ::_thesis: verum
end;
theorem Th24: :: LATTICE8:24
for A being non empty set
for L being lower-bounded LATTICE
for d being BiFunction of A,L
for O1, O2 being Ordinal
for q being QuadrSeq of d st O1 c= O2 holds
ConsecutiveDelta2 (q,O1) c= ConsecutiveDelta2 (q,O2)
proof
let A be non empty set ; ::_thesis: for L being lower-bounded LATTICE
for d being BiFunction of A,L
for O1, O2 being Ordinal
for q being QuadrSeq of d st O1 c= O2 holds
ConsecutiveDelta2 (q,O1) c= ConsecutiveDelta2 (q,O2)
let L be lower-bounded LATTICE; ::_thesis: for d being BiFunction of A,L
for O1, O2 being Ordinal
for q being QuadrSeq of d st O1 c= O2 holds
ConsecutiveDelta2 (q,O1) c= ConsecutiveDelta2 (q,O2)
let d be BiFunction of A,L; ::_thesis: for O1, O2 being Ordinal
for q being QuadrSeq of d st O1 c= O2 holds
ConsecutiveDelta2 (q,O1) c= ConsecutiveDelta2 (q,O2)
let O1, O2 be Ordinal; ::_thesis: for q being QuadrSeq of d st O1 c= O2 holds
ConsecutiveDelta2 (q,O1) c= ConsecutiveDelta2 (q,O2)
let q be QuadrSeq of d; ::_thesis: ( O1 c= O2 implies ConsecutiveDelta2 (q,O1) c= ConsecutiveDelta2 (q,O2) )
defpred S1[ Ordinal] means ( O1 c= $1 implies ConsecutiveDelta2 (q,O1) c= ConsecutiveDelta2 (q,$1) );
A1: for O1 being Ordinal st O1 <> {} & O1 is limit_ordinal & ( for O2 being Ordinal st O2 in O1 holds
S1[O2] ) holds
S1[O1]
proof
deffunc H1( Ordinal) -> BiFunction of (ConsecutiveSet2 (A,$1)),L = ConsecutiveDelta2 (q,$1);
let O2 be Ordinal; ::_thesis: ( O2 <> {} & O2 is limit_ordinal & ( for O2 being Ordinal st O2 in O2 holds
S1[O2] ) implies S1[O2] )
assume that
A2: ( O2 <> {} & O2 is limit_ordinal ) and
for O3 being Ordinal st O3 in O2 & O1 c= O3 holds
ConsecutiveDelta2 (q,O1) c= ConsecutiveDelta2 (q,O3) ; ::_thesis: S1[O2]
consider L being T-Sequence such that
A3: ( dom L = O2 & ( for O3 being Ordinal st O3 in O2 holds
L . O3 = H1(O3) ) ) from ORDINAL2:sch_2();
A4: ConsecutiveDelta2 (q,O2) = union (rng L) by A2, A3, Th20;
assume A5: O1 c= O2 ; ::_thesis: ConsecutiveDelta2 (q,O1) c= ConsecutiveDelta2 (q,O2)
percases ( O1 = O2 or O1 <> O2 ) ;
suppose O1 = O2 ; ::_thesis: ConsecutiveDelta2 (q,O1) c= ConsecutiveDelta2 (q,O2)
hence ConsecutiveDelta2 (q,O1) c= ConsecutiveDelta2 (q,O2) ; ::_thesis: verum
end;
suppose O1 <> O2 ; ::_thesis: ConsecutiveDelta2 (q,O1) c= ConsecutiveDelta2 (q,O2)
then A6: O1 c< O2 by A5, XBOOLE_0:def_8;
then O1 in O2 by ORDINAL1:11;
then A7: L . O1 in rng L by A3, FUNCT_1:def_3;
L . O1 = ConsecutiveDelta2 (q,O1) by A3, A6, ORDINAL1:11;
hence ConsecutiveDelta2 (q,O1) c= ConsecutiveDelta2 (q,O2) by A4, A7, ZFMISC_1:74; ::_thesis: verum
end;
end;
end;
A8: for O1 being Ordinal st S1[O1] holds
S1[ succ O1]
proof
let O2 be Ordinal; ::_thesis: ( S1[O2] implies S1[ succ O2] )
assume A9: ( O1 c= O2 implies ConsecutiveDelta2 (q,O1) c= ConsecutiveDelta2 (q,O2) ) ; ::_thesis: S1[ succ O2]
assume A10: O1 c= succ O2 ; ::_thesis: ConsecutiveDelta2 (q,O1) c= ConsecutiveDelta2 (q,(succ O2))
percases ( O1 = succ O2 or O1 <> succ O2 ) ;
suppose O1 = succ O2 ; ::_thesis: ConsecutiveDelta2 (q,O1) c= ConsecutiveDelta2 (q,(succ O2))
hence ConsecutiveDelta2 (q,O1) c= ConsecutiveDelta2 (q,(succ O2)) ; ::_thesis: verum
end;
supposeA11: O1 <> succ O2 ; ::_thesis: ConsecutiveDelta2 (q,O1) c= ConsecutiveDelta2 (q,(succ O2))
ConsecutiveDelta2 (q,(succ O2)) = new_bi_fun2 ((BiFun ((ConsecutiveDelta2 (q,O2)),(ConsecutiveSet2 (A,O2)),L)),(Quadr2 (q,O2))) by Th19
.= new_bi_fun2 ((ConsecutiveDelta2 (q,O2)),(Quadr2 (q,O2))) by LATTICE5:def_15 ;
then A12: ConsecutiveDelta2 (q,O2) c= ConsecutiveDelta2 (q,(succ O2)) by Th13;
O1 c< succ O2 by A10, A11, XBOOLE_0:def_8;
then O1 in succ O2 by ORDINAL1:11;
hence ConsecutiveDelta2 (q,O1) c= ConsecutiveDelta2 (q,(succ O2)) by A9, A12, ORDINAL1:22, XBOOLE_1:1; ::_thesis: verum
end;
end;
end;
A13: S1[ {} ] ;
for O being Ordinal holds S1[O] from ORDINAL2:sch_1(A13, A8, A1);
hence ( O1 c= O2 implies ConsecutiveDelta2 (q,O1) c= ConsecutiveDelta2 (q,O2) ) ; ::_thesis: verum
end;
theorem Th25: :: LATTICE8:25
for A being non empty set
for L being lower-bounded LATTICE
for d being BiFunction of A,L st d is zeroed holds
for q being QuadrSeq of d
for O being Ordinal holds ConsecutiveDelta2 (q,O) is zeroed
proof
let A be non empty set ; ::_thesis: for L being lower-bounded LATTICE
for d being BiFunction of A,L st d is zeroed holds
for q being QuadrSeq of d
for O being Ordinal holds ConsecutiveDelta2 (q,O) is zeroed
let L be lower-bounded LATTICE; ::_thesis: for d being BiFunction of A,L st d is zeroed holds
for q being QuadrSeq of d
for O being Ordinal holds ConsecutiveDelta2 (q,O) is zeroed
let d be BiFunction of A,L; ::_thesis: ( d is zeroed implies for q being QuadrSeq of d
for O being Ordinal holds ConsecutiveDelta2 (q,O) is zeroed )
assume A1: d is zeroed ; ::_thesis: for q being QuadrSeq of d
for O being Ordinal holds ConsecutiveDelta2 (q,O) is zeroed
let q be QuadrSeq of d; ::_thesis: for O being Ordinal holds ConsecutiveDelta2 (q,O) is zeroed
let O be Ordinal; ::_thesis: ConsecutiveDelta2 (q,O) is zeroed
defpred S1[ Ordinal] means ConsecutiveDelta2 (q,$1) is zeroed ;
A2: for O1 being Ordinal st S1[O1] holds
S1[ succ O1]
proof
let O1 be Ordinal; ::_thesis: ( S1[O1] implies S1[ succ O1] )
assume ConsecutiveDelta2 (q,O1) is zeroed ; ::_thesis: S1[ succ O1]
then A3: new_bi_fun2 ((ConsecutiveDelta2 (q,O1)),(Quadr2 (q,O1))) is zeroed by Th10;
let z be Element of ConsecutiveSet2 (A,(succ O1)); :: according to LATTICE5:def_6 ::_thesis: (ConsecutiveDelta2 (q,(succ O1))) . (z,z) = Bottom L
reconsider z9 = z as Element of new_set2 (ConsecutiveSet2 (A,O1)) by Th15;
ConsecutiveDelta2 (q,(succ O1)) = new_bi_fun2 ((BiFun ((ConsecutiveDelta2 (q,O1)),(ConsecutiveSet2 (A,O1)),L)),(Quadr2 (q,O1))) by Th19
.= new_bi_fun2 ((ConsecutiveDelta2 (q,O1)),(Quadr2 (q,O1))) by LATTICE5:def_15 ;
hence (ConsecutiveDelta2 (q,(succ O1))) . (z,z) = (new_bi_fun2 ((ConsecutiveDelta2 (q,O1)),(Quadr2 (q,O1)))) . (z9,z9)
.= Bottom L by A3, LATTICE5:def_6 ;
::_thesis: verum
end;
A4: for O1 being Ordinal st O1 <> {} & O1 is limit_ordinal & ( for O2 being Ordinal st O2 in O1 holds
S1[O2] ) holds
S1[O1]
proof
deffunc H1( Ordinal) -> BiFunction of (ConsecutiveSet2 (A,$1)),L = ConsecutiveDelta2 (q,$1);
let O2 be Ordinal; ::_thesis: ( O2 <> {} & O2 is limit_ordinal & ( for O2 being Ordinal st O2 in O2 holds
S1[O2] ) implies S1[O2] )
assume that
A5: ( O2 <> {} & O2 is limit_ordinal ) and
A6: for O1 being Ordinal st O1 in O2 holds
ConsecutiveDelta2 (q,O1) is zeroed ; ::_thesis: S1[O2]
set CS = ConsecutiveSet2 (A,O2);
consider Ls being T-Sequence such that
A7: ( dom Ls = O2 & ( for O1 being Ordinal st O1 in O2 holds
Ls . O1 = H1(O1) ) ) from ORDINAL2:sch_2();
ConsecutiveDelta2 (q,O2) = union (rng Ls) by A5, A7, Th20;
then reconsider f = union (rng Ls) as BiFunction of (ConsecutiveSet2 (A,O2)),L ;
deffunc H2( Ordinal) -> set = ConsecutiveSet2 (A,$1);
consider Ts being T-Sequence such that
A8: ( dom Ts = O2 & ( for O1 being Ordinal st O1 in O2 holds
Ts . O1 = H2(O1) ) ) from ORDINAL2:sch_2();
A9: ConsecutiveSet2 (A,O2) = union (rng Ts) by A5, A8, Th16;
f is zeroed
proof
let x be Element of ConsecutiveSet2 (A,O2); :: according to LATTICE5:def_6 ::_thesis: f . (x,x) = Bottom L
consider y being set such that
A10: x in y and
A11: y in rng Ts by A9, TARSKI:def_4;
consider o being set such that
A12: o in dom Ts and
A13: y = Ts . o by A11, FUNCT_1:def_3;
reconsider o = o as Ordinal by A12;
A14: Ls . o = ConsecutiveDelta2 (q,o) by A7, A8, A12;
then reconsider h = Ls . o as BiFunction of (ConsecutiveSet2 (A,o)),L ;
reconsider x9 = x as Element of ConsecutiveSet2 (A,o) by A8, A10, A12, A13;
A15: dom h = [:(ConsecutiveSet2 (A,o)),(ConsecutiveSet2 (A,o)):] by FUNCT_2:def_1;
A16: h is zeroed
proof
let z be Element of ConsecutiveSet2 (A,o); :: according to LATTICE5:def_6 ::_thesis: h . (z,z) = Bottom L
A17: ConsecutiveDelta2 (q,o) is zeroed by A6, A8, A12;
thus h . (z,z) = (ConsecutiveDelta2 (q,o)) . (z,z) by A7, A8, A12
.= Bottom L by A17, LATTICE5:def_6 ; ::_thesis: verum
end;
ConsecutiveDelta2 (q,o) in rng Ls by A7, A8, A12, A14, FUNCT_1:def_3;
then A18: h c= f by A14, ZFMISC_1:74;
x in ConsecutiveSet2 (A,o) by A8, A10, A12, A13;
then [x,x] in dom h by A15, ZFMISC_1:87;
hence f . (x,x) = h . (x9,x9) by A18, GRFUNC_1:2
.= Bottom L by A16, LATTICE5:def_6 ;
::_thesis: verum
end;
hence S1[O2] by A5, A7, Th20; ::_thesis: verum
end;
A19: S1[ {} ]
proof
let z be Element of ConsecutiveSet2 (A,{}); :: according to LATTICE5:def_6 ::_thesis: (ConsecutiveDelta2 (q,{})) . (z,z) = Bottom L
reconsider z9 = z as Element of A by Th14;
thus (ConsecutiveDelta2 (q,{})) . (z,z) = d . (z9,z9) by Th18
.= Bottom L by A1, LATTICE5:def_6 ; ::_thesis: verum
end;
for O being Ordinal holds S1[O] from ORDINAL2:sch_1(A19, A2, A4);
hence ConsecutiveDelta2 (q,O) is zeroed ; ::_thesis: verum
end;
theorem Th26: :: LATTICE8:26
for A being non empty set
for L being lower-bounded LATTICE
for d being BiFunction of A,L st d is symmetric holds
for q being QuadrSeq of d
for O being Ordinal holds ConsecutiveDelta2 (q,O) is symmetric
proof
let A be non empty set ; ::_thesis: for L being lower-bounded LATTICE
for d being BiFunction of A,L st d is symmetric holds
for q being QuadrSeq of d
for O being Ordinal holds ConsecutiveDelta2 (q,O) is symmetric
let L be lower-bounded LATTICE; ::_thesis: for d being BiFunction of A,L st d is symmetric holds
for q being QuadrSeq of d
for O being Ordinal holds ConsecutiveDelta2 (q,O) is symmetric
let d be BiFunction of A,L; ::_thesis: ( d is symmetric implies for q being QuadrSeq of d
for O being Ordinal holds ConsecutiveDelta2 (q,O) is symmetric )
assume A1: d is symmetric ; ::_thesis: for q being QuadrSeq of d
for O being Ordinal holds ConsecutiveDelta2 (q,O) is symmetric
let q be QuadrSeq of d; ::_thesis: for O being Ordinal holds ConsecutiveDelta2 (q,O) is symmetric
let O be Ordinal; ::_thesis: ConsecutiveDelta2 (q,O) is symmetric
defpred S1[ Ordinal] means ConsecutiveDelta2 (q,$1) is symmetric ;
A2: for O1 being Ordinal st S1[O1] holds
S1[ succ O1]
proof
let O1 be Ordinal; ::_thesis: ( S1[O1] implies S1[ succ O1] )
assume ConsecutiveDelta2 (q,O1) is symmetric ; ::_thesis: S1[ succ O1]
then A3: new_bi_fun2 ((ConsecutiveDelta2 (q,O1)),(Quadr2 (q,O1))) is symmetric by Th11;
let x, y be Element of ConsecutiveSet2 (A,(succ O1)); :: according to LATTICE5:def_5 ::_thesis: (ConsecutiveDelta2 (q,(succ O1))) . (x,y) = (ConsecutiveDelta2 (q,(succ O1))) . (y,x)
reconsider x9 = x, y9 = y as Element of new_set2 (ConsecutiveSet2 (A,O1)) by Th15;
A4: ConsecutiveDelta2 (q,(succ O1)) = new_bi_fun2 ((BiFun ((ConsecutiveDelta2 (q,O1)),(ConsecutiveSet2 (A,O1)),L)),(Quadr2 (q,O1))) by Th19
.= new_bi_fun2 ((ConsecutiveDelta2 (q,O1)),(Quadr2 (q,O1))) by LATTICE5:def_15 ;
hence (ConsecutiveDelta2 (q,(succ O1))) . (x,y) = (new_bi_fun2 ((ConsecutiveDelta2 (q,O1)),(Quadr2 (q,O1)))) . (y9,x9) by A3, LATTICE5:def_5
.= (ConsecutiveDelta2 (q,(succ O1))) . (y,x) by A4 ;
::_thesis: verum
end;
A5: for O1 being Ordinal st O1 <> {} & O1 is limit_ordinal & ( for O2 being Ordinal st O2 in O1 holds
S1[O2] ) holds
S1[O1]
proof
deffunc H1( Ordinal) -> BiFunction of (ConsecutiveSet2 (A,$1)),L = ConsecutiveDelta2 (q,$1);
let O2 be Ordinal; ::_thesis: ( O2 <> {} & O2 is limit_ordinal & ( for O2 being Ordinal st O2 in O2 holds
S1[O2] ) implies S1[O2] )
assume that
A6: ( O2 <> {} & O2 is limit_ordinal ) and
A7: for O1 being Ordinal st O1 in O2 holds
ConsecutiveDelta2 (q,O1) is symmetric ; ::_thesis: S1[O2]
set CS = ConsecutiveSet2 (A,O2);
consider Ls being T-Sequence such that
A8: ( dom Ls = O2 & ( for O1 being Ordinal st O1 in O2 holds
Ls . O1 = H1(O1) ) ) from ORDINAL2:sch_2();
ConsecutiveDelta2 (q,O2) = union (rng Ls) by A6, A8, Th20;
then reconsider f = union (rng Ls) as BiFunction of (ConsecutiveSet2 (A,O2)),L ;
deffunc H2( Ordinal) -> set = ConsecutiveSet2 (A,$1);
consider Ts being T-Sequence such that
A9: ( dom Ts = O2 & ( for O1 being Ordinal st O1 in O2 holds
Ts . O1 = H2(O1) ) ) from ORDINAL2:sch_2();
A10: ConsecutiveSet2 (A,O2) = union (rng Ts) by A6, A9, Th16;
f is symmetric
proof
let x, y be Element of ConsecutiveSet2 (A,O2); :: according to LATTICE5:def_5 ::_thesis: f . (x,y) = f . (y,x)
consider x1 being set such that
A11: x in x1 and
A12: x1 in rng Ts by A10, TARSKI:def_4;
consider o1 being set such that
A13: o1 in dom Ts and
A14: x1 = Ts . o1 by A12, FUNCT_1:def_3;
consider y1 being set such that
A15: y in y1 and
A16: y1 in rng Ts by A10, TARSKI:def_4;
consider o2 being set such that
A17: o2 in dom Ts and
A18: y1 = Ts . o2 by A16, FUNCT_1:def_3;
reconsider o1 = o1, o2 = o2 as Ordinal by A13, A17;
A19: x in ConsecutiveSet2 (A,o1) by A9, A11, A13, A14;
A20: Ls . o1 = ConsecutiveDelta2 (q,o1) by A8, A9, A13;
then reconsider h1 = Ls . o1 as BiFunction of (ConsecutiveSet2 (A,o1)),L ;
A21: h1 is symmetric
proof
let x, y be Element of ConsecutiveSet2 (A,o1); :: according to LATTICE5:def_5 ::_thesis: h1 . (x,y) = h1 . (y,x)
A22: ConsecutiveDelta2 (q,o1) is symmetric by A7, A9, A13;
thus h1 . (x,y) = (ConsecutiveDelta2 (q,o1)) . (x,y) by A8, A9, A13
.= (ConsecutiveDelta2 (q,o1)) . (y,x) by A22, LATTICE5:def_5
.= h1 . (y,x) by A8, A9, A13 ; ::_thesis: verum
end;
A23: dom h1 = [:(ConsecutiveSet2 (A,o1)),(ConsecutiveSet2 (A,o1)):] by FUNCT_2:def_1;
A24: y in ConsecutiveSet2 (A,o2) by A9, A15, A17, A18;
A25: Ls . o2 = ConsecutiveDelta2 (q,o2) by A8, A9, A17;
then reconsider h2 = Ls . o2 as BiFunction of (ConsecutiveSet2 (A,o2)),L ;
A26: h2 is symmetric
proof
let x, y be Element of ConsecutiveSet2 (A,o2); :: according to LATTICE5:def_5 ::_thesis: h2 . (x,y) = h2 . (y,x)
A27: ConsecutiveDelta2 (q,o2) is symmetric by A7, A9, A17;
thus h2 . (x,y) = (ConsecutiveDelta2 (q,o2)) . (x,y) by A8, A9, A17
.= (ConsecutiveDelta2 (q,o2)) . (y,x) by A27, LATTICE5:def_5
.= h2 . (y,x) by A8, A9, A17 ; ::_thesis: verum
end;
A28: dom h2 = [:(ConsecutiveSet2 (A,o2)),(ConsecutiveSet2 (A,o2)):] by FUNCT_2:def_1;
percases ( o1 c= o2 or o2 c= o1 ) ;
suppose o1 c= o2 ; ::_thesis: f . (x,y) = f . (y,x)
then A29: ConsecutiveSet2 (A,o1) c= ConsecutiveSet2 (A,o2) by Th21;
then A30: [y,x] in dom h2 by A19, A24, A28, ZFMISC_1:87;
ConsecutiveDelta2 (q,o2) in rng Ls by A8, A9, A17, A25, FUNCT_1:def_3;
then A31: h2 c= f by A25, ZFMISC_1:74;
reconsider x9 = x, y9 = y as Element of ConsecutiveSet2 (A,o2) by A9, A15, A17, A18, A19, A29;
[x,y] in dom h2 by A19, A24, A28, A29, ZFMISC_1:87;
hence f . (x,y) = h2 . (x9,y9) by A31, GRFUNC_1:2
.= h2 . (y9,x9) by A26, LATTICE5:def_5
.= f . (y,x) by A31, A30, GRFUNC_1:2 ;
::_thesis: verum
end;
suppose o2 c= o1 ; ::_thesis: f . (x,y) = f . (y,x)
then A32: ConsecutiveSet2 (A,o2) c= ConsecutiveSet2 (A,o1) by Th21;
then A33: [y,x] in dom h1 by A19, A24, A23, ZFMISC_1:87;
ConsecutiveDelta2 (q,o1) in rng Ls by A8, A9, A13, A20, FUNCT_1:def_3;
then A34: h1 c= f by A20, ZFMISC_1:74;
reconsider x9 = x, y9 = y as Element of ConsecutiveSet2 (A,o1) by A9, A11, A13, A14, A24, A32;
[x,y] in dom h1 by A19, A24, A23, A32, ZFMISC_1:87;
hence f . (x,y) = h1 . (x9,y9) by A34, GRFUNC_1:2
.= h1 . (y9,x9) by A21, LATTICE5:def_5
.= f . (y,x) by A34, A33, GRFUNC_1:2 ;
::_thesis: verum
end;
end;
end;
hence S1[O2] by A6, A8, Th20; ::_thesis: verum
end;
A35: S1[ {} ]
proof
let x, y be Element of ConsecutiveSet2 (A,{}); :: according to LATTICE5:def_5 ::_thesis: (ConsecutiveDelta2 (q,{})) . (x,y) = (ConsecutiveDelta2 (q,{})) . (y,x)
reconsider x9 = x, y9 = y as Element of A by Th14;
thus (ConsecutiveDelta2 (q,{})) . (x,y) = d . (x9,y9) by Th18
.= d . (y9,x9) by A1, LATTICE5:def_5
.= (ConsecutiveDelta2 (q,{})) . (y,x) by Th18 ; ::_thesis: verum
end;
for O being Ordinal holds S1[O] from ORDINAL2:sch_1(A35, A2, A5);
hence ConsecutiveDelta2 (q,O) is symmetric ; ::_thesis: verum
end;
theorem Th27: :: LATTICE8:27
for A being non empty set
for L being lower-bounded LATTICE st L is modular holds
for d being BiFunction of A,L st d is symmetric & d is u.t.i. holds
for O being Ordinal
for q being QuadrSeq of d st O c= DistEsti d holds
ConsecutiveDelta2 (q,O) is u.t.i.
