:: Translations, Endomorphisms, and Stable Equational Theories
:: by Grzegorz Bancerek
::
:: Received February 9, 1996
:: Copyright (c) 1996-2021 Association of Mizar Users

definition
let S be non empty ManySortedSign ;
let A be MSAlgebra over S;
let s be SortSymbol of S;
mode Element of A,s is Element of the Sorts of A . s;
end;

definition
let I be set ;
let A be ManySortedSet of I;
let h1, h2 be ManySortedFunction of A,A;
:: original: **
redefine func h2 ** h1 -> ManySortedFunction of A,A;
coherence
h2 ** h1 is ManySortedFunction of A,A
proof end;
end;

theorem Th1: :: MSUALG_6:1
for S being non empty non void ManySortedSign
for A being MSAlgebra over S
for o being OperSymbol of S
for a being set st a in Args (o,A) holds
a is Function
proof end;

theorem Th2: :: MSUALG_6:2
for S being non empty non void ManySortedSign
for A being MSAlgebra over S
for o being OperSymbol of S
for a being Function st a in Args (o,A) holds
( dom a = dom () & ( for i being set st i in dom () holds
a . i in the Sorts of A . (() /. i) ) )
proof end;

definition
let S be non empty non void ManySortedSign ;
let A be MSAlgebra over S;
attr A is feasible means :Def1: :: MSUALG_6:def 1
for o being OperSymbol of S st Args (o,A) <> {} holds
Result (o,A) <> {} ;
end;

:: deftheorem Def1 defines feasible MSUALG_6:def 1 :
for S being non empty non void ManySortedSign
for A being MSAlgebra over S holds
( A is feasible iff for o being OperSymbol of S st Args (o,A) <> {} holds
Result (o,A) <> {} );

theorem Th3: :: MSUALG_6:3
for S being non empty non void ManySortedSign
for o being OperSymbol of S
for A being MSAlgebra over S holds
( Args (o,A) <> {} iff for i being Element of NAT st i in dom () holds
the Sorts of A . (() /. i) <> {} )
proof end;

registration
let S be non empty non void ManySortedSign ;
cluster non-empty -> feasible for MSAlgebra over S;
coherence
for b1 being MSAlgebra over S st b1 is non-empty holds
b1 is feasible
;
end;

registration
let S be non empty non void ManySortedSign ;
cluster non-empty for MSAlgebra over S;
existence
ex b1 being MSAlgebra over S st b1 is non-empty
proof end;
end;

definition
let S be non empty non void ManySortedSign ;
let A be MSAlgebra over S;
mode Endomorphism of A -> ManySortedFunction of A,A means :Def2: :: MSUALG_6:def 2
it is_homomorphism A,A;
existence
ex b1 being ManySortedFunction of A,A st b1 is_homomorphism A,A
proof end;
end;

:: deftheorem Def2 defines Endomorphism MSUALG_6:def 2 :
for S being non empty non void ManySortedSign
for A being MSAlgebra over S
for b3 being ManySortedFunction of A,A holds
( b3 is Endomorphism of A iff b3 is_homomorphism A,A );

theorem Th4: :: MSUALG_6:4
for S being non empty non void ManySortedSign
for A being MSAlgebra over S holds id the Sorts of A is Endomorphism of A
proof end;

theorem Th5: :: MSUALG_6:5
for S being non empty non void ManySortedSign
for A being MSAlgebra over S
for h1, h2 being ManySortedFunction of A,A
for o being OperSymbol of S
for a being Element of Args (o,A) st a in Args (o,A) holds
h2 # (h1 # a) = (h2 ** h1) # a
proof end;

theorem Th6: :: MSUALG_6:6
for S being non empty non void ManySortedSign
for A being MSAlgebra over S
for h1, h2 being Endomorphism of A holds h2 ** h1 is Endomorphism of A
proof end;

definition
let S be non empty non void ManySortedSign ;
let A be MSAlgebra over S;
let h1, h2 be Endomorphism of A;
:: original: **
redefine func h2 ** h1 -> Endomorphism of A;
coherence
h2 ** h1 is Endomorphism of A
by Th6;
end;