proof
let A be non empty set ; ::_thesis: for L being lower-bounded LATTICE st L is modular holds
for d being BiFunction of A,L st d is symmetric & d is u.t.i. holds
for O being Ordinal
for q being QuadrSeq of d st O c= DistEsti d holds
ConsecutiveDelta2 (q,O) is u.t.i.
let L be lower-bounded LATTICE; ::_thesis: ( L is modular implies for d being BiFunction of A,L st d is symmetric & d is u.t.i. holds
for O being Ordinal
for q being QuadrSeq of d st O c= DistEsti d holds
ConsecutiveDelta2 (q,O) is u.t.i. )
assume A1: L is modular ; ::_thesis: for d being BiFunction of A,L st d is symmetric & d is u.t.i. holds
for O being Ordinal
for q being QuadrSeq of d st O c= DistEsti d holds
ConsecutiveDelta2 (q,O) is u.t.i.
let d be BiFunction of A,L; ::_thesis: ( d is symmetric & d is u.t.i. implies for O being Ordinal
for q being QuadrSeq of d st O c= DistEsti d holds
ConsecutiveDelta2 (q,O) is u.t.i. )
assume that
A2: d is symmetric and
A3: d is u.t.i. ; ::_thesis: for O being Ordinal
for q being QuadrSeq of d st O c= DistEsti d holds
ConsecutiveDelta2 (q,O) is u.t.i.
let O be Ordinal; ::_thesis: for q being QuadrSeq of d st O c= DistEsti d holds
ConsecutiveDelta2 (q,O) is u.t.i.
let q be QuadrSeq of d; ::_thesis: ( O c= DistEsti d implies ConsecutiveDelta2 (q,O) is u.t.i. )
defpred S1[ Ordinal] means ( $1 c= DistEsti d implies ConsecutiveDelta2 (q,$1) is u.t.i. );
A4: for O1 being Ordinal st S1[O1] holds
S1[ succ O1]
proof
let O1 be Ordinal; ::_thesis: ( S1[O1] implies S1[ succ O1] )
assume that
A5: ( O1 c= DistEsti d implies ConsecutiveDelta2 (q,O1) is u.t.i. ) and
A6: succ O1 c= DistEsti d ; ::_thesis: ConsecutiveDelta2 (q,(succ O1)) is u.t.i.
A7: O1 in DistEsti d by A6, ORDINAL1:21;
then A8: O1 in dom q by LATTICE5:25;
then q . O1 in rng q by FUNCT_1:def_3;
then A9: q . O1 in { [u,v,a9,b9] where u, v is Element of A, a9, b9 is Element of L : d . (u,v) <= a9 "\/" b9 } by LATTICE5:def_13;
let x, y, z be Element of ConsecutiveSet2 (A,(succ O1)); :: according to LATTICE5:def_7 ::_thesis: (ConsecutiveDelta2 (q,(succ O1))) . (x,z) <= ((ConsecutiveDelta2 (q,(succ O1))) . (x,y)) "\/" ((ConsecutiveDelta2 (q,(succ O1))) . (y,z))
A10: ConsecutiveDelta2 (q,O1) is symmetric by A2, Th26;
reconsider x9 = x, y9 = y, z9 = z as Element of new_set2 (ConsecutiveSet2 (A,O1)) by Th15;
set f = new_bi_fun2 ((ConsecutiveDelta2 (q,O1)),(Quadr2 (q,O1)));
set X = (Quadr2 (q,O1)) `1_4 ;
set Y = (Quadr2 (q,O1)) `2_4 ;
reconsider a = (Quadr2 (q,O1)) `3_4 , b = (Quadr2 (q,O1)) `4_4 as Element of L ;
A11: ( dom d = [:A,A:] & d c= ConsecutiveDelta2 (q,O1) ) by Th23, FUNCT_2:def_1;
consider u, v being Element of A, a9, b9 being Element of L such that
A12: q . O1 = [u,v,a9,b9] and
A13: d . (u,v) <= a9 "\/" b9 by A9;
A14: Quadr2 (q,O1) = [u,v,a9,b9] by A8, A12, Def6;
then A15: ( u = (Quadr2 (q,O1)) `1_4 & v = (Quadr2 (q,O1)) `2_4 ) by MCART_1:def_8, MCART_1:def_9;
A16: ( a9 = a & b9 = b ) by A14, MCART_1:def_10, MCART_1:def_11;
d . (u,v) = d . [u,v]
.= (ConsecutiveDelta2 (q,O1)) . (((Quadr2 (q,O1)) `1_4),((Quadr2 (q,O1)) `2_4)) by A15, A11, GRFUNC_1:2 ;
then new_bi_fun2 ((ConsecutiveDelta2 (q,O1)),(Quadr2 (q,O1))) is u.t.i. by A1, A5, A7, A10, A13, A16, Th12, ORDINAL1:def_2;
then A17: (new_bi_fun2 ((ConsecutiveDelta2 (q,O1)),(Quadr2 (q,O1)))) . (x9,z9) <= ((new_bi_fun2 ((ConsecutiveDelta2 (q,O1)),(Quadr2 (q,O1)))) . (x9,y9)) "\/" ((new_bi_fun2 ((ConsecutiveDelta2 (q,O1)),(Quadr2 (q,O1)))) . (y9,z9)) by LATTICE5:def_7;
ConsecutiveDelta2 (q,(succ O1)) = new_bi_fun2 ((BiFun ((ConsecutiveDelta2 (q,O1)),(ConsecutiveSet2 (A,O1)),L)),(Quadr2 (q,O1))) by Th19
.= new_bi_fun2 ((ConsecutiveDelta2 (q,O1)),(Quadr2 (q,O1))) by LATTICE5:def_15 ;
hence (ConsecutiveDelta2 (q,(succ O1))) . (x,z) <= ((ConsecutiveDelta2 (q,(succ O1))) . (x,y)) "\/" ((ConsecutiveDelta2 (q,(succ O1))) . (y,z)) by A17; ::_thesis: verum
end;
A18: for O1 being Ordinal st O1 <> {} & O1 is limit_ordinal & ( for O2 being Ordinal st O2 in O1 holds
S1[O2] ) holds
S1[O1]
proof
deffunc H1( Ordinal) -> BiFunction of (ConsecutiveSet2 (A,$1)),L = ConsecutiveDelta2 (q,$1);
let O2 be Ordinal; ::_thesis: ( O2 <> {} & O2 is limit_ordinal & ( for O2 being Ordinal st O2 in O2 holds
S1[O2] ) implies S1[O2] )
assume that
A19: ( O2 <> {} & O2 is limit_ordinal ) and
A20: for O1 being Ordinal st O1 in O2 & O1 c= DistEsti d holds
ConsecutiveDelta2 (q,O1) is u.t.i. and
A21: O2 c= DistEsti d ; ::_thesis: ConsecutiveDelta2 (q,O2) is u.t.i.
set CS = ConsecutiveSet2 (A,O2);
consider Ls being T-Sequence such that
A22: ( dom Ls = O2 & ( for O1 being Ordinal st O1 in O2 holds
Ls . O1 = H1(O1) ) ) from ORDINAL2:sch_2();
ConsecutiveDelta2 (q,O2) = union (rng Ls) by A19, A22, Th20;
then reconsider f = union (rng Ls) as BiFunction of (ConsecutiveSet2 (A,O2)),L ;
deffunc H2( Ordinal) -> set = ConsecutiveSet2 (A,$1);
consider Ts being T-Sequence such that
A23: ( dom Ts = O2 & ( for O1 being Ordinal st O1 in O2 holds
Ts . O1 = H2(O1) ) ) from ORDINAL2:sch_2();
A24: ConsecutiveSet2 (A,O2) = union (rng Ts) by A19, A23, Th16;
f is u.t.i.
proof
let x, y, z be Element of ConsecutiveSet2 (A,O2); :: according to LATTICE5:def_7 ::_thesis: f . (x,z) <= (f . (x,y)) "\/" (f . (y,z))
consider X being set such that
A25: x in X and
A26: X in rng Ts by A24, TARSKI:def_4;
consider o1 being set such that
A27: o1 in dom Ts and
A28: X = Ts . o1 by A26, FUNCT_1:def_3;
consider Y being set such that
A29: y in Y and
A30: Y in rng Ts by A24, TARSKI:def_4;
consider o2 being set such that
A31: o2 in dom Ts and
A32: Y = Ts . o2 by A30, FUNCT_1:def_3;
consider Z being set such that
A33: z in Z and
A34: Z in rng Ts by A24, TARSKI:def_4;
consider o3 being set such that
A35: o3 in dom Ts and
A36: Z = Ts . o3 by A34, FUNCT_1:def_3;
reconsider o1 = o1, o2 = o2, o3 = o3 as Ordinal by A27, A31, A35;
A37: x in ConsecutiveSet2 (A,o1) by A23, A25, A27, A28;
A38: Ls . o3 = ConsecutiveDelta2 (q,o3) by A22, A23, A35;
then reconsider h3 = Ls . o3 as BiFunction of (ConsecutiveSet2 (A,o3)),L ;
A39: h3 is u.t.i.
proof
let x, y, z be Element of ConsecutiveSet2 (A,o3); :: according to LATTICE5:def_7 ::_thesis: h3 . (x,z) <= (h3 . (x,y)) "\/" (h3 . (y,z))
o3 c= DistEsti d by A21, A23, A35, ORDINAL1:def_2;
then A40: ConsecutiveDelta2 (q,o3) is u.t.i. by A20, A23, A35;
ConsecutiveDelta2 (q,o3) = h3 by A22, A23, A35;
hence h3 . (x,z) <= (h3 . (x,y)) "\/" (h3 . (y,z)) by A40, LATTICE5:def_7; ::_thesis: verum
end;
A41: dom h3 = [:(ConsecutiveSet2 (A,o3)),(ConsecutiveSet2 (A,o3)):] by FUNCT_2:def_1;
A42: z in ConsecutiveSet2 (A,o3) by A23, A33, A35, A36;
A43: Ls . o2 = ConsecutiveDelta2 (q,o2) by A22, A23, A31;
then reconsider h2 = Ls . o2 as BiFunction of (ConsecutiveSet2 (A,o2)),L ;
A44: h2 is u.t.i.
proof
let x, y, z be Element of ConsecutiveSet2 (A,o2); :: according to LATTICE5:def_7 ::_thesis: h2 . (x,z) <= (h2 . (x,y)) "\/" (h2 . (y,z))
o2 c= DistEsti d by A21, A23, A31, ORDINAL1:def_2;
then A45: ConsecutiveDelta2 (q,o2) is u.t.i. by A20, A23, A31;
ConsecutiveDelta2 (q,o2) = h2 by A22, A23, A31;
hence h2 . (x,z) <= (h2 . (x,y)) "\/" (h2 . (y,z)) by A45, LATTICE5:def_7; ::_thesis: verum
end;
A46: dom h2 = [:(ConsecutiveSet2 (A,o2)),(ConsecutiveSet2 (A,o2)):] by FUNCT_2:def_1;
A47: Ls . o1 = ConsecutiveDelta2 (q,o1) by A22, A23, A27;
then reconsider h1 = Ls . o1 as BiFunction of (ConsecutiveSet2 (A,o1)),L ;
A48: h1 is u.t.i.
proof
let x, y, z be Element of ConsecutiveSet2 (A,o1); :: according to LATTICE5:def_7 ::_thesis: h1 . (x,z) <= (h1 . (x,y)) "\/" (h1 . (y,z))
o1 c= DistEsti d by A21, A23, A27, ORDINAL1:def_2;
then A49: ConsecutiveDelta2 (q,o1) is u.t.i. by A20, A23, A27;
ConsecutiveDelta2 (q,o1) = h1 by A22, A23, A27;
hence h1 . (x,z) <= (h1 . (x,y)) "\/" (h1 . (y,z)) by A49, LATTICE5:def_7; ::_thesis: verum
end;
A50: dom h1 = [:(ConsecutiveSet2 (A,o1)),(ConsecutiveSet2 (A,o1)):] by FUNCT_2:def_1;
A51: y in ConsecutiveSet2 (A,o2) by A23, A29, A31, A32;
percases ( o1 c= o3 or o3 c= o1 ) ;
supposeA52: o1 c= o3 ; ::_thesis: f . (x,z) <= (f . (x,y)) "\/" (f . (y,z))
then A53: ConsecutiveSet2 (A,o1) c= ConsecutiveSet2 (A,o3) by Th21;
thus (f . (x,y)) "\/" (f . (y,z)) >= f . (x,z) ::_thesis: verum
proof
percases ( o2 c= o3 or o3 c= o2 ) ;
supposeA54: o2 c= o3 ; ::_thesis: (f . (x,y)) "\/" (f . (y,z)) >= f . (x,z)
reconsider z9 = z as Element of ConsecutiveSet2 (A,o3) by A23, A33, A35, A36;
reconsider x9 = x as Element of ConsecutiveSet2 (A,o3) by A37, A53;
ConsecutiveDelta2 (q,o3) in rng Ls by A22, A23, A35, A38, FUNCT_1:def_3;
then A55: h3 c= f by A38, ZFMISC_1:74;
A56: ConsecutiveSet2 (A,o2) c= ConsecutiveSet2 (A,o3) by A54, Th21;
then reconsider y9 = y as Element of ConsecutiveSet2 (A,o3) by A51;
[y,z] in dom h3 by A51, A42, A41, A56, ZFMISC_1:87;
then A57: f . (y,z) = h3 . (y9,z9) by A55, GRFUNC_1:2;
[x,z] in dom h3 by A37, A42, A41, A53, ZFMISC_1:87;
then A58: f . (x,z) = h3 . (x9,z9) by A55, GRFUNC_1:2;
[x,y] in dom h3 by A37, A51, A41, A53, A56, ZFMISC_1:87;
then f . (x,y) = h3 . (x9,y9) by A55, GRFUNC_1:2;
hence (f . (x,y)) "\/" (f . (y,z)) >= f . (x,z) by A39, A57, A58, LATTICE5:def_7; ::_thesis: verum
end;
supposeA59: o3 c= o2 ; ::_thesis: (f . (x,y)) "\/" (f . (y,z)) >= f . (x,z)
reconsider y9 = y as Element of ConsecutiveSet2 (A,o2) by A23, A29, A31, A32;
ConsecutiveDelta2 (q,o2) in rng Ls by A22, A23, A31, A43, FUNCT_1:def_3;
then A60: h2 c= f by A43, ZFMISC_1:74;
A61: ConsecutiveSet2 (A,o3) c= ConsecutiveSet2 (A,o2) by A59, Th21;
then reconsider z9 = z as Element of ConsecutiveSet2 (A,o2) by A42;
[y,z] in dom h2 by A51, A42, A46, A61, ZFMISC_1:87;
then A62: f . (y,z) = h2 . (y9,z9) by A60, GRFUNC_1:2;
ConsecutiveSet2 (A,o1) c= ConsecutiveSet2 (A,o3) by A52, Th21;
then A63: ConsecutiveSet2 (A,o1) c= ConsecutiveSet2 (A,o2) by A61, XBOOLE_1:1;
then reconsider x9 = x as Element of ConsecutiveSet2 (A,o2) by A37;
[x,y] in dom h2 by A37, A51, A46, A63, ZFMISC_1:87;
then A64: f . (x,y) = h2 . (x9,y9) by A60, GRFUNC_1:2;
[x,z] in dom h2 by A37, A42, A46, A61, A63, ZFMISC_1:87;
then f . (x,z) = h2 . (x9,z9) by A60, GRFUNC_1:2;
hence (f . (x,y)) "\/" (f . (y,z)) >= f . (x,z) by A44, A64, A62, LATTICE5:def_7; ::_thesis: verum
end;
end;
end;
end;
supposeA65: o3 c= o1 ; ::_thesis: f . (x,z) <= (f . (x,y)) "\/" (f . (y,z))
then A66: ConsecutiveSet2 (A,o3) c= ConsecutiveSet2 (A,o1) by Th21;
thus (f . (x,y)) "\/" (f . (y,z)) >= f . (x,z) ::_thesis: verum
proof
percases ( o1 c= o2 or o2 c= o1 ) ;
supposeA67: o1 c= o2 ; ::_thesis: (f . (x,y)) "\/" (f . (y,z)) >= f . (x,z)
reconsider y9 = y as Element of ConsecutiveSet2 (A,o2) by A23, A29, A31, A32;
ConsecutiveDelta2 (q,o2) in rng Ls by A22, A23, A31, A43, FUNCT_1:def_3;
then A68: h2 c= f by A43, ZFMISC_1:74;
A69: ConsecutiveSet2 (A,o1) c= ConsecutiveSet2 (A,o2) by A67, Th21;
then reconsider x9 = x as Element of ConsecutiveSet2 (A,o2) by A37;
[x,y] in dom h2 by A37, A51, A46, A69, ZFMISC_1:87;
then A70: f . (x,y) = h2 . (x9,y9) by A68, GRFUNC_1:2;
ConsecutiveSet2 (A,o3) c= ConsecutiveSet2 (A,o1) by A65, Th21;
then A71: ConsecutiveSet2 (A,o3) c= ConsecutiveSet2 (A,o2) by A69, XBOOLE_1:1;
then reconsider z9 = z as Element of ConsecutiveSet2 (A,o2) by A42;
[y,z] in dom h2 by A51, A42, A46, A71, ZFMISC_1:87;
then A72: f . (y,z) = h2 . (y9,z9) by A68, GRFUNC_1:2;
[x,z] in dom h2 by A37, A42, A46, A69, A71, ZFMISC_1:87;
then f . (x,z) = h2 . (x9,z9) by A68, GRFUNC_1:2;
hence (f . (x,y)) "\/" (f . (y,z)) >= f . (x,z) by A44, A70, A72, LATTICE5:def_7; ::_thesis: verum
end;
supposeA73: o2 c= o1 ; ::_thesis: (f . (x,y)) "\/" (f . (y,z)) >= f . (x,z)
reconsider x9 = x as Element of ConsecutiveSet2 (A,o1) by A23, A25, A27, A28;
reconsider z9 = z as Element of ConsecutiveSet2 (A,o1) by A42, A66;
ConsecutiveDelta2 (q,o1) in rng Ls by A22, A23, A27, A47, FUNCT_1:def_3;
then A74: h1 c= f by A47, ZFMISC_1:74;
A75: ConsecutiveSet2 (A,o2) c= ConsecutiveSet2 (A,o1) by A73, Th21;
then reconsider y9 = y as Element of ConsecutiveSet2 (A,o1) by A51;
[x,y] in dom h1 by A37, A51, A50, A75, ZFMISC_1:87;
then A76: f . (x,y) = h1 . (x9,y9) by A74, GRFUNC_1:2;
[x,z] in dom h1 by A37, A42, A50, A66, ZFMISC_1:87;
then A77: f . (x,z) = h1 . (x9,z9) by A74, GRFUNC_1:2;
[y,z] in dom h1 by A51, A42, A50, A66, A75, ZFMISC_1:87;
then f . (y,z) = h1 . (y9,z9) by A74, GRFUNC_1:2;
hence (f . (x,y)) "\/" (f . (y,z)) >= f . (x,z) by A48, A76, A77, LATTICE5:def_7; ::_thesis: verum
end;
end;
end;
end;
end;
end;
hence ConsecutiveDelta2 (q,O2) is u.t.i. by A19, A22, Th20; ::_thesis: verum
end;
A78: S1[ {} ]
proof
assume {} c= DistEsti d ; ::_thesis: ConsecutiveDelta2 (q,{}) is u.t.i.
let x, y, z be Element of ConsecutiveSet2 (A,{}); :: according to LATTICE5:def_7 ::_thesis: (ConsecutiveDelta2 (q,{})) . (x,z) <= ((ConsecutiveDelta2 (q,{})) . (x,y)) "\/" ((ConsecutiveDelta2 (q,{})) . (y,z))
reconsider x9 = x, y9 = y, z9 = z as Element of A by Th14;
( ConsecutiveDelta2 (q,{}) = d & d . (x9,z9) <= (d . (x9,y9)) "\/" (d . (y9,z9)) ) by A3, Th18, LATTICE5:def_7;
hence (ConsecutiveDelta2 (q,{})) . (x,z) <= ((ConsecutiveDelta2 (q,{})) . (x,y)) "\/" ((ConsecutiveDelta2 (q,{})) . (y,z)) ; ::_thesis: verum
end;
for O being Ordinal holds S1[O] from ORDINAL2:sch_1(A78, A4, A18);
hence ( O c= DistEsti d implies ConsecutiveDelta2 (q,O) is u.t.i. ) ; ::_thesis: verum
end;
theorem :: LATTICE8:28
for A being non empty set
for L being lower-bounded modular LATTICE
for d being distance_function of A,L
for O being Ordinal
for q being QuadrSeq of d st O c= DistEsti d holds
ConsecutiveDelta2 (q,O) is distance_function of (ConsecutiveSet2 (A,O)),L by Th25, Th26, Th27;
definition
let A be non empty set ;
let L be lower-bounded LATTICE;
let d be BiFunction of A,L;
func NextSet2 d -> set equals :: LATTICE8:def 8
ConsecutiveSet2 (A,(DistEsti d));
correctness
coherence
ConsecutiveSet2 (A,(DistEsti d)) is set ;
;
end;
:: deftheorem defines NextSet2 LATTICE8:def_8_:_
for A being non empty set
for L being lower-bounded LATTICE
for d being BiFunction of A,L holds NextSet2 d = ConsecutiveSet2 (A,(DistEsti d));
registration
let A be non empty set ;
let L be lower-bounded LATTICE;
let d be BiFunction of A,L;
cluster NextSet2 d -> non empty ;
coherence
not NextSet2 d is empty ;
end;
definition
let A be non empty set ;
let L be lower-bounded LATTICE;
let d be BiFunction of A,L;
let q be QuadrSeq of d;
func NextDelta2 q -> set equals :: LATTICE8:def 9
ConsecutiveDelta2 (q,(DistEsti d));
correctness
coherence
ConsecutiveDelta2 (q,(DistEsti d)) is set ;
;
end;
:: deftheorem defines NextDelta2 LATTICE8:def_9_:_
for A being non empty set
for L being lower-bounded LATTICE
for d being BiFunction of A,L
for q being QuadrSeq of d holds NextDelta2 q = ConsecutiveDelta2 (q,(DistEsti d));
definition
let A be non empty set ;
let L be lower-bounded modular LATTICE;
let d be distance_function of A,L;
let q be QuadrSeq of d;
:: original: NextDelta2
redefine func NextDelta2 q -> distance_function of (NextSet2 d),L;
coherence
NextDelta2 q is distance_function of (NextSet2 d),L by Th25, Th26, Th27;
end;
definition
let A be non empty set ;
let L be lower-bounded LATTICE;
let d be distance_function of A,L;
let Aq be non empty set ;
let dq be distance_function of Aq,L;
predAq,dq is_extension2_of A,d means :Def10: :: LATTICE8:def 10
ex q being QuadrSeq of d st
( Aq = NextSet2 d & dq = NextDelta2 q );
end;
:: deftheorem Def10 defines is_extension2_of LATTICE8:def_10_:_
for A being non empty set
for L being lower-bounded LATTICE
for d being distance_function of A,L
for Aq being non empty set
for dq being distance_function of Aq,L holds
( Aq,dq is_extension2_of A,d iff ex q being QuadrSeq of d st
( Aq = NextSet2 d & dq = NextDelta2 q ) );
theorem Th29: :: LATTICE8:29
for A being non empty set
for L being lower-bounded LATTICE
for d being distance_function of A,L
for Aq being non empty set
for dq being distance_function of Aq,L st Aq,dq is_extension2_of A,d holds
for x, y being Element of A
for a, b being Element of L st d . (x,y) <= a "\/" b holds
ex z1, z2 being Element of Aq st
( dq . (x,z1) = a & dq . (z1,z2) = ((d . (x,y)) "\/" a) "/\" b & dq . (z2,y) = a )
proof
let A be non empty set ; ::_thesis: for L being lower-bounded LATTICE
for d being distance_function of A,L
for Aq being non empty set
for dq being distance_function of Aq,L st Aq,dq is_extension2_of A,d holds
for x, y being Element of A
for a, b being Element of L st d . (x,y) <= a "\/" b holds
ex z1, z2 being Element of Aq st
( dq . (x,z1) = a & dq . (z1,z2) = ((d . (x,y)) "\/" a) "/\" b & dq . (z2,y) = a )
let L be lower-bounded LATTICE; ::_thesis: for d being distance_function of A,L
for Aq being non empty set
for dq being distance_function of Aq,L st Aq,dq is_extension2_of A,d holds
for x, y being Element of A
for a, b being Element of L st d . (x,y) <= a "\/" b holds
ex z1, z2 being Element of Aq st
( dq . (x,z1) = a & dq . (z1,z2) = ((d . (x,y)) "\/" a) "/\" b & dq . (z2,y) = a )
let d be distance_function of A,L; ::_thesis: for Aq being non empty set
for dq being distance_function of Aq,L st Aq,dq is_extension2_of A,d holds
for x, y being Element of A
for a, b being Element of L st d . (x,y) <= a "\/" b holds
ex z1, z2 being Element of Aq st
( dq . (x,z1) = a & dq . (z1,z2) = ((d . (x,y)) "\/" a) "/\" b & dq . (z2,y) = a )
let Aq be non empty set ; ::_thesis: for dq being distance_function of Aq,L st Aq,dq is_extension2_of A,d holds
for x, y being Element of A
for a, b being Element of L st d . (x,y) <= a "\/" b holds
ex z1, z2 being Element of Aq st
( dq . (x,z1) = a & dq . (z1,z2) = ((d . (x,y)) "\/" a) "/\" b & dq . (z2,y) = a )
let dq be distance_function of Aq,L; ::_thesis: ( Aq,dq is_extension2_of A,d implies for x, y being Element of A
for a, b being Element of L st d . (x,y) <= a "\/" b holds
ex z1, z2 being Element of Aq st
( dq . (x,z1) = a & dq . (z1,z2) = ((d . (x,y)) "\/" a) "/\" b & dq . (z2,y) = a ) )
assume Aq,dq is_extension2_of A,d ; ::_thesis: for x, y being Element of A
for a, b being Element of L st d . (x,y) <= a "\/" b holds
ex z1, z2 being Element of Aq st
( dq . (x,z1) = a & dq . (z1,z2) = ((d . (x,y)) "\/" a) "/\" b & dq . (z2,y) = a )
then consider q being QuadrSeq of d such that
A1: Aq = NextSet2 d and
A2: dq = NextDelta2 q by Def10;
let x, y be Element of A; ::_thesis: for a, b being Element of L st d . (x,y) <= a "\/" b holds
ex z1, z2 being Element of Aq st
( dq . (x,z1) = a & dq . (z1,z2) = ((d . (x,y)) "\/" a) "/\" b & dq . (z2,y) = a )
let a, b be Element of L; ::_thesis: ( d . (x,y) <= a "\/" b implies ex z1, z2 being Element of Aq st
( dq . (x,z1) = a & dq . (z1,z2) = ((d . (x,y)) "\/" a) "/\" b & dq . (z2,y) = a ) )
A3: rng q = { [x9,y9,a9,b9] where x9, y9 is Element of A, a9, b9 is Element of L : d . (x9,y9) <= a9 "\/" b9 } by LATTICE5:def_13;
assume d . (x,y) <= a "\/" b ; ::_thesis: ex z1, z2 being Element of Aq st
( dq . (x,z1) = a & dq . (z1,z2) = ((d . (x,y)) "\/" a) "/\" b & dq . (z2,y) = a )
then [x,y,a,b] in rng q by A3;
then consider o being set such that
A4: o in dom q and
A5: q . o = [x,y,a,b] by FUNCT_1:def_3;
reconsider o = o as Ordinal by A4;
A6: q . o = Quadr2 (q,o) by A4, Def6;
then A7: x = (Quadr2 (q,o)) `1_4 by A5, MCART_1:74;
A8: b = (Quadr2 (q,o)) `4_4 by A5, A6, MCART_1:74;
A9: y = (Quadr2 (q,o)) `2_4 by A5, A6, MCART_1:74;
reconsider B = ConsecutiveSet2 (A,o) as non empty set ;
{B} in {{B},{{B}}} by TARSKI:def_2;
then A10: {B} in B \/ {{B},{{B}}} by XBOOLE_0:def_3;
o in DistEsti d by A4, LATTICE5:25;
then A11: succ o c= DistEsti d by ORDINAL1:21;
then A12: ConsecutiveDelta2 (q,(succ o)) c= ConsecutiveDelta2 (q,(DistEsti d)) by Th24;
reconsider cd = ConsecutiveDelta2 (q,o) as BiFunction of B,L ;
reconsider Q = Quadr2 (q,o) as Element of [:B,B, the carrier of L, the carrier of L:] ;
A13: ( x in A & y in A ) ;
A14: {{B}} in {{B},{{B}}} by TARSKI:def_2;
then A15: {{B}} in new_set2 B by XBOOLE_0:def_3;
ConsecutiveSet2 (A,(succ o)) = new_set2 B by Th15;
then new_set2 B c= ConsecutiveSet2 (A,(DistEsti d)) by A11, Th21;
then reconsider z1 = {B}, z2 = {{B}} as Element of Aq by A1, A10, A15;
take z1 ; ::_thesis: ex z2 being Element of Aq st
( dq . (x,z1) = a & dq . (z1,z2) = ((d . (x,y)) "\/" a) "/\" b & dq . (z2,y) = a )
take z2 ; ::_thesis: ( dq . (x,z1) = a & dq . (z1,z2) = ((d . (x,y)) "\/" a) "/\" b & dq . (z2,y) = a )
A16: cd is zeroed by Th25;
A c= B by Th17;
then reconsider xo = x, yo = y as Element of B by A13;
A17: B c= new_set2 B by XBOOLE_1:7;
( xo in B & yo in B ) ;
then reconsider x1 = xo, y1 = yo as Element of new_set2 B by A17;
A18: ConsecutiveDelta2 (q,(succ o)) = new_bi_fun2 ((BiFun ((ConsecutiveDelta2 (q,o)),(ConsecutiveSet2 (A,o)),L)),(Quadr2 (q,o))) by Th19
.= new_bi_fun2 (cd,Q) by LATTICE5:def_15 ;
dom d = [:A,A:] by FUNCT_2:def_1;
then A19: [xo,yo] in dom d by ZFMISC_1:87;
d c= cd by Th23;
then A20: cd . (xo,yo) = d . (x,y) by A19, GRFUNC_1:2;
A21: a = (Quadr2 (q,o)) `3_4 by A5, A6, MCART_1:74;
A22: dom (new_bi_fun2 (cd,Q)) = [:(new_set2 B),(new_set2 B):] by FUNCT_2:def_1;
then [x1,{B}] in dom (new_bi_fun2 (cd,Q)) by A10, ZFMISC_1:87;
hence dq . (x,z1) = (new_bi_fun2 (cd,Q)) . (x1,{B}) by A2, A12, A18, GRFUNC_1:2
.= (cd . (xo,xo)) "\/" a by A7, A21, Def4
.= (Bottom L) "\/" a by A16, LATTICE5:def_6
.= a by WAYBEL_1:3 ;
::_thesis: ( dq . (z1,z2) = ((d . (x,y)) "\/" a) "/\" b & dq . (z2,y) = a )
[{B},{{B}}] in dom (new_bi_fun2 (cd,Q)) by A10, A15, A22, ZFMISC_1:87;
hence dq . (z1,z2) = (new_bi_fun2 (cd,Q)) . ({B},{{B}}) by A2, A12, A18, GRFUNC_1:2
.= ((d . (x,y)) "\/" a) "/\" b by A7, A9, A21, A8, A20, Def4 ;
::_thesis: dq . (z2,y) = a
{{B}} in B \/ {{B},{{B}}} by A14, XBOOLE_0:def_3;
then [{{B}},y1] in dom (new_bi_fun2 (cd,Q)) by A22, ZFMISC_1:87;
hence dq . (z2,y) = (new_bi_fun2 (cd,Q)) . ({{B}},y1) by A2, A12, A18, GRFUNC_1:2
.= (cd . (yo,yo)) "\/" a by A9, A21, Def4
.= (Bottom L) "\/" a by A16, LATTICE5:def_6
.= a by WAYBEL_1:3 ;
::_thesis: verum
end;
definition
let A be non empty set ;
let L be lower-bounded modular LATTICE;
let d be distance_function of A,L;
mode ExtensionSeq2 of A,d -> Function means :Def11: :: LATTICE8:def 11
( dom it = NAT & it . 0 = [A,d] & ( for n being Element of NAT ex A9 being non empty set ex d9 being distance_function of A9,L ex Aq being non empty set ex dq being distance_function of Aq,L st
( Aq,dq is_extension2_of A9,d9 & it . n = [A9,d9] & it . (n + 1) = [Aq,dq] ) ) );
existence
ex b1 being Function st
( dom b1 = NAT & b1 . 0 = [A,d] & ( for n being Element of NAT ex A9 being non empty set ex d9 being distance_function of A9,L ex Aq being non empty set ex dq being distance_function of Aq,L st
( Aq,dq is_extension2_of A9,d9 & b1 . n = [A9,d9] & b1 . (n + 1) = [Aq,dq] ) ) )
proof
defpred S1[ set , set , set ] means ( ex A9 being non empty set ex d9 being distance_function of A9,L ex Aq being non empty set ex dq being distance_function of Aq,L st
( Aq,dq is_extension2_of A9,d9 & $2 = [A9,d9] & $3 = [Aq,dq] ) or ( $3 = 0 & ( for A9 being non empty set
for d9 being distance_function of A9,L
for Aq being non empty set
for dq being distance_function of Aq,L holds
( not Aq,dq is_extension2_of A9,d9 or not $2 = [A9,d9] ) ) ) );
A1: for n being Element of NAT
for x being set ex y being set st S1[n,x,y]
proof
let n be Element of NAT ; ::_thesis: for x being set ex y being set st S1[n,x,y]
let x be set ; ::_thesis: ex y being set st S1[n,x,y]
percases ( ex A9 being non empty set ex d9 being distance_function of A9,L ex Aq being non empty set ex dq being distance_function of Aq,L st
( Aq,dq is_extension2_of A9,d9 & x = [A9,d9] ) or for A9 being non empty set
for d9 being distance_function of A9,L
for Aq being non empty set
for dq being distance_function of Aq,L holds
( not Aq,dq is_extension2_of A9,d9 or not x = [A9,d9] ) ) ;
suppose ex A9 being non empty set ex d9 being distance_function of A9,L ex Aq being non empty set ex dq being distance_function of Aq,L st
( Aq,dq is_extension2_of A9,d9 & x = [A9,d9] ) ; ::_thesis: ex y being set st S1[n,x,y]
then consider A9 being non empty set , d9 being distance_function of A9,L, Aq being non empty set , dq being distance_function of Aq,L such that
A2: ( Aq,dq is_extension2_of A9,d9 & x = [A9,d9] ) ;
take [Aq,dq] ; ::_thesis: S1[n,x,[Aq,dq]]
thus S1[n,x,[Aq,dq]] by A2; ::_thesis: verum
end;
supposeA3: for A9 being non empty set
for d9 being distance_function of A9,L
for Aq being non empty set
for dq being distance_function of Aq,L holds
( not Aq,dq is_extension2_of A9,d9 or not x = [A9,d9] ) ; ::_thesis: ex y being set st S1[n,x,y]
take 0 ; ::_thesis: S1[n,x, 0 ]
thus S1[n,x, 0 ] by A3; ::_thesis: verum
end;
end;
end;
consider f being Function such that
A4: dom f = NAT and
A5: f . 0 = [A,d] and
A6: for n being Element of NAT holds S1[n,f . n,f . (n + 1)] from RECDEF_1:sch_1(A1);
take f ; ::_thesis: ( dom f = NAT & f . 0 = [A,d] & ( for n being Element of NAT ex A9 being non empty set ex d9 being distance_function of A9,L ex Aq being non empty set ex dq being distance_function of Aq,L st
( Aq,dq is_extension2_of A9,d9 & f . n = [A9,d9] & f . (n + 1) = [Aq,dq] ) ) )
thus dom f = NAT by A4; ::_thesis: ( f . 0 = [A,d] & ( for n being Element of NAT ex A9 being non empty set ex d9 being distance_function of A9,L ex Aq being non empty set ex dq being distance_function of Aq,L st
( Aq,dq is_extension2_of A9,d9 & f . n = [A9,d9] & f . (n + 1) = [Aq,dq] ) ) )
thus f . 0 = [A,d] by A5; ::_thesis: for n being Element of NAT ex A9 being non empty set ex d9 being distance_function of A9,L ex Aq being non empty set ex dq being distance_function of Aq,L st
( Aq,dq is_extension2_of A9,d9 & f . n = [A9,d9] & f . (n + 1) = [Aq,dq] )
defpred S2[ Element of NAT ] means ex A9 being non empty set ex d9 being distance_function of A9,L ex Aq being non empty set ex dq being distance_function of Aq,L st
( Aq,dq is_extension2_of A9,d9 & f . $1 = [A9,d9] & f . ($1 + 1) = [Aq,dq] );
A7: for k being Element of NAT st S2[k] holds
S2[k + 1]
proof
let k be Element of NAT ; ::_thesis: ( S2[k] implies S2[k + 1] )
given A9 being non empty set , d9 being distance_function of A9,L, Aq being non empty set , dq being distance_function of Aq,L such that Aq,dq is_extension2_of A9,d9 and
f . k = [A9,d9] and
A8: f . (k + 1) = [Aq,dq] ; ::_thesis: S2[k + 1]
ex A1 being non empty set ex d1 being distance_function of A1,L ex AQ being non empty set ex DQ being distance_function of AQ,L st
( AQ,DQ is_extension2_of A1,d1 & f . (k + 1) = [A1,d1] )
proof
set Q = the QuadrSeq of dq;
set AQ = NextSet2 dq;
take Aq ; ::_thesis: ex d1 being distance_function of Aq,L ex AQ being non empty set ex DQ being distance_function of AQ,L st
( AQ,DQ is_extension2_of Aq,d1 & f . (k + 1) = [Aq,d1] )
take dq ; ::_thesis: ex AQ being non empty set ex DQ being distance_function of AQ,L st
( AQ,DQ is_extension2_of Aq,dq & f . (k + 1) = [Aq,dq] )
set DQ = NextDelta2 the QuadrSeq of dq;
take NextSet2 dq ; ::_thesis: ex DQ being distance_function of (NextSet2 dq),L st
( NextSet2 dq,DQ is_extension2_of Aq,dq & f . (k + 1) = [Aq,dq] )
take NextDelta2 the QuadrSeq of dq ; ::_thesis: ( NextSet2 dq, NextDelta2 the QuadrSeq of dq is_extension2_of Aq,dq & f . (k + 1) = [Aq,dq] )
thus NextSet2 dq, NextDelta2 the QuadrSeq of dq is_extension2_of Aq,dq by Def10; ::_thesis: f . (k + 1) = [Aq,dq]
thus f . (k + 1) = [Aq,dq] by A8; ::_thesis: verum
end;
hence S2[k + 1] by A6; ::_thesis: verum
end;
ex A9 being non empty set ex d9 being distance_function of A9,L ex Aq being non empty set ex dq being distance_function of Aq,L st
( Aq,dq is_extension2_of A9,d9 & f . 0 = [A9,d9] )
proof
set Aq = NextSet2 d;
set q = the QuadrSeq of d;
take A ; ::_thesis: ex d9 being distance_function of A,L ex Aq being non empty set ex dq being distance_function of Aq,L st
( Aq,dq is_extension2_of A,d9 & f . 0 = [A,d9] )
take d ; ::_thesis: ex Aq being non empty set ex dq being distance_function of Aq,L st
( Aq,dq is_extension2_of A,d & f . 0 = [A,d] )
consider dq being distance_function of (NextSet2 d),L such that
A9: dq = NextDelta2 the QuadrSeq of d ;
take NextSet2 d ; ::_thesis: ex dq being distance_function of (NextSet2 d),L st
( NextSet2 d,dq is_extension2_of A,d & f . 0 = [A,d] )
take dq ; ::_thesis: ( NextSet2 d,dq is_extension2_of A,d & f . 0 = [A,d] )
thus NextSet2 d,dq is_extension2_of A,d by A9, Def10; ::_thesis: f . 0 = [A,d]
thus f . 0 = [A,d] by A5; ::_thesis: verum
end;
then A10: S2[ 0 ] by A6;
thus for k being Element of NAT holds S2[k] from NAT_1:sch_1(A10, A7); ::_thesis: verum
end;
end;
:: deftheorem Def11 defines ExtensionSeq2 LATTICE8:def_11_:_
for A being non empty set
for L being lower-bounded modular LATTICE
for d being distance_function of A,L
for b4 being Function holds
( b4 is ExtensionSeq2 of A,d iff ( dom b4 = NAT & b4 . 0 = [A,d] & ( for n being Element of NAT ex A9 being non empty set ex d9 being distance_function of A9,L ex Aq being non empty set ex dq being distance_function of Aq,L st
( Aq,dq is_extension2_of A9,d9 & b4 . n = [A9,d9] & b4 . (n + 1) = [Aq,dq] ) ) ) );
theorem Th30: :: LATTICE8:30
for A being non empty set
for L being lower-bounded modular LATTICE
for d being distance_function of A,L
for S being ExtensionSeq2 of A,d
for k, l being Element of NAT st k <= l holds
(S . k) `1 c= (S . l) `1
proof
let A be non empty set ; ::_thesis: for L being lower-bounded modular LATTICE
for d being distance_function of A,L
for S being ExtensionSeq2 of A,d
for k, l being Element of NAT st k <= l holds
(S . k) `1 c= (S . l) `1
let L be lower-bounded modular LATTICE; ::_thesis: for d being distance_function of A,L
for S being ExtensionSeq2 of A,d
for k, l being Element of NAT st k <= l holds
(S . k) `1 c= (S . l) `1
let d be distance_function of A,L; ::_thesis: for S being ExtensionSeq2 of A,d
for k, l being Element of NAT st k <= l holds
(S . k) `1 c= (S . l) `1
let S be ExtensionSeq2 of A,d; ::_thesis: for k, l being Element of NAT st k <= l holds
(S . k) `1 c= (S . l) `1
let k be Element of NAT ; ::_thesis: for l being Element of NAT st k <= l holds
(S . k) `1 c= (S . l) `1
defpred S1[ Element of NAT ] means ( k <= $1 implies (S . k) `1 c= (S . $1) `1 );
A1: for i being Element of NAT st S1[i] holds
S1[i + 1]
proof
let i be Element of NAT ; ::_thesis: ( S1[i] implies S1[i + 1] )
assume that
A2: ( k <= i implies (S . k) `1 c= (S . i) `1 ) and
A3: k <= i + 1 ; ::_thesis: (S . k) `1 c= (S . (i + 1)) `1
percases ( k = i + 1 or k <= i ) by A3, NAT_1:8;
suppose k = i + 1 ; ::_thesis: (S . k) `1 c= (S . (i + 1)) `1
hence (S . k) `1 c= (S . (i + 1)) `1 ; ::_thesis: verum
end;
supposeA4: k <= i ; ::_thesis: (S . k) `1 c= (S . (i + 1)) `1
consider A9 being non empty set , d9 being distance_function of A9,L, Aq being non empty set , dq being distance_function of Aq,L such that
A5: Aq,dq is_extension2_of A9,d9 and
A6: S . i = [A9,d9] and
A7: S . (i + 1) = [Aq,dq] by Def11;
[A9,d9] `1 = A9 ;
then A8: (S . i) `1 c= ConsecutiveSet2 (A9,(DistEsti d9)) by Th17, A6;
ex q being QuadrSeq of d9 st
( Aq = NextSet2 d9 & dq = NextDelta2 q ) by A5, Def10;
then [Aq,dq] `1 = ConsecutiveSet2 (A9,(DistEsti d9)) ;
hence (S . k) `1 c= (S . (i + 1)) `1 by A2, A4, A8, A7, XBOOLE_1:1; ::_thesis: verum
end;
end;
end;
A9: S1[ 0 ] by NAT_1:3;
thus for l being Element of NAT holds S1[l] from NAT_1:sch_1(A9, A1); ::_thesis: verum
end;
theorem Th31: :: LATTICE8:31
for A being non empty set
for L being lower-bounded modular LATTICE
for d being distance_function of A,L
for S being ExtensionSeq2 of A,d
for k, l being Element of NAT st k <= l holds
(S . k) `2 c= (S . l) `2
proof
let A be non empty set ; ::_thesis: for L being lower-bounded modular LATTICE
for d being distance_function of A,L
for S being ExtensionSeq2 of A,d
for k, l being Element of NAT st k <= l holds
(S . k) `2 c= (S . l) `2
let L be lower-bounded modular LATTICE; ::_thesis: for d being distance_function of A,L
for S being ExtensionSeq2 of A,d
for k, l being Element of NAT st k <= l holds
(S . k) `2 c= (S . l) `2
let d be distance_function of A,L; ::_thesis: for S being ExtensionSeq2 of A,d
for k, l being Element of NAT st k <= l holds
(S . k) `2 c= (S . l) `2
let S be ExtensionSeq2 of A,d; ::_thesis: for k, l being Element of NAT st k <= l holds
(S . k) `2 c= (S . l) `2
let k be Element of NAT ; ::_thesis: for l being Element of NAT st k <= l holds
(S . k) `2 c= (S . l) `2
defpred S1[ Element of NAT ] means ( k <= $1 implies (S . k) `2 c= (S . $1) `2 );
A1: for i being Element of NAT st S1[i] holds
S1[i + 1]
proof
let i be Element of NAT ; ::_thesis: ( S1[i] implies S1[i + 1] )
assume that
A2: ( k <= i implies (S . k) `2 c= (S . i) `2 ) and
A3: k <= i + 1 ; ::_thesis: (S . k) `2 c= (S . (i + 1)) `2
percases ( k = i + 1 or k <= i ) by A3, NAT_1:8;
suppose k = i + 1 ; ::_thesis: (S . k) `2 c= (S . (i + 1)) `2
hence (S . k) `2 c= (S . (i + 1)) `2 ; ::_thesis: verum
end;
supposeA4: k <= i ; ::_thesis: (S . k) `2 c= (S . (i + 1)) `2
consider A9 being non empty set , d9 being distance_function of A9,L, Aq being non empty set , dq being distance_function of Aq,L such that
A5: Aq,dq is_extension2_of A9,d9 and
A6: S . i = [A9,d9] and
A7: S . (i + 1) = [Aq,dq] by Def11;
consider q being QuadrSeq of d9 such that
Aq = NextSet2 d9 and
A8: dq = NextDelta2 q by A5, Def10;
[A9,d9] `2 = d9 ;
then A9: (S . i) `2 c= ConsecutiveDelta2 (q,(DistEsti d9)) by Th23, A6;
[Aq,dq] `2 = ConsecutiveDelta2 (q,(DistEsti d9)) by A8;
hence (S . k) `2 c= (S . (i + 1)) `2 by A2, A4, A9, A7, XBOOLE_1:1; ::_thesis: verum
end;
end;
end;
A10: S1[ 0 ] by NAT_1:3;
thus for l being Element of NAT holds S1[l] from NAT_1:sch_1(A10, A1); ::_thesis: verum
end;
theorem Th32: :: LATTICE8:32
for L being lower-bounded modular LATTICE
for S being ExtensionSeq2 of the carrier of L, BasicDF L
for FS being non empty set st FS = union { ((S . i) `1) where i is Element of NAT : verum } holds
union { ((S . i) `2) where i is Element of NAT : verum } is distance_function of FS,L
proof
let L be lower-bounded modular LATTICE; ::_thesis: for S being ExtensionSeq2 of the carrier of L, BasicDF L
for FS being non empty set st FS = union { ((S . i) `1) where i is Element of NAT : verum } holds
union { ((S . i) `2) where i is Element of NAT : verum } is distance_function of FS,L
let S be ExtensionSeq2 of the carrier of L, BasicDF L; ::_thesis: for FS being non empty set st FS = union { ((S . i) `1) where i is Element of NAT : verum } holds
union { ((S . i) `2) where i is Element of NAT : verum } is distance_function of FS,L
let FS be non empty set ; ::_thesis: ( FS = union { ((S . i) `1) where i is Element of NAT : verum } implies union { ((S . i) `2) where i is Element of NAT : verum } is distance_function of FS,L )
assume A1: FS = union { ((S . i) `1) where i is Element of NAT : verum } ; ::_thesis: union { ((S . i) `2) where i is Element of NAT : verum } is distance_function of FS,L
reconsider FS = FS as non empty set ;
set A = the carrier of L;
set FD = union { ((S . i) `2) where i is Element of NAT : verum } ;
now__::_thesis:_for_x,_y_being_set_st_x_in__{__((S_._i)_`2)_where_i_is_Element_of_NAT_:_verum__}__&_y_in__{__((S_._i)_`2)_where_i_is_Element_of_NAT_:_verum__}__holds_
x,y_are_c=-comparable
let x, y be set ; ::_thesis: ( x in { ((S . i) `2) where i is Element of NAT : verum } & y in { ((S . i) `2) where i is Element of NAT : verum } implies x,y are_c=-comparable )
assume that
A2: x in { ((S . i) `2) where i is Element of NAT : verum } and
A3: y in { ((S . i) `2) where i is Element of NAT : verum } ; ::_thesis: x,y are_c=-comparable
consider k being Element of NAT such that
A4: x = (S . k) `2 by A2;
consider l being Element of NAT such that
A5: y = (S . l) `2 by A3;
( k <= l or l <= k ) ;
then ( x c= y or y c= x ) by A4, A5, Th31;
hence x,y are_c=-comparable by XBOOLE_0:def_9; ::_thesis: verum
end;
then A6: { ((S . i) `2) where i is Element of NAT : verum } is c=-linear by ORDINAL1:def_8;
{ ((S . i) `2) where i is Element of NAT : verum } c= PFuncs ([:FS,FS:], the carrier of L)
proof
let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in { ((S . i) `2) where i is Element of NAT : verum } or z in PFuncs ([:FS,FS:], the carrier of L) )
assume z in { ((S . i) `2) where i is Element of NAT : verum } ; ::_thesis: z in PFuncs ([:FS,FS:], the carrier of L)
then consider j being Element of NAT such that
A7: z = (S . j) `2 ;
consider A9 being non empty set , d9 being distance_function of A9,L, Aq being non empty set , dq being distance_function of Aq,L such that
Aq,dq is_extension2_of A9,d9 and
A8: S . j = [A9,d9] and
S . (j + 1) = [Aq,dq] by Def11;
B8: A9 = [A9,d9] `1 ;
c8: d9 = [A9,d9] `2 ;
A9 in { ((S . i) `1) where i is Element of NAT : verum } by B8, A8;
then ( dom d9 = [:A9,A9:] & A9 c= FS ) by A1, FUNCT_2:def_1, ZFMISC_1:74;
then A9: ( rng d9 c= the carrier of L & dom d9 c= [:FS,FS:] ) by ZFMISC_1:96;
z = d9 by A7, A8, c8;
hence z in PFuncs ([:FS,FS:], the carrier of L) by A9, PARTFUN1:def_3; ::_thesis: verum
end;
then union { ((S . i) `2) where i is Element of NAT : verum } in PFuncs ([:FS,FS:], the carrier of L) by A6, TREES_2:40;
then A10: ex g being Function st
( union { ((S . i) `2) where i is Element of NAT : verum } = g & dom g c= [:FS,FS:] & rng g c= the carrier of L ) by PARTFUN1:def_3;
(S . 0) `1 in { ((S . i) `1) where i is Element of NAT : verum } ;
then reconsider X = { ((S . i) `1) where i is Element of NAT : verum } as non empty set ;
set LL = { [:I,I:] where I is Element of X : I in X } ;
set PP = { [:((S . i) `1),((S . i) `1):] where i is Element of NAT : verum } ;
defpred S1[ set , set ] means $2 = (S . $1) `2 ;
A11: { [:I,I:] where I is Element of X : I in X } = { [:((S . i) `1),((S . i) `1):] where i is Element of NAT : verum }
proof
thus { [:I,I:] where I is Element of X : I in X } c= { [:((S . i) `1),((S . i) `1):] where i is Element of NAT : verum } :: according to XBOOLE_0:def_10 ::_thesis: { [:((S . i) `1),((S . i) `1):] where i is Element of NAT : verum } c= { [:I,I:] where I is Element of X : I in X }
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { [:I,I:] where I is Element of X : I in X } or x in { [:((S . i) `1),((S . i) `1):] where i is Element of NAT : verum } )
assume x in { [:I,I:] where I is Element of X : I in X } ; ::_thesis: x in { [:((S . i) `1),((S . i) `1):] where i is Element of NAT : verum }
then consider J being Element of X such that
A12: x = [:J,J:] and
A13: J in X ;
ex j being Element of NAT st J = (S . j) `1 by A13;
hence x in { [:((S . i) `1),((S . i) `1):] where i is Element of NAT : verum } by A12; ::_thesis: verum
end;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { [:((S . i) `1),((S . i) `1):] where i is Element of NAT : verum } or x in { [:I,I:] where I is Element of X : I in X } )
assume x in { [:((S . i) `1),((S . i) `1):] where i is Element of NAT : verum } ; ::_thesis: x in { [:I,I:] where I is Element of X : I in X }
then consider j being Element of NAT such that
A14: x = [:((S . j) `1),((S . j) `1):] ;
(S . j) `1 in X ;
hence x in { [:I,I:] where I is Element of X : I in X } by A14; ::_thesis: verum
end;
reconsider FD = union { ((S . i) `2) where i is Element of NAT : verum } as Function by A10;
A15: for x being set st x in NAT holds
ex y being set st S1[x,y] ;
consider F being Function such that
A16: dom F = NAT and
A17: for x being set st x in NAT holds
S1[x,F . x] from CLASSES1:sch_1(A15);
A18: rng F = { ((S . i) `2) where i is Element of NAT : verum }
proof
thus rng F c= { ((S . i) `2) where i is Element of NAT : verum } :: according to XBOOLE_0:def_10 ::_thesis: { ((S . i) `2) where i is Element of NAT : verum } c= rng F
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in rng F or x in { ((S . i) `2) where i is Element of NAT : verum } )
assume x in rng F ; ::_thesis: x in { ((S . i) `2) where i is Element of NAT : verum }
then consider j being set such that
A19: j in dom F and
A20: F . j = x by FUNCT_1:def_3;
reconsider j = j as Element of NAT by A16, A19;
x = (S . j) `2 by A17, A20;
hence x in { ((S . i) `2) where i is Element of NAT : verum } ; ::_thesis: verum
end;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { ((S . i) `2) where i is Element of NAT : verum } or x in rng F )
assume x in { ((S . i) `2) where i is Element of NAT : verum } ; ::_thesis: x in rng F
then consider j being Element of NAT such that
A21: x = (S . j) `2 ;
x = F . j by A17, A21;
hence x in rng F by A16, FUNCT_1:def_3; ::_thesis: verum
end;
F is Function-yielding
proof
let x be set ; :: according to FUNCOP_1:def_6 ::_thesis: ( not x in proj1 F or F . x is set )
assume x in dom F ; ::_thesis: F . x is set
then reconsider j = x as Element of NAT by A16;
consider A1 being non empty set , d1 being distance_function of A1,L, Aq1 being non empty set , dq1 being distance_function of Aq1,L such that
Aq1,dq1 is_extension2_of A1,d1 and
A22: S . j = [A1,d1] and
S . (j + 1) = [Aq1,dq1] by Def11;
[A1,d1] `2 = d1 ;
hence F . x is set by A17, A22; ::_thesis: verum
end;
then reconsider F = F as Function-yielding Function ;
A23: rng (doms F) = { [:((S . i) `1),((S . i) `1):] where i is Element of NAT : verum }
proof
thus rng (doms F) c= { [:((S . i) `1),((S . i) `1):] where i is Element of NAT : verum } :: according to XBOOLE_0:def_10 ::_thesis: { [:((S . i) `1),((S . i) `1):] where i is Element of NAT : verum } c= rng (doms F)
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in rng (doms F) or x in { [:((S . i) `1),((S . i) `1):] where i is Element of NAT : verum } )
assume x in rng (doms F) ; ::_thesis: x in { [:((S . i) `1),((S . i) `1):] where i is Element of NAT : verum }
then consider j being set such that
A24: j in dom (doms F) and
A25: x = (doms F) . j by FUNCT_1:def_3;
A26: j in dom F by A24, FUNCT_6:59;
reconsider j = j as Element of NAT by A16, A24, FUNCT_6:59;
consider A1 being non empty set , d1 being distance_function of A1,L, Aq1 being non empty set , dq1 being distance_function of Aq1,L such that
Aq1,dq1 is_extension2_of A1,d1 and
A27: S . j = [A1,d1] and
S . (j + 1) = [Aq1,dq1] by Def11;
A28: [A1,d1] `2 = d1 ;
A29: [A1,d1] `1 = A1 ;
x = dom (F . j) by A25, A26, FUNCT_6:22
.= dom d1 by A17, A28, A27
.= [:((S . j) `1),((S . j) `1):] by A29, A27, FUNCT_2:def_1 ;
hence x in { [:((S . i) `1),((S . i) `1):] where i is Element of NAT : verum } ; ::_thesis: verum
end;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { [:((S . i) `1),((S . i) `1):] where i is Element of NAT : verum } or x in rng (doms F) )
assume x in { [:((S . i) `1),((S . i) `1):] where i is Element of NAT : verum } ; ::_thesis: x in rng (doms F)
then consider j being Element of NAT such that
A30: x = [:((S . j) `1),((S . j) `1):] ;
consider A1 being non empty set , d1 being distance_function of A1,L, Aq1 being non empty set , dq1 being distance_function of Aq1,L such that
Aq1,dq1 is_extension2_of A1,d1 and
A31: S . j = [A1,d1] and
S . (j + 1) = [Aq1,dq1] by Def11;
A32: [A1,d1] `2 = d1 ;
j in NAT ;
then A33: j in dom (doms F) by A16, FUNCT_6:59;
[A1,d1] `1 = A1 ;
then x = dom d1 by A30, A31, FUNCT_2:def_1
.= dom (F . j) by A17, A32, A31
.= (doms F) . j by A16, FUNCT_6:22 ;
hence x in rng (doms F) by A33, FUNCT_1:def_3; ::_thesis: verum
end;
now__::_thesis:_for_x,_y_being_set_st_x_in_X_&_y_in_X_holds_
x,y_are_c=-comparable
let x, y be set ; ::_thesis: ( x in X & y in X implies x,y are_c=-comparable )
assume that
A34: x in X and
A35: y in X ; ::_thesis: x,y are_c=-comparable
consider k being Element of NAT such that
A36: x = (S . k) `1 by A34;
consider l being Element of NAT such that
A37: y = (S . l) `1 by A35;
( k <= l or l <= k ) ;
then ( x c= y or y c= x ) by A36, A37, Th30;
hence x,y are_c=-comparable by XBOOLE_0:def_9; ::_thesis: verum
end;
then X is c=-linear by ORDINAL1:def_8;
then [:FS,FS:] = union (rng (doms F)) by A1, A23, A11, LATTICE5:3
.= dom FD by A18, LATTICE5:1 ;
then reconsider FD = FD as BiFunction of FS,L by A10, FUNCT_2:def_1, RELSET_1:4;
A38: FD is symmetric
proof
let x, y be Element of FS; :: according to LATTICE5:def_5 ::_thesis: FD . (x,y) = FD . (y,x)
consider x1 being set such that
A39: x in x1 and
A40: x1 in X by A1, TARSKI:def_4;
consider k being Element of NAT such that
A41: x1 = (S . k) `1 by A40;
consider A1 being non empty set , d1 being distance_function of A1,L, Aq1 being non empty set , dq1 being distance_function of Aq1,L such that
Aq1,dq1 is_extension2_of A1,d1 and
A42: S . k = [A1,d1] and
S . (k + 1) = [Aq1,dq1] by Def11;
A44: [A1,d1] `1 = A1 ;
then A43: x in A1 by A39, A41, A42;
[A1,d1] `2 = d1 ;
then d1 in { ((S . i) `2) where i is Element of NAT : verum } by A42;
then A45: d1 c= FD by ZFMISC_1:74;
consider y1 being set such that
A46: y in y1 and
A47: y1 in X by A1, TARSKI:def_4;
consider l being Element of NAT such that
A48: y1 = (S . l) `1 by A47;
consider A2 being non empty set , d2 being distance_function of A2,L, Aq2 being non empty set , dq2 being distance_function of Aq2,L such that
Aq2,dq2 is_extension2_of A2,d2 and
A49: S . l = [A2,d2] and
S . (l + 1) = [Aq2,dq2] by Def11;
A51: [A2,d2] `1 = A2 ;
then A50: y in A2 by A46, A48, A49;
[A2,d2] `2 = d2 ;
then d2 in { ((S . i) `2) where i is Element of NAT : verum } by A49;
then A52: d2 c= FD by ZFMISC_1:74;
percases ( k <= l or l <= k ) ;
suppose k <= l ; ::_thesis: FD . (x,y) = FD . (y,x)
then A1 c= A2 by A44, A51, Th30, A42, A49;
then reconsider x9 = x, y9 = y as Element of A2 by A43, A46, A48, A49, A51;
A53: dom d2 = [:A2,A2:] by FUNCT_2:def_1;
hence FD . (x,y) = d2 . [x9,y9] by A52, GRFUNC_1:2
.= d2 . (x9,y9)
.= d2 . (y9,x9) by LATTICE5:def_5
.= FD . [y9,x9] by A52, A53, GRFUNC_1:2
.= FD . (y,x) ;
::_thesis: verum
end;
suppose l <= k ; ::_thesis: FD . (x,y) = FD . (y,x)
then A2 c= A1 by A44, A51, Th30, A42, A49;
then reconsider x9 = x, y9 = y as Element of A1 by A39, A41, A42, A50, A44;
A54: dom d1 = [:A1,A1:] by FUNCT_2:def_1;
hence FD . (x,y) = d1 . [x9,y9] by A45, GRFUNC_1:2
.= d1 . (x9,y9)
.= d1 . (y9,x9) by LATTICE5:def_5
.= FD . [y9,x9] by A45, A54, GRFUNC_1:2
.= FD . (y,x) ;
::_thesis: verum
end;
end;
end;
A55: FD is u.t.i.
proof
let x, y, z be Element of FS; :: according to LATTICE5:def_7 ::_thesis: FD . (x,z) <= (FD . (x,y)) "\/" (FD . (y,z))
consider x1 being set such that
A56: x in x1 and
A57: x1 in X by A1, TARSKI:def_4;
consider k being Element of NAT such that
A58: x1 = (S . k) `1 by A57;
consider A1 being non empty set , d1 being distance_function of A1,L, Aq1 being non empty set , dq1 being distance_function of Aq1,L such that
Aq1,dq1 is_extension2_of A1,d1 and
A59: S . k = [A1,d1] and
S . (k + 1) = [Aq1,dq1] by Def11;
A63: [A1,d1] `1 = A1 ;
then A60: x in A1 by A56, A58, A59;
[A1,d1] `2 = d1 ;
then d1 in { ((S . i) `2) where i is Element of NAT : verum } by A59;
then A61: d1 c= FD by ZFMISC_1:74;
A62: dom d1 = [:A1,A1:] by FUNCT_2:def_1;
consider y1 being set such that
A64: y in y1 and
A65: y1 in X by A1, TARSKI:def_4;
consider l being Element of NAT such that
A66: y1 = (S . l) `1 by A65;
consider A2 being non empty set , d2 being distance_function of A2,L, Aq2 being non empty set , dq2 being distance_function of Aq2,L such that
Aq2,dq2 is_extension2_of A2,d2 and
A67: S . l = [A2,d2] and
S . (l + 1) = [Aq2,dq2] by Def11;
A71: [A2,d2] `1 = A2 ;
then A68: y in A2 by A64, A66, A67;
[A2,d2] `2 = d2 ;
then d2 in { ((S . i) `2) where i is Element of NAT : verum } by A67;
then A69: d2 c= FD by ZFMISC_1:74;
A70: dom d2 = [:A2,A2:] by FUNCT_2:def_1;
consider z1 being set such that
A72: z in z1 and
A73: z1 in X by A1, TARSKI:def_4;
consider n being Element of NAT such that
A74: z1 = (S . n) `1 by A73;
consider A3 being non empty set , d3 being distance_function of A3,L, Aq3 being non empty set , dq3 being distance_function of Aq3,L such that
Aq3,dq3 is_extension2_of A3,d3 and
A75: S . n = [A3,d3] and
S . (n + 1) = [Aq3,dq3] by Def11;
A77: [A3,d3] `1 = A3 ;
then A76: z in A3 by A72, A74, A75;
[A3,d3] `2 = d3 ;
then d3 in { ((S . i) `2) where i is Element of NAT : verum } by A75;
then A78: d3 c= FD by ZFMISC_1:74;
A79: dom d3 = [:A3,A3:] by FUNCT_2:def_1;
percases ( k <= l or l <= k ) ;
suppose k <= l ; ::_thesis: FD . (x,z) <= (FD . (x,y)) "\/" (FD . (y,z))
then A80: A1 c= A2 by A63, A71, Th30, A59, A67;
thus (FD . (x,y)) "\/" (FD . (y,z)) >= FD . (x,z) ::_thesis: verum
proof
percases ( l <= n or n <= l ) ;
suppose l <= n ; ::_thesis: (FD . (x,y)) "\/" (FD . (y,z)) >= FD . (x,z)
then A81: A2 c= A3 by A71, A77, Th30, A67, A75;
then A1 c= A3 by A80, XBOOLE_1:1;
then reconsider x9 = x, y9 = y, z9 = z as Element of A3 by A60, A68, A72, A74, A75, A81, A77;
A82: FD . (y,z) = d3 . [y9,z9] by A78, A79, GRFUNC_1:2
.= d3 . (y9,z9) ;
A83: FD . (x,z) = d3 . [x9,z9] by A78, A79, GRFUNC_1:2
.= d3 . (x9,z9) ;
FD . (x,y) = d3 . [x9,y9] by A78, A79, GRFUNC_1:2
.= d3 . (x9,y9) ;
hence (FD . (x,y)) "\/" (FD . (y,z)) >= FD . (x,z) by A82, A83, LATTICE5:def_7; ::_thesis: verum
end;
suppose n <= l ; ::_thesis: (FD . (x,y)) "\/" (FD . (y,z)) >= FD . (x,z)
then A3 c= A2 by A71, A77, Th30, A67, A75;
then reconsider x9 = x, y9 = y, z9 = z as Element of A2 by A60, A64, A66, A67, A76, A80, A71;
A84: FD . (y,z) = d2 . [y9,z9] by A69, A70, GRFUNC_1:2
.= d2 . (y9,z9) ;
A85: FD . (x,z) = d2 . [x9,z9] by A69, A70, GRFUNC_1:2
.= d2 . (x9,z9) ;
FD . (x,y) = d2 . [x9,y9] by A69, A70, GRFUNC_1:2
.= d2 . (x9,y9) ;
hence (FD . (x,y)) "\/" (FD . (y,z)) >= FD . (x,z) by A84, A85, LATTICE5:def_7; ::_thesis: verum
end;
end;
end;
end;
suppose l <= k ; ::_thesis: FD . (x,z) <= (FD . (x,y)) "\/" (FD . (y,z))
then A86: A2 c= A1 by A63, A71, Th30, A59, A67;
thus (FD . (x,y)) "\/" (FD . (y,z)) >= FD . (x,z) ::_thesis: verum
proof
percases ( k <= n or n <= k ) ;
suppose k <= n ; ::_thesis: (FD . (x,y)) "\/" (FD . (y,z)) >= FD . (x,z)
then A87: A1 c= A3 by A63, A77, Th30, A59, A75;
then A2 c= A3 by A86, XBOOLE_1:1;
then reconsider x9 = x, y9 = y, z9 = z as Element of A3 by A60, A68, A72, A74, A75, A87, A77;
A88: FD . (y,z) = d3 . [y9,z9] by A78, A79, GRFUNC_1:2
.= d3 . (y9,z9) ;
A89: FD . (x,z) = d3 . [x9,z9] by A78, A79, GRFUNC_1:2
.= d3 . (x9,z9) ;
FD . (x,y) = d3 . [x9,y9] by A78, A79, GRFUNC_1:2
.= d3 . (x9,y9) ;
hence (FD . (x,y)) "\/" (FD . (y,z)) >= FD . (x,z) by A88, A89, LATTICE5:def_7; ::_thesis: verum
end;
suppose n <= k ; ::_thesis: (FD . (x,y)) "\/" (FD . (y,z)) >= FD . (x,z)
then A3 c= A1 by A63, A77, Th30, A59, A75;
then reconsider x9 = x, y9 = y, z9 = z as Element of A1 by A56, A58, A59, A68, A76, A86, A63;
A90: FD . (y,z) = d1 . [y9,z9] by A61, A62, GRFUNC_1:2
.= d1 . (y9,z9) ;
A91: FD . (x,z) = d1 . [x9,z9] by A61, A62, GRFUNC_1:2
.= d1 . (x9,z9) ;
FD . (x,y) = d1 . [x9,y9] by A61, A62, GRFUNC_1:2
.= d1 . (x9,y9) ;
hence (FD . (x,y)) "\/" (FD . (y,z)) >= FD . (x,z) by A90, A91, LATTICE5:def_7; ::_thesis: verum
end;
end;
end;
end;
end;
end;
FD is zeroed
proof
let x be Element of FS; :: according to LATTICE5:def_6 ::_thesis: FD . (x,x) = Bottom L
consider y being set such that
A92: x in y and
A93: y in X by A1, TARSKI:def_4;
consider j being Element of NAT such that
A94: y = (S . j) `1 by A93;
consider A1 being non empty set , d1 being distance_function of A1,L, Aq1 being non empty set , dq1 being distance_function of Aq1,L such that
Aq1,dq1 is_extension2_of A1,d1 and
A95: S . j = [A1,d1] and
S . (j + 1) = [Aq1,dq1] by Def11;
[A1,d1] `2 = d1 ;
then d1 in { ((S . i) `2) where i is Element of NAT : verum } by A95;
then A96: d1 c= FD by ZFMISC_1:74;
[A1,d1] `1 = A1 ;
then reconsider x9 = x as Element of A1 by A92, A94, A95;
dom d1 = [:A1,A1:] by FUNCT_2:def_1;
hence FD . (x,x) = d1 . [x9,x9] by A96, GRFUNC_1:2
.= d1 . (x9,x9)
.= Bottom L by LATTICE5:def_6 ;
::_thesis: verum
end;
hence union { ((S . i) `2) where i is Element of NAT : verum } is distance_function of FS,L by A38, A55; ::_thesis: verum
end;
theorem Th33: :: LATTICE8:33
for L being lower-bounded modular LATTICE
for S being ExtensionSeq2 of the carrier of L, BasicDF L
for FS being non empty set
for FD being distance_function of FS,L
for x, y being Element of FS
for a, b being Element of L st FS = union { ((S . i) `1) where i is Element of NAT : verum } & FD = union { ((S . i) `2) where i is Element of NAT : verum } & FD . (x,y) <= a "\/" b holds
ex z1, z2 being Element of FS st
( FD . (x,z1) = a & FD . (z1,z2) = ((FD . (x,y)) "\/" a) "/\" b & FD . (z2,y) = a )
proof
let L be lower-bounded modular LATTICE; ::_thesis: for S being ExtensionSeq2 of the carrier of L, BasicDF L
for FS being non empty set
for FD being distance_function of FS,L
for x, y being Element of FS
for a, b being Element of L st FS = union { ((S . i) `1) where i is Element of NAT : verum } & FD = union { ((S . i) `2) where i is Element of NAT : verum } & FD . (x,y) <= a "\/" b holds
ex z1, z2 being Element of FS st
( FD . (x,z1) = a & FD . (z1,z2) = ((FD . (x,y)) "\/" a) "/\" b & FD . (z2,y) = a )
let S be ExtensionSeq2 of the carrier of L, BasicDF L; ::_thesis: for FS being non empty set
for FD being distance_function of FS,L
for x, y being Element of FS
for a, b being Element of L st FS = union { ((S . i) `1) where i is Element of NAT : verum } & FD = union { ((S . i) `2) where i is Element of NAT : verum } & FD . (x,y) <= a "\/" b holds
ex z1, z2 being Element of FS st
( FD . (x,z1) = a & FD . (z1,z2) = ((FD . (x,y)) "\/" a) "/\" b & FD . (z2,y) = a )
let FS be non empty set ; ::_thesis: for FD being distance_function of FS,L
for x, y being Element of FS
for a, b being Element of L st FS = union { ((S . i) `1) where i is Element of NAT : verum } & FD = union { ((S . i) `2) where i is Element of NAT : verum } & FD . (x,y) <= a "\/" b holds
ex z1, z2 being Element of FS st
( FD . (x,z1) = a & FD . (z1,z2) = ((FD . (x,y)) "\/" a) "/\" b & FD . (z2,y) = a )
let FD be distance_function of FS,L; ::_thesis: for x, y being Element of FS
for a, b being Element of L st FS = union { ((S . i) `1) where i is Element of NAT : verum } & FD = union { ((S . i) `2) where i is Element of NAT : verum } & FD . (x,y) <= a "\/" b holds
ex z1, z2 being Element of FS st
( FD . (x,z1) = a & FD . (z1,z2) = ((FD . (x,y)) "\/" a) "/\" b & FD . (z2,y) = a )
let x, y be Element of FS; ::_thesis: for a, b being Element of L st FS = union { ((S . i) `1) where i is Element of NAT : verum } & FD = union { ((S . i) `2) where i is Element of NAT : verum } & FD . (x,y) <= a "\/" b holds
ex z1, z2 being Element of FS st
( FD . (x,z1) = a & FD . (z1,z2) = ((FD . (x,y)) "\/" a) "/\" b & FD . (z2,y) = a )
let a, b be Element of L; ::_thesis: ( FS = union { ((S . i) `1) where i is Element of NAT : verum } & FD = union { ((S . i) `2) where i is Element of NAT : verum } & FD . (x,y) <= a "\/" b implies ex z1, z2 being Element of FS st
( FD . (x,z1) = a & FD . (z1,z2) = ((FD . (x,y)) "\/" a) "/\" b & FD . (z2,y) = a ) )
assume that
A1: FS = union { ((S . i) `1) where i is Element of NAT : verum } and
A2: FD = union { ((S . i) `2) where i is Element of NAT : verum } and
A3: FD . (x,y) <= a "\/" b ; ::_thesis: ex z1, z2 being Element of FS st
( FD . (x,z1) = a & FD . (z1,z2) = ((FD . (x,y)) "\/" a) "/\" b & FD . (z2,y) = a )
(S . 0) `1 in { ((S . i) `1) where i is Element of NAT : verum } ;
then reconsider X = { ((S . i) `1) where i is Element of NAT : verum } as non empty set ;
consider x1 being set such that
A4: x in x1 and
A5: x1 in X by A1, TARSKI:def_4;
consider k being Element of NAT such that
A6: x1 = (S . k) `1 by A5;
consider A1 being non empty set , d1 being distance_function of A1,L, Aq1 being non empty set , dq1 being distance_function of Aq1,L such that
A7: Aq1,dq1 is_extension2_of A1,d1 and
A8: S . k = [A1,d1] and
A9: S . (k + 1) = [Aq1,dq1] by Def11;
A12: [A1,d1] `1 = A1 ;
then A10: x in A1 by A4, A6, A8;
[A1,d1] `2 = d1 ;
then d1 in { ((S . i) `2) where i is Element of NAT : verum } by A8;
then A11: d1 c= FD by A2, ZFMISC_1:74;
A13: [Aq1,dq1] `1 = Aq1 ;
then Aq1 in { ((S . i) `1) where i is Element of NAT : verum } by A9;
then A14: Aq1 c= FS by A1, ZFMISC_1:74;
[Aq1,dq1] `2 = dq1 ;
then dq1 in { ((S . i) `2) where i is Element of NAT : verum } by A9;
then A15: dq1 c= FD by A2, ZFMISC_1:74;
consider y1 being set such that
A16: y in y1 and
A17: y1 in X by A1, TARSKI:def_4;
consider l being Element of NAT such that
A18: y1 = (S . l) `1 by A17;
consider A2 being non empty set , d2 being distance_function of A2,L, Aq2 being non empty set , dq2 being distance_function of Aq2,L such that
A19: Aq2,dq2 is_extension2_of A2,d2 and
A20: S . l = [A2,d2] and
A21: S . (l + 1) = [Aq2,dq2] by Def11;
A24: [A2,d2] `1 = A2 ;
then A22: y in A2 by A16, A18, A20;
[A2,d2] `2 = d2 ;
then d2 in { ((S . i) `2) where i is Element of NAT : verum } by A20;
then A23: d2 c= FD by A2, ZFMISC_1:74;
A25: [Aq2,dq2] `1 = Aq2 ;
then Aq2 in { ((S . i) `1) where i is Element of NAT : verum } by A21;
then A26: Aq2 c= FS by A1, ZFMISC_1:74;
[Aq2,dq2] `2 = dq2 ;
then dq2 in { ((S . i) `2) where i is Element of NAT : verum } by A21;
then A27: dq2 c= FD by A2, ZFMISC_1:74;
percases ( k <= l or l <= k ) ;
suppose k <= l ; ::_thesis: ex z1, z2 being Element of FS st
( FD . (x,z1) = a & FD . (z1,z2) = ((FD . (x,y)) "\/" a) "/\" b & FD . (z2,y) = a )
then A1 c= A2 by A12, A24, Th30, A8, A20;
then reconsider x9 = x, y9 = y as Element of A2 by A10, A16, A18, A20, A24;
A28: ( x9 in A2 & y9 in A2 ) ;
A2 c= Aq2 by A24, A25, Th30, A20, A21, NAT_1:11;
then reconsider x99 = x9, y99 = y9 as Element of Aq2 by A28;
dom d2 = [:A2,A2:] by FUNCT_2:def_1;
then A29: FD . (x,y) = d2 . [x9,y9] by A23, GRFUNC_1:2
.= d2 . (x9,y9) ;
then consider z1, z2 being Element of Aq2 such that
A30: dq2 . (x,z1) = a and
A31: dq2 . (z1,z2) = ((d2 . (x9,y9)) "\/" a) "/\" b and
A32: dq2 . (z2,y) = a by A3, A19, Th29;
( z1 in Aq2 & z2 in Aq2 ) ;
then reconsider z19 = z1, z29 = z2 as Element of FS by A26;
take z19 ; ::_thesis: ex z2 being Element of FS st
( FD . (x,z19) = a & FD . (z19,z2) = ((FD . (x,y)) "\/" a) "/\" b & FD . (z2,y) = a )
take z29 ; ::_thesis: ( FD . (x,z19) = a & FD . (z19,z29) = ((FD . (x,y)) "\/" a) "/\" b & FD . (z29,y) = a )
A33: dom dq2 = [:Aq2,Aq2:] by FUNCT_2:def_1;
hence FD . (x,z19) = dq2 . [x99,z1] by A27, GRFUNC_1:2
.= a by A30 ;
::_thesis: ( FD . (z19,z29) = ((FD . (x,y)) "\/" a) "/\" b & FD . (z29,y) = a )
thus FD . (z19,z29) = dq2 . [z1,z2] by A27, A33, GRFUNC_1:2
.= ((FD . (x,y)) "\/" a) "/\" b by A29, A31 ; ::_thesis: FD . (z29,y) = a
thus FD . (z29,y) = dq2 . [z2,y99] by A27, A33, GRFUNC_1:2
.= a by A32 ; ::_thesis: verum
end;
suppose l <= k ; ::_thesis: ex z1, z2 being Element of FS st
( FD . (x,z1) = a & FD . (z1,z2) = ((FD . (x,y)) "\/" a) "/\" b & FD . (z2,y) = a )
then A2 c= A1 by A12, A24, Th30, A8, A20;
then reconsider x9 = x, y9 = y as Element of A1 by A4, A6, A8, A22, A12;
A34: ( x9 in A1 & y9 in A1 ) ;
A1 c= Aq1 by A12, A13, Th30, A8, A9, NAT_1:11;
then reconsider x99 = x9, y99 = y9 as Element of Aq1 by A34;
dom d1 = [:A1,A1:] by FUNCT_2:def_1;
then A35: FD . (x,y) = d1 . [x9,y9] by A11, GRFUNC_1:2
.= d1 . (x9,y9) ;
then consider z1, z2 being Element of Aq1 such that
A36: dq1 . (x,z1) = a and
A37: dq1 . (z1,z2) = ((d1 . (x9,y9)) "\/" a) "/\" b and
A38: dq1 . (z2,y) = a by A3, A7, Th29;
( z1 in Aq1 & z2 in Aq1 ) ;
then reconsider z19 = z1, z29 = z2 as Element of FS by A14;
take z19 ; ::_thesis: ex z2 being Element of FS st
( FD . (x,z19) = a & FD . (z19,z2) = ((FD . (x,y)) "\/" a) "/\" b & FD . (z2,y) = a )
take z29 ; ::_thesis: ( FD . (x,z19) = a & FD . (z19,z29) = ((FD . (x,y)) "\/" a) "/\" b & FD . (z29,y) = a )
A39: dom dq1 = [:Aq1,Aq1:] by FUNCT_2:def_1;
hence FD . (x,z19) = dq1 . [x99,z1] by A15, GRFUNC_1:2
.= a by A36 ;
::_thesis: ( FD . (z19,z29) = ((FD . (x,y)) "\/" a) "/\" b & FD . (z29,y) = a )
thus FD . (z19,z29) = dq1 . [z1,z2] by A15, A39, GRFUNC_1:2
.= ((FD . (x,y)) "\/" a) "/\" b by A35, A37 ; ::_thesis: FD . (z29,y) = a
thus FD . (z29,y) = dq1 . [z2,y99] by A15, A39, GRFUNC_1:2
.= a by A38 ; ::_thesis: verum
end;
end;
end;
Lm3: for m being Element of NAT holds
( not m in Seg 4 or m = 1 or m = 2 or m = 3 or m = 4 )
proof
let m be Element of NAT ; ::_thesis: ( not m in Seg 4 or m = 1 or m = 2 or m = 3 or m = 4 )
assume A1: m in Seg 4 ; ::_thesis: ( m = 1 or m = 2 or m = 3 or m = 4 )
then m <= 4 by FINSEQ_1:1;
then ( m = 0 or m = 1 or m = 2 or m = 3 or m = 4 ) by NAT_1:28;
hence ( m = 1 or m = 2 or m = 3 or m = 4 ) by A1, FINSEQ_1:1; ::_thesis: verum
end;
Lm4: for j being Element of NAT st 1 <= j & j < 4 & not j = 1 & not j = 2 holds
j = 3
proof
let j be Element of NAT ; ::_thesis: ( 1 <= j & j < 4 & not j = 1 & not j = 2 implies j = 3 )
assume that
A1: 1 <= j and
A2: j < 4 ; ::_thesis: ( j = 1 or j = 2 or j = 3 )
j < 3 + 1 by A2;
then j <= 3 by NAT_1:13;
then ( j = 0 or j = 1 or j = 2 or j = 3 ) by NAT_1:27;
hence ( j = 1 or j = 2 or j = 3 ) by A1; ::_thesis: verum
end;
theorem Th34: :: LATTICE8:34
for L being lower-bounded modular LATTICE
for S being ExtensionSeq2 of the carrier of L, BasicDF L
for FS being non empty set
for FD being distance_function of FS,L
for f being Homomorphism of L,(EqRelLATT FS)
for e1, e2 being Equivalence_Relation of FS
for x, y being set st f = alpha FD & FS = union { ((S . i) `1) where i is Element of NAT : verum } & FD = union { ((S . i) `2) where i is Element of NAT : verum } & e1 in the carrier of (Image f) & e2 in the carrier of (Image f) & [x,y] in e1 "\/" e2 holds
ex F being non empty FinSequence of FS st
( len F = 2 + 2 & x,y are_joint_by F,e1,e2 )
proof
let L be lower-bounded modular LATTICE; ::_thesis: for S being ExtensionSeq2 of the carrier of L, BasicDF L
for FS being non empty set
for FD being distance_function of FS,L
for f being Homomorphism of L,(EqRelLATT FS)
for e1, e2 being Equivalence_Relation of FS
for x, y being set st f = alpha FD & FS = union { ((S . i) `1) where i is Element of NAT : verum } & FD = union { ((S . i) `2) where i is Element of NAT : verum } & e1 in the carrier of (Image f) & e2 in the carrier of (Image f) & [x,y] in e1 "\/" e2 holds
ex F being non empty FinSequence of FS st
( len F = 2 + 2 & x,y are_joint_by F,e1,e2 )
let S be ExtensionSeq2 of the carrier of L, BasicDF L; ::_thesis: for FS being non empty set
for FD being distance_function of FS,L
for f being Homomorphism of L,(EqRelLATT FS)
for e1, e2 being Equivalence_Relation of FS
for x, y being set st f = alpha FD & FS = union { ((S . i) `1) where i is Element of NAT : verum } & FD = union { ((S . i) `2) where i is Element of NAT : verum } & e1 in the carrier of (Image f) & e2 in the carrier of (Image f) & [x,y] in e1 "\/" e2 holds
ex F being non empty FinSequence of FS st
( len F = 2 + 2 & x,y are_joint_by F,e1,e2 )
let FS be non empty set ; ::_thesis: for FD being distance_function of FS,L
for f being Homomorphism of L,(EqRelLATT FS)
for e1, e2 being Equivalence_Relation of FS
for x, y being set st f = alpha FD & FS = union { ((S . i) `1) where i is Element of NAT : verum } & FD = union { ((S . i) `2) where i is Element of NAT : verum } & e1 in the carrier of (Image f) & e2 in the carrier of (Image f) & [x,y] in e1 "\/" e2 holds
ex F being non empty FinSequence of FS st
( len F = 2 + 2 & x,y are_joint_by F,e1,e2 )
let FD be distance_function of FS,L; ::_thesis: for f being Homomorphism of L,(EqRelLATT FS)
for e1, e2 being Equivalence_Relation of FS
for x, y being set st f = alpha FD & FS = union { ((S . i) `1) where i is Element of NAT : verum } & FD = union { ((S . i) `2) where i is Element of NAT : verum } & e1 in the carrier of (Image f) & e2 in the carrier of (Image f) & [x,y] in e1 "\/" e2 holds
ex F being non empty FinSequence of FS st
( len F = 2 + 2 & x,y are_joint_by F,e1,e2 )
let f be Homomorphism of L,(EqRelLATT FS); ::_thesis: for e1, e2 being Equivalence_Relation of FS
for x, y being set st f = alpha FD & FS = union { ((S . i) `1) where i is Element of NAT : verum } & FD = union { ((S . i) `2) where i is Element of NAT : verum } & e1 in the carrier of (Image f) & e2 in the carrier of (Image f) & [x,y] in e1 "\/" e2 holds
ex F being non empty FinSequence of FS st
( len F = 2 + 2 & x,y are_joint_by F,e1,e2 )
let e1, e2 be Equivalence_Relation of FS; ::_thesis: for x, y being set st f = alpha FD & FS = union { ((S . i) `1) where i is Element of NAT : verum } & FD = union { ((S . i) `2) where i is Element of NAT : verum } & e1 in the carrier of (Image f) & e2 in the carrier of (Image f) & [x,y] in e1 "\/" e2 holds
ex F being non empty FinSequence of FS st
( len F = 2 + 2 & x,y are_joint_by F,e1,e2 )
let x, y be set ; ::_thesis: ( f = alpha FD & FS = union { ((S . i) `1) where i is Element of NAT : verum } & FD = union { ((S . i) `2) where i is Element of NAT : verum } & e1 in the carrier of (Image f) & e2 in the carrier of (Image f) & [x,y] in e1 "\/" e2 implies ex F being non empty FinSequence of FS st
( len F = 2 + 2 & x,y are_joint_by F,e1,e2 ) )
assume that
A1: f = alpha FD and
A2: ( FS = union { ((S . i) `1) where i is Element of NAT : verum } & FD = union { ((S . i) `2) where i is Element of NAT : verum } ) and
A3: e1 in the carrier of (Image f) and
A4: e2 in the carrier of (Image f) and
A5: [x,y] in e1 "\/" e2 ; ::_thesis: ex F being non empty FinSequence of FS st
( len F = 2 + 2 & x,y are_joint_by F,e1,e2 )
A6: the carrier of (Image f) = rng f by YELLOW_0:def_15;
then consider a being set such that
A7: a in dom f and
A8: e1 = f . a by A3, FUNCT_1:def_3;
consider b being set such that
A9: b in dom f and
A10: e2 = f . b by A4, A6, FUNCT_1:def_3;
reconsider a = a, b = b as Element of L by A7, A9;
reconsider a = a, b = b as Element of L ;
consider e being Equivalence_Relation of FS such that
A11: e = f . (a "\/" b) and
A12: for u, v being Element of FS holds
( [u,v] in e iff FD . (u,v) <= a "\/" b ) by A1, LATTICE5:def_8;
consider e29 being Equivalence_Relation of FS such that
A13: e29 = f . b and
A14: for u, v being Element of FS holds
( [u,v] in e29 iff FD . (u,v) <= b ) by A1, LATTICE5:def_8;
consider e19 being Equivalence_Relation of FS such that
A15: e19 = f . a and
A16: for u, v being Element of FS holds
( [u,v] in e19 iff FD . (u,v) <= a ) by A1, LATTICE5:def_8;
field (e1 "\/" e2) = FS by ORDERS_1:12;
then reconsider u = x, v = y as Element of FS by A5, RELAT_1:15;
A17: Seg 4 = { n where n is Element of NAT : ( 1 <= n & n <= 4 ) } by FINSEQ_1:def_1;
then A18: 1 in Seg 4 ;
e = (f . a) "\/" (f . b) by A11, WAYBEL_6:2
.= e1 "\/" e2 by A8, A10, LATTICE5:10 ;
then A19: FD . (u,v) <= a "\/" b by A5, A12;
then consider z1, z2 being Element of FS such that
A20: FD . (u,z1) = a and
A21: FD . (z1,z2) = ((FD . (u,v)) "\/" a) "/\" b and
A22: FD . (z2,v) = a by A2, Th33;
defpred S1[ set , set ] means ( ( $1 = 1 implies $2 = u ) & ( $1 = 2 implies $2 = z1 ) & ( $1 = 3 implies $2 = z2 ) & ( $1 = 4 implies $2 = v ) );
A23: for m being Nat st m in Seg 4 holds
ex w being set st S1[m,w]
proof
let m be Nat; ::_thesis: ( m in Seg 4 implies ex w being set st S1[m,w] )
assume A24: m in Seg 4 ; ::_thesis: ex w being set st S1[m,w]
percases ( m = 1 or m = 2 or m = 3 or m = 4 ) by A24, Lm3;
supposeA25: m = 1 ; ::_thesis: ex w being set st S1[m,w]
take x ; ::_thesis: S1[m,x]
thus S1[m,x] by A25; ::_thesis: verum
end;
supposeA26: m = 2 ; ::_thesis: ex w being set st S1[m,w]
take z1 ; ::_thesis: S1[m,z1]
thus S1[m,z1] by A26; ::_thesis: verum
end;
supposeA27: m = 3 ; ::_thesis: ex w being set st S1[m,w]
take z2 ; ::_thesis: S1[m,z2]
thus S1[m,z2] by A27; ::_thesis: verum
end;
supposeA28: m = 4 ; ::_thesis: ex w being set st S1[m,w]
take y ; ::_thesis: S1[m,y]
thus S1[m,y] by A28; ::_thesis: verum
end;
end;
end;
ex p being FinSequence st
( dom p = Seg 4 & ( for k being Nat st k in Seg 4 holds
S1[k,p . k] ) ) from FINSEQ_1:sch_1(A23);
then consider h being FinSequence such that
A29: dom h = Seg 4 and
A30: for m being Nat st m in Seg 4 holds
( ( m = 1 implies h . m = u ) & ( m = 2 implies h . m = z1 ) & ( m = 3 implies h . m = z2 ) & ( m = 4 implies h . m = v ) ) ;
A31: len h = 4 by A29, FINSEQ_1:def_3;
A32: 3 in Seg 4 by A17;
A33: 4 in Seg 4 by A17;
A34: 2 in Seg 4 by A17;
rng h c= FS
proof
let w be set ; :: according to TARSKI:def_3 ::_thesis: ( not w in rng h or w in FS )
assume w in rng h ; ::_thesis: w in FS
then consider j being set such that
A35: j in dom h and
A36: w = h . j by FUNCT_1:def_3;
percases ( j = 1 or j = 2 or j = 3 or j = 4 ) by A29, A35, Lm3;
suppose j = 1 ; ::_thesis: w in FS
then h . j = u by A30, A18;
hence w in FS by A36; ::_thesis: verum
end;
suppose j = 2 ; ::_thesis: w in FS
then h . j = z1 by A30, A34;
hence w in FS by A36; ::_thesis: verum
end;
suppose j = 3 ; ::_thesis: w in FS
then h . j = z2 by A30, A32;
hence w in FS by A36; ::_thesis: verum
end;
suppose j = 4 ; ::_thesis: w in FS
then h . j = v by A30, A33;
hence w in FS by A36; ::_thesis: verum
end;
end;
end;
then reconsider h = h as FinSequence of FS by FINSEQ_1:def_4;
len h <> 0 by A29, FINSEQ_1:def_3;
then reconsider h = h as non empty FinSequence of FS ;
A37: h . 1 = x by A30, A18;
A38: for j being Element of NAT st 1 <= j & j < len h holds
( ( j is odd implies [(h . j),(h . (j + 1))] in e1 ) & ( j is even implies [(h . j),(h . (j + 1))] in e2 ) )
proof
let j be Element of NAT ; ::_thesis: ( 1 <= j & j < len h implies ( ( j is odd implies [(h . j),(h . (j + 1))] in e1 ) & ( j is even implies [(h . j),(h . (j + 1))] in e2 ) ) )
assume A39: ( 1 <= j & j < len h ) ; ::_thesis: ( ( j is odd implies [(h . j),(h . (j + 1))] in e1 ) & ( j is even implies [(h . j),(h . (j + 1))] in e2 ) )
percases ( j = 1 or j = 2 or j = 3 ) by A31, A39, Lm4;
supposeA40: j = 1 ; ::_thesis: ( ( j is odd implies [(h . j),(h . (j + 1))] in e1 ) & ( j is even implies [(h . j),(h . (j + 1))] in e2 ) )
[u,z1] in e19 by A16, A20;
then [(h . 1),z1] in e19 by A30, A18;
hence ( ( j is odd implies [(h . j),(h . (j + 1))] in e1 ) & ( j is even implies [(h . j),(h . (j + 1))] in e2 ) ) by A8, A15, A30, A34, A40; ::_thesis: verum
end;
supposeA41: j = 2 ; ::_thesis: ( ( j is odd implies [(h . j),(h . (j + 1))] in e1 ) & ( j is even implies [(h . j),(h . (j + 1))] in e2 ) )
(FD . (u,v)) "\/" a <= (a "\/" b) "\/" a by A19, WAYBEL_1:2;
then (FD . (u,v)) "\/" a <= b "\/" (a "\/" a) by LATTICE3:14;
then (FD . (u,v)) "\/" a <= b "\/" a by YELLOW_5:1;
then ((FD . (u,v)) "\/" a) "/\" b <= b "/\" (b "\/" a) by WAYBEL_1:1;
then ((FD . (u,v)) "\/" a) "/\" b <= b by LATTICE3:18;
then [z1,z2] in e29 by A14, A21;
then A42: [(h . 2),z2] in e29 by A30, A34;
consider i being Element of NAT such that
A43: i = 1 ;
2 * i = j by A41, A43;
hence ( ( j is odd implies [(h . j),(h . (j + 1))] in e1 ) & ( j is even implies [(h . j),(h . (j + 1))] in e2 ) ) by A10, A13, A30, A32, A41, A42; ::_thesis: verum
end;
supposeA44: j = 3 ; ::_thesis: ( ( j is odd implies [(h . j),(h . (j + 1))] in e1 ) & ( j is even implies [(h . j),(h . (j + 1))] in e2 ) )
[z2,v] in e19 by A16, A22;
then A45: [(h . 3),v] in e19 by A30, A32;
consider i being Element of NAT such that
A46: i = 1 ;
(2 * i) + 1 = j by A44, A46;
hence ( ( j is odd implies [(h . j),(h . (j + 1))] in e1 ) & ( j is even implies [(h . j),(h . (j + 1))] in e2 ) ) by A8, A15, A30, A33, A44, A45; ::_thesis: verum
end;
end;
end;
take h ; ::_thesis: ( len h = 2 + 2 & x,y are_joint_by h,e1,e2 )
thus len h = 2 + 2 by A29, FINSEQ_1:def_3; ::_thesis: x,y are_joint_by h,e1,e2
h . (len h) = h . 4 by A29, FINSEQ_1:def_3
.= y by A30, A33 ;
hence x,y are_joint_by h,e1,e2 by A37, A38, LATTICE5:def_3; ::_thesis: verum
end;
theorem Th35: :: LATTICE8:35
for L being lower-bounded modular LATTICE holds L has_a_representation_of_type<= 2
proof
let L be lower-bounded modular LATTICE; ::_thesis: L has_a_representation_of_type<= 2
set A = the carrier of L;
set D = BasicDF L;
set S = the ExtensionSeq2 of the carrier of L, BasicDF L;
set FS = union { (( the ExtensionSeq2 of the carrier of L, BasicDF L . i) `1) where i is Element of NAT : verum } ;
A1: ( the ExtensionSeq2 of the carrier of L, BasicDF L . 0) `1 in { (( the ExtensionSeq2 of the carrier of L, BasicDF L . i) `1) where i is Element of NAT : verum } ;
A2: the ExtensionSeq2 of the carrier of L, BasicDF L . 0 = [ the carrier of L,(BasicDF L)] by Def11;
[ the carrier of L,(BasicDF L)] `1 = the carrier of L ;
then the carrier of L c= union { (( the ExtensionSeq2 of the carrier of L, BasicDF L . i) `1) where i is Element of NAT : verum } by A1, A2, ZFMISC_1:74;
then reconsider FS = union { (( the ExtensionSeq2 of the carrier of L, BasicDF L . i) `1) where i is Element of NAT : verum } as non empty set ;
reconsider FD = union { (( the ExtensionSeq2 of the carrier of L, BasicDF L . i) `2) where i is Element of NAT : verum } as distance_function of FS,L by Th32;
alpha FD is join-preserving
proof
set f = alpha FD;
let a, b be Element of L; :: according to WAYBEL_0:def_35 ::_thesis: alpha FD preserves_sup_of {a,b}
A3: ex_sup_of (alpha FD) .: {a,b}, EqRelLATT FS by YELLOW_0:17;
consider e2 being Equivalence_Relation of FS such that
A4: e2 = (alpha FD) . b and
A5: for x, y being Element of FS holds
( [x,y] in e2 iff FD . (x,y) <= b ) by LATTICE5:def_8;
consider e1 being Equivalence_Relation of FS such that
A6: e1 = (alpha FD) . a and
A7: for x, y being Element of FS holds
( [x,y] in e1 iff FD . (x,y) <= a ) by LATTICE5:def_8;
consider e3 being Equivalence_Relation of FS such that
A8: e3 = (alpha FD) . (a "\/" b) and
A9: for x, y being Element of FS holds
( [x,y] in e3 iff FD . (x,y) <= a "\/" b ) by LATTICE5:def_8;
A10: field e2 = FS by ORDERS_1:12;
now__::_thesis:_for_x,_y_being_set_st_[x,y]_in_e2_holds_
[x,y]_in_e3
let x, y be set ; ::_thesis: ( [x,y] in e2 implies [x,y] in e3 )
A11: b <= b "\/" a by YELLOW_0:22;
assume A12: [x,y] in e2 ; ::_thesis: [x,y] in e3
then reconsider x9 = x, y9 = y as Element of FS by A10, RELAT_1:15;
FD . (x9,y9) <= b by A5, A12;
then FD . (x9,y9) <= b "\/" a by A11, ORDERS_2:3;
hence [x,y] in e3 by A9; ::_thesis: verum
end;
then A13: e2 c= e3 by RELAT_1:def_3;
A14: field e3 = FS by ORDERS_1:12;
for u, v being set st [u,v] in e3 holds
[u,v] in e1 "\/" e2
proof
let u, v be set ; ::_thesis: ( [u,v] in e3 implies [u,v] in e1 "\/" e2 )
A15: Seg 4 = { n where n is Element of NAT : ( 1 <= n & n <= 4 ) } by FINSEQ_1:def_1;
then A16: 3 in Seg 4 ;
assume A17: [u,v] in e3 ; ::_thesis: [u,v] in e1 "\/" e2
then reconsider x = u, y = v as Element of FS by A14, RELAT_1:15;
A18: FD . (x,y) <= a "\/" b by A9, A17;
then consider z1, z2 being Element of FS such that
A19: FD . (x,z1) = a and
A20: FD . (z1,z2) = ((FD . (x,y)) "\/" a) "/\" b and
A21: FD . (z2,y) = a by Th33;
A22: u in FS by A14, A17, RELAT_1:15;
defpred S1[ set , set ] means ( ( $1 = 1 implies $2 = x ) & ( $1 = 2 implies $2 = z1 ) & ( $1 = 3 implies $2 = z2 ) & ( $1 = 4 implies $2 = y ) );
A23: for m being Nat st m in Seg 4 holds
ex w being set st S1[m,w]
proof
let m be Nat; ::_thesis: ( m in Seg 4 implies ex w being set st S1[m,w] )
assume A24: m in Seg 4 ; ::_thesis: ex w being set st S1[m,w]
percases ( m = 1 or m = 2 or m = 3 or m = 4 ) by A24, Lm3;
supposeA25: m = 1 ; ::_thesis: ex w being set st S1[m,w]
take x ; ::_thesis: S1[m,x]
thus S1[m,x] by A25; ::_thesis: verum
end;
supposeA26: m = 2 ; ::_thesis: ex w being set st S1[m,w]
take z1 ; ::_thesis: S1[m,z1]
thus S1[m,z1] by A26; ::_thesis: verum
end;
supposeA27: m = 3 ; ::_thesis: ex w being set st S1[m,w]
take z2 ; ::_thesis: S1[m,z2]
thus S1[m,z2] by A27; ::_thesis: verum
end;
supposeA28: m = 4 ; ::_thesis: ex w being set st S1[m,w]
take y ; ::_thesis: S1[m,y]
thus S1[m,y] by A28; ::_thesis: verum
end;
end;
end;
ex p being FinSequence st
( dom p = Seg 4 & ( for k being Nat st k in Seg 4 holds
S1[k,p . k] ) ) from FINSEQ_1:sch_1(A23);
then consider h being FinSequence such that
A29: dom h = Seg 4 and
A30: for m being Nat st m in Seg 4 holds
( ( m = 1 implies h . m = x ) & ( m = 2 implies h . m = z1 ) & ( m = 3 implies h . m = z2 ) & ( m = 4 implies h . m = y ) ) ;
A31: len h = 4 by A29, FINSEQ_1:def_3;
A32: 4 in Seg 4 by A15;
A33: 1 in Seg 4 by A15;
then A34: u = h . 1 by A30;
A35: 2 in Seg 4 by A15;
A36: for j being Element of NAT st 1 <= j & j < len h holds
[(h . j),(h . (j + 1))] in e1 \/ e2
proof
let j be Element of NAT ; ::_thesis: ( 1 <= j & j < len h implies [(h . j),(h . (j + 1))] in e1 \/ e2 )
assume A37: ( 1 <= j & j < len h ) ; ::_thesis: [(h . j),(h . (j + 1))] in e1 \/ e2
percases ( j = 1 or j = 2 or j = 3 ) by A31, A37, Lm4;
supposeA38: j = 1 ; ::_thesis: [(h . j),(h . (j + 1))] in e1 \/ e2
[x,z1] in e1 by A7, A19;
then [(h . 1),z1] in e1 by A30, A33;
then [(h . 1),(h . 2)] in e1 by A30, A35;
hence [(h . j),(h . (j + 1))] in e1 \/ e2 by A38, XBOOLE_0:def_3; ::_thesis: verum
end;
supposeA39: j = 2 ; ::_thesis: [(h . j),(h . (j + 1))] in e1 \/ e2
(FD . (x,y)) "\/" a <= (a "\/" b) "\/" a by A18, WAYBEL_1:2;
then (FD . (x,y)) "\/" a <= b "\/" (a "\/" a) by LATTICE3:14;
then (FD . (x,y)) "\/" a <= b "\/" a by YELLOW_5:1;
then ((FD . (x,y)) "\/" a) "/\" b <= b "/\" (b "\/" a) by WAYBEL_1:1;
then ((FD . (x,y)) "\/" a) "/\" b <= b by LATTICE3:18;
then [z1,z2] in e2 by A5, A20;
then [(h . 2),z2] in e2 by A30, A35;
then [(h . 2),(h . 3)] in e2 by A30, A16;
hence [(h . j),(h . (j + 1))] in e1 \/ e2 by A39, XBOOLE_0:def_3; ::_thesis: verum
end;
supposeA40: j = 3 ; ::_thesis: [(h . j),(h . (j + 1))] in e1 \/ e2
[z2,y] in e1 by A7, A21;
then [(h . 3),y] in e1 by A30, A16;
then [(h . 3),(h . 4)] in e1 by A30, A32;
hence [(h . j),(h . (j + 1))] in e1 \/ e2 by A40, XBOOLE_0:def_3; ::_thesis: verum
end;
end;
end;
v = h . 4 by A30, A32
.= h . (len h) by A29, FINSEQ_1:def_3 ;
hence [u,v] in e1 "\/" e2 by A22, A31, A34, A36, EQREL_1:28; ::_thesis: verum
end;
then A41: e3 c= e1 "\/" e2 by RELAT_1:def_3;
A42: field e1 = FS by ORDERS_1:12;
now__::_thesis:_for_x,_y_being_set_st_[x,y]_in_e1_holds_
[x,y]_in_e3
let x, y be set ; ::_thesis: ( [x,y] in e1 implies [x,y] in e3 )
A43: a <= a "\/" b by YELLOW_0:22;
assume A44: [x,y] in e1 ; ::_thesis: [x,y] in e3
then reconsider x9 = x, y9 = y as Element of FS by A42, RELAT_1:15;
FD . (x9,y9) <= a by A7, A44;
then FD . (x9,y9) <= a "\/" b by A43, ORDERS_2:3;
hence [x,y] in e3 by A9; ::_thesis: verum
end;
then e1 c= e3 by RELAT_1:def_3;
then e1 \/ e2 c= e3 by A13, XBOOLE_1:8;
then A45: e1 "\/" e2 c= e3 by EQREL_1:def_2;
dom (alpha FD) = the carrier of L by FUNCT_2:def_1;
then sup ((alpha FD) .: {a,b}) = sup {((alpha FD) . a),((alpha FD) . b)} by FUNCT_1:60
.= ((alpha FD) . a) "\/" ((alpha FD) . b) by YELLOW_0:41
.= e1 "\/" e2 by A6, A4, LATTICE5:10
.= (alpha FD) . (a "\/" b) by A8, A45, A41, XBOOLE_0:def_10
.= (alpha FD) . (sup {a,b}) by YELLOW_0:41 ;
hence alpha FD preserves_sup_of {a,b} by A3, WAYBEL_0:def_31; ::_thesis: verum
end;
then reconsider f = alpha FD as Homomorphism of L,(EqRelLATT FS) by LATTICE5:14;
A46: dom f = the carrier of L by FUNCT_2:def_1;
A47: ex e being Equivalence_Relation of FS st
( e in the carrier of (Image f) & e <> id FS )
proof
A48: { the carrier of L} <> {{ the carrier of L}}
proof
assume { the carrier of L} = {{ the carrier of L}} ; ::_thesis: contradiction
then { the carrier of L} in { the carrier of L} by TARSKI:def_1;
hence contradiction ; ::_thesis: verum
end;
consider A9 being non empty set , d9 being distance_function of A9,L, Aq9 being non empty set , dq9 being distance_function of Aq9,L such that
A49: Aq9,dq9 is_extension2_of A9,d9 and
A50: the ExtensionSeq2 of the carrier of L, BasicDF L . 0 = [A9,d9] and
A51: the ExtensionSeq2 of the carrier of L, BasicDF L . (0 + 1) = [Aq9,dq9] by Def11;
( A9 = the carrier of L & d9 = BasicDF L ) by A2, A50, XTUPLE_0:1;
then consider q being QuadrSeq of BasicDF L such that
A52: Aq9 = NextSet2 (BasicDF L) and
A53: dq9 = NextDelta2 q by A49, Def10;
ConsecutiveSet2 ( the carrier of L,{}) = the carrier of L by Th14;
then reconsider Q = Quadr2 (q,{}) as Element of [: the carrier of L, the carrier of L, the carrier of L, the carrier of L:] ;
A54: ( the ExtensionSeq2 of the carrier of L, BasicDF L . 1) `2 in { (( the ExtensionSeq2 of the carrier of L, BasicDF L . i) `2) where i is Element of NAT : verum } ;
succ {} c= DistEsti (BasicDF L) by Th1;
then {} in DistEsti (BasicDF L) by ORDINAL1:21;
then A55: {} in dom q by LATTICE5:25;
then q . {} in rng q by FUNCT_1:def_3;
then q . {} in { [u,v,a9,b9] where u, v is Element of the carrier of L, a9, b9 is Element of L : (BasicDF L) . (u,v) <= a9 "\/" b9 } by LATTICE5:def_13;
then consider u, v being Element of the carrier of L, a, b being Element of L such that
A56: q . {} = [u,v,a,b] and
(BasicDF L) . (u,v) <= a "\/" b ;
consider e being Equivalence_Relation of FS such that
A57: e = f . b and
A58: for x, y being Element of FS holds
( [x,y] in e iff FD . (x,y) <= b ) by LATTICE5:def_8;
Quadr2 (q,{}) = [u,v,a,b] by A55, A56, Def6;
then A59: b = Q `4_4 by MCART_1:def_11;
[Aq9,dq9] `2 = NextDelta2 q by A53;
then A60: NextDelta2 q c= FD by A54, A51, ZFMISC_1:74;
A61: {{ the carrier of L}} in {{ the carrier of L},{{ the carrier of L}}} by TARSKI:def_2;
then A62: {{ the carrier of L}} in the carrier of L \/ {{ the carrier of L},{{ the carrier of L}}} by XBOOLE_0:def_3;
take e ; ::_thesis: ( e in the carrier of (Image f) & e <> id FS )
e in rng f by A46, A57, FUNCT_1:def_3;
hence e in the carrier of (Image f) by YELLOW_0:def_15; ::_thesis: e <> id FS
A63: ( the ExtensionSeq2 of the carrier of L, BasicDF L . 1) `1 in { (( the ExtensionSeq2 of the carrier of L, BasicDF L . i) `1) where i is Element of NAT : verum } ;
[Aq9,dq9] `1 = NextSet2 (BasicDF L) by A52;
then A64: NextSet2 (BasicDF L) c= FS by A63, A51, ZFMISC_1:74;
new_set2 the carrier of L = new_set2 (ConsecutiveSet2 ( the carrier of L,{})) by Th14
.= ConsecutiveSet2 ( the carrier of L,(succ {})) by Th15 ;
then new_set2 the carrier of L c= NextSet2 (BasicDF L) by Th1, Th21;
then A65: new_set2 the carrier of L c= FS by A64, XBOOLE_1:1;
A66: {{ the carrier of L}} in new_set2 the carrier of L by A61, XBOOLE_0:def_3;
A67: { the carrier of L} in {{ the carrier of L},{{ the carrier of L}}} by TARSKI:def_2;
then { the carrier of L} in the carrier of L \/ {{ the carrier of L},{{ the carrier of L}}} by XBOOLE_0:def_3;
then reconsider W = { the carrier of L}, V = {{ the carrier of L}} as Element of FS by A65, A66;
A68: ( ConsecutiveSet2 ( the carrier of L,{}) = the carrier of L & ConsecutiveDelta2 (q,{}) = BasicDF L ) by Th14, Th18;
ConsecutiveDelta2 (q,(succ {})) = new_bi_fun2 ((BiFun ((ConsecutiveDelta2 (q,{})),(ConsecutiveSet2 ( the carrier of L,{})),L)),(Quadr2 (q,{}))) by Th19
.= new_bi_fun2 ((BasicDF L),Q) by A68, LATTICE5:def_15 ;
then new_bi_fun2 ((BasicDF L),Q) c= NextDelta2 q by Th1, Th24;
then A69: new_bi_fun2 ((BasicDF L),Q) c= FD by A60, XBOOLE_1:1;
( dom (new_bi_fun2 ((BasicDF L),Q)) = [:(new_set2 the carrier of L),(new_set2 the carrier of L):] & { the carrier of L} in new_set2 the carrier of L ) by A67, FUNCT_2:def_1, XBOOLE_0:def_3;
then [{ the carrier of L},{{ the carrier of L}}] in dom (new_bi_fun2 ((BasicDF L),Q)) by A62, ZFMISC_1:87;
then FD . (W,V) = (new_bi_fun2 ((BasicDF L),Q)) . ({ the carrier of L},{{ the carrier of L}}) by A69, GRFUNC_1:2
.= (((BasicDF L) . ((Q `1_4),(Q `2_4))) "\/" (Q `3_4)) "/\" (Q `4_4) by Def4 ;
then FD . (W,V) <= b by A59, YELLOW_0:23;
then [{ the carrier of L},{{ the carrier of L}}] in e by A58;
hence e <> id FS by A48, RELAT_1:def_10; ::_thesis: verum
end;
A70: now__::_thesis:_not_FS_is_trivial
consider e being Equivalence_Relation of FS such that
e in the carrier of (Image f) and
A71: e <> id FS by A47;
assume FS is trivial ; ::_thesis: contradiction
then consider x being set such that
A72: FS = {x} by ZFMISC_1:131;
A73: ( [:{x},{x}:] = {[x,x]} & id {x} = {[x,x]} ) by SYSREL:13, ZFMISC_1:29;
field e = {x} by A72, EQREL_1:9;
then id FS c= e by A72, RELAT_2:1;
hence contradiction by A72, A71, A73, XBOOLE_0:def_10; ::_thesis: verum
end;
BasicDF L is onto by LATTICE5:40;
then A74: rng (BasicDF L) = the carrier of L by FUNCT_2:def_3;
for w being set st w in the carrier of L holds
ex z being set st
( z in [:FS,FS:] & w = FD . z )
proof
let w be set ; ::_thesis: ( w in the carrier of L implies ex z being set st
( z in [:FS,FS:] & w = FD . z ) )
A75: ( the ExtensionSeq2 of the carrier of L, BasicDF L . 0) `1 in { (( the ExtensionSeq2 of the carrier of L, BasicDF L . i) `1) where i is Element of NAT : verum } ;
A76: ( the ExtensionSeq2 of the carrier of L, BasicDF L . 0) `2 in { (( the ExtensionSeq2 of the carrier of L, BasicDF L . i) `2) where i is Element of NAT : verum } ;
A77: the ExtensionSeq2 of the carrier of L, BasicDF L . 0 = [ the carrier of L,(BasicDF L)] by Def11;
BasicDF L = [ the carrier of L,(BasicDF L)] `2 ;
then A78: BasicDF L c= FD by A76, A77, ZFMISC_1:74;
assume w in the carrier of L ; ::_thesis: ex z being set st
( z in [:FS,FS:] & w = FD . z )
then consider z being set such that
A79: z in [: the carrier of L, the carrier of L:] and
A80: (BasicDF L) . z = w by A74, FUNCT_2:11;
take z ; ::_thesis: ( z in [:FS,FS:] & w = FD . z )
the carrier of L = [ the carrier of L,(BasicDF L)] `1 ;
then the carrier of L c= FS by A75, A77, ZFMISC_1:74;
then [: the carrier of L, the carrier of L:] c= [:FS,FS:] by ZFMISC_1:96;
hence z in [:FS,FS:] by A79; ::_thesis: w = FD . z
z in dom (BasicDF L) by A79, FUNCT_2:def_1;
hence w = FD . z by A80, A78, GRFUNC_1:2; ::_thesis: verum
end;
then rng FD = the carrier of L by FUNCT_2:10;
then A81: FD is onto by FUNCT_2:def_3;
reconsider FS = FS as non trivial set by A70;
take FS ; :: according to LATTICE8:def_2 ::_thesis: ex f being Homomorphism of L,(EqRelLATT FS) st
( f is V14() & Image f is finitely_typed & ex e being Equivalence_Relation of FS st
( e in the carrier of (Image f) & e <> id FS ) & type_of (Image f) <= 2 )
reconsider f = f as Homomorphism of L,(EqRelLATT FS) ;
take f ; ::_thesis: ( f is V14() & Image f is finitely_typed & ex e being Equivalence_Relation of FS st
( e in the carrier of (Image f) & e <> id FS ) & type_of (Image f) <= 2 )
thus f is V14() by A81, LATTICE5:15; ::_thesis: ( Image f is finitely_typed & ex e being Equivalence_Relation of FS st
( e in the carrier of (Image f) & e <> id FS ) & type_of (Image f) <= 2 )
reconsider FD = FD as distance_function of FS,L ;
thus Image f is finitely_typed ::_thesis: ( ex e being Equivalence_Relation of FS st
( e in the carrier of (Image f) & e <> id FS ) & type_of (Image f) <= 2 )
proof
take FS ; :: according to LATTICE8:def_1 ::_thesis: ( ( for e being set st e in the carrier of (Image f) holds
e is Equivalence_Relation of FS ) & ex o being Element of NAT st
for e1, e2 being Equivalence_Relation of FS
for x, y being set st e1 in the carrier of (Image f) & e2 in the carrier of (Image f) & [x,y] in e1 "\/" e2 holds
ex F being non empty FinSequence of FS st
( len F = o & x,y are_joint_by F,e1,e2 ) )
thus for e being set st e in the carrier of (Image f) holds
e is Equivalence_Relation of FS ::_thesis: ex o being Element of NAT st
for e1, e2 being Equivalence_Relation of FS
for x, y being set st e1 in the carrier of (Image f) & e2 in the carrier of (Image f) & [x,y] in e1 "\/" e2 holds
ex F being non empty FinSequence of FS st
( len F = o & x,y are_joint_by F,e1,e2 )
proof
let e be set ; ::_thesis: ( e in the carrier of (Image f) implies e is Equivalence_Relation of FS )
assume e in the carrier of (Image f) ; ::_thesis: e is Equivalence_Relation of FS
then e in rng f by YELLOW_0:def_15;
then consider x being set such that
A82: x in dom f and
A83: e = f . x by FUNCT_1:def_3;
reconsider x = x as Element of L by A82;
ex E being Equivalence_Relation of FS st
( E = f . x & ( for u, v being Element of FS holds
( [u,v] in E iff FD . (u,v) <= x ) ) ) by LATTICE5:def_8;
hence e is Equivalence_Relation of FS by A83; ::_thesis: verum
end;
take 2 + 2 ; ::_thesis: for e1, e2 being Equivalence_Relation of FS
for x, y being set st e1 in the carrier of (Image f) & e2 in the carrier of (Image f) & [x,y] in e1 "\/" e2 holds
ex F being non empty FinSequence of FS st
( len F = 2 + 2 & x,y are_joint_by F,e1,e2 )
thus for e1, e2 being Equivalence_Relation of FS
for x, y being set st e1 in the carrier of (Image f) & e2 in the carrier of (Image f) & [x,y] in e1 "\/" e2 holds
ex F being non empty FinSequence of FS st
( len F = 2 + 2 & x,y are_joint_by F,e1,e2 ) by Th34; ::_thesis: verum
end;
thus ex e being Equivalence_Relation of FS st
( e in the carrier of (Image f) & e <> id FS ) ::_thesis: type_of (Image f) <= 2
proof
A84: { the carrier of L} <> {{ the carrier of L}}
proof
assume { the carrier of L} = {{ the carrier of L}} ; ::_thesis: contradiction
then { the carrier of L} in { the carrier of L} by TARSKI:def_1;
hence contradiction ; ::_thesis: verum
end;
consider A9 being non empty set , d9 being distance_function of A9,L, Aq9 being non empty set , dq9 being distance_function of Aq9,L such that
A85: Aq9,dq9 is_extension2_of A9,d9 and
A86: the ExtensionSeq2 of the carrier of L, BasicDF L . 0 = [A9,d9] and
A87: the ExtensionSeq2 of the carrier of L, BasicDF L . (0 + 1) = [Aq9,dq9] by Def11;
( A9 = the carrier of L & d9 = BasicDF L ) by A2, A86, XTUPLE_0:1;
then consider q being QuadrSeq of BasicDF L such that
A88: Aq9 = NextSet2 (BasicDF L) and
A89: dq9 = NextDelta2 q by A85, Def10;
ConsecutiveSet2 ( the carrier of L,{}) = the carrier of L by Th14;
then reconsider Q = Quadr2 (q,{}) as Element of [: the carrier of L, the carrier of L, the carrier of L, the carrier of L:] ;
A90: ( the ExtensionSeq2 of the carrier of L, BasicDF L . 1) `2 in { (( the ExtensionSeq2 of the carrier of L, BasicDF L . i) `2) where i is Element of NAT : verum } ;
succ {} c= DistEsti (BasicDF L) by Th1;
then {} in DistEsti (BasicDF L) by ORDINAL1:21;
then A91: {} in dom q by LATTICE5:25;
then q . {} in rng q by FUNCT_1:def_3;
then q . {} in { [u,v,a9,b9] where u, v is Element of the carrier of L, a9, b9 is Element of L : (BasicDF L) . (u,v) <= a9 "\/" b9 } by LATTICE5:def_13;
then consider u, v being Element of the carrier of L, a, b being Element of L such that
A92: q . {} = [u,v,a,b] and
(BasicDF L) . (u,v) <= a "\/" b ;
consider e being Equivalence_Relation of FS such that
A93: e = f . b and
A94: for x, y being Element of FS holds
( [x,y] in e iff FD . (x,y) <= b ) by LATTICE5:def_8;
Quadr2 (q,{}) = [u,v,a,b] by A91, A92, Def6;
then A95: b = Q `4_4 by MCART_1:def_11;
[Aq9,dq9] `2 = NextDelta2 q by A89;
then A96: NextDelta2 q c= FD by A90, A87, ZFMISC_1:74;
A97: {{ the carrier of L}} in {{ the carrier of L},{{ the carrier of L}}} by TARSKI:def_2;
then A98: {{ the carrier of L}} in the carrier of L \/ {{ the carrier of L},{{ the carrier of L}}} by XBOOLE_0:def_3;
take e ; ::_thesis: ( e in the carrier of (Image f) & e <> id FS )
e in rng f by A46, A93, FUNCT_1:def_3;
hence e in the carrier of (Image f) by YELLOW_0:def_15; ::_thesis: e <> id FS
A99: ( the ExtensionSeq2 of the carrier of L, BasicDF L . 1) `1 in { (( the ExtensionSeq2 of the carrier of L, BasicDF L . i) `1) where i is Element of NAT : verum } ;
[Aq9,dq9] `1 = NextSet2 (BasicDF L) by A88;
then A100: NextSet2 (BasicDF L) c= FS by A99, A87, ZFMISC_1:74;
new_set2 the carrier of L = new_set2 (ConsecutiveSet2 ( the carrier of L,{})) by Th14
.= ConsecutiveSet2 ( the carrier of L,(succ {})) by Th15 ;
then new_set2 the carrier of L c= NextSet2 (BasicDF L) by Th1, Th21;
then A101: new_set2 the carrier of L c= FS by A100, XBOOLE_1:1;
A102: {{ the carrier of L}} in new_set2 the carrier of L by A97, XBOOLE_0:def_3;
A103: { the carrier of L} in {{ the carrier of L},{{ the carrier of L}}} by TARSKI:def_2;
then { the carrier of L} in the carrier of L \/ {{ the carrier of L},{{ the carrier of L}}} by XBOOLE_0:def_3;
then reconsider W = { the carrier of L}, V = {{ the carrier of L}} as Element of FS by A101, A102;
A104: ( ConsecutiveSet2 ( the carrier of L,{}) = the carrier of L & ConsecutiveDelta2 (q,{}) = BasicDF L ) by Th14, Th18;
ConsecutiveDelta2 (q,(succ {})) = new_bi_fun2 ((BiFun ((ConsecutiveDelta2 (q,{})),(ConsecutiveSet2 ( the carrier of L,{})),L)),(Quadr2 (q,{}))) by Th19
.= new_bi_fun2 ((BasicDF L),Q) by A104, LATTICE5:def_15 ;
then new_bi_fun2 ((BasicDF L),Q) c= NextDelta2 q by Th1, Th24;
then A105: new_bi_fun2 ((BasicDF L),Q) c= FD by A96, XBOOLE_1:1;
( dom (new_bi_fun2 ((BasicDF L),Q)) = [:(new_set2 the carrier of L),(new_set2 the carrier of L):] & { the carrier of L} in new_set2 the carrier of L ) by A103, FUNCT_2:def_1, XBOOLE_0:def_3;
then [{ the carrier of L},{{ the carrier of L}}] in dom (new_bi_fun2 ((BasicDF L),Q)) by A98, ZFMISC_1:87;
then FD . (W,V) = (new_bi_fun2 ((BasicDF L),Q)) . ({ the carrier of L},{{ the carrier of L}}) by A105, GRFUNC_1:2
.= (((BasicDF L) . ((Q `1_4),(Q `2_4))) "\/" (Q `3_4)) "/\" (Q `4_4) by Def4 ;
then FD . (W,V) <= b by A95, YELLOW_0:23;
then [{ the carrier of L},{{ the carrier of L}}] in e by A94;
hence e <> id FS by A84, RELAT_1:def_10; ::_thesis: verum
end;
for e1, e2 being Equivalence_Relation of FS
for x, y being set st e1 in the carrier of (Image f) & e2 in the carrier of (Image f) & [x,y] in e1 "\/" e2 holds
ex F being non empty FinSequence of FS st
( len F = 2 + 2 & x,y are_joint_by F,e1,e2 ) by Th34;
hence type_of (Image f) <= 2 by A47, LATTICE5:13; ::_thesis: verum
end;
theorem :: LATTICE8:36
for L being lower-bounded LATTICE holds
( L has_a_representation_of_type<= 2 iff L is modular ) by Th9, Th35;