definition
let S be non empty non void ManySortedSign ;
func TranslationRel S -> Relation of the carrier of S means :Def3: :: MSUALG_6:def 3
for s1, s2 being SortSymbol of S holds
( [s1,s2] in it iff ex o being OperSymbol of S st
( the_result_sort_of o = s2 & ex i being Element of NAT st
( i in dom () & () /. i = s1 ) ) );
existence
ex b1 being Relation of the carrier of S st
for s1, s2 being SortSymbol of S holds
( [s1,s2] in b1 iff ex o being OperSymbol of S st
( the_result_sort_of o = s2 & ex i being Element of NAT st
( i in dom () & () /. i = s1 ) ) )
proof end;
correctness
uniqueness
for b1, b2 being Relation of the carrier of S st ( for s1, s2 being SortSymbol of S holds
( [s1,s2] in b1 iff ex o being OperSymbol of S st
( the_result_sort_of o = s2 & ex i being Element of NAT st
( i in dom () & () /. i = s1 ) ) ) ) & ( for s1, s2 being SortSymbol of S holds
( [s1,s2] in b2 iff ex o being OperSymbol of S st
( the_result_sort_of o = s2 & ex i being Element of NAT st
( i in dom () & () /. i = s1 ) ) ) ) holds
b1 = b2
;
proof end;
end;

:: deftheorem Def3 defines TranslationRel MSUALG_6:def 3 :
for S being non empty non void ManySortedSign
for b2 being Relation of the carrier of S holds
( b2 = TranslationRel S iff for s1, s2 being SortSymbol of S holds
( [s1,s2] in b2 iff ex o being OperSymbol of S st
( the_result_sort_of o = s2 & ex i being Element of NAT st
( i in dom () & () /. i = s1 ) ) ) );

theorem Th7: :: MSUALG_6:7
for S being non empty non void ManySortedSign
for o being OperSymbol of S
for A being MSAlgebra over S
for a being Function st a in Args (o,A) holds
for i being Nat
for x being Element of A,(() /. i) holds a +* (i,x) in Args (o,A)
proof end;

theorem Th8: :: MSUALG_6:8
for S being non empty non void ManySortedSign
for A1, A2 being MSAlgebra over S
for h being ManySortedFunction of A1,A2
for o being OperSymbol of S st Args (o,A1) <> {} & Args (o,A2) <> {} holds
for i being Element of NAT st i in dom () holds
for x being Element of A1,(() /. i) holds (h . (() /. i)) . x in the Sorts of A2 . (() /. i)
proof end;

theorem Th9: :: MSUALG_6:9
for S being non empty non void ManySortedSign
for o being OperSymbol of S
for i being Element of NAT st i in dom () holds
for A1, A2 being MSAlgebra over S
for h being ManySortedFunction of A1,A2
for a, b being Element of Args (o,A1) st a in Args (o,A1) & h # a in Args (o,A2) holds
for f, g1, g2 being Function st f = a & g1 = h # a & g2 = h # b holds
for x being Element of A1,(() /. i) st b = f +* (i,x) holds
( g2 . i = (h . (() /. i)) . x & h # b = g1 +* (i,(g2 . i)) )
proof end;

definition
let S be non empty non void ManySortedSign ;
let o be OperSymbol of S;
let i be Nat;
let A be MSAlgebra over S;
let a be Function;
func transl (o,i,a,A) -> Function means :Def4: :: MSUALG_6:def 4
( dom it = the Sorts of A . (() /. i) & ( for x being object st x in the Sorts of A . (() /. i) holds
it . x = (Den (o,A)) . (a +* (i,x)) ) );
existence
ex b1 being Function st
( dom b1 = the Sorts of A . (() /. i) & ( for x being object st x in the Sorts of A . (() /. i) holds
b1 . x = (Den (o,A)) . (a +* (i,x)) ) )
proof end;
uniqueness
for b1, b2 being Function st dom b1 = the Sorts of A . (() /. i) & ( for x being object st x in the Sorts of A . (() /. i) holds
b1 . x = (Den (o,A)) . (a +* (i,x)) ) & dom b2 = the Sorts of A . (() /. i) & ( for x being object st x in the Sorts of A . (() /. i) holds
b2 . x = (Den (o,A)) . (a +* (i,x)) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def4 defines transl MSUALG_6:def 4 :
for S being non empty non void ManySortedSign
for o being OperSymbol of S
for i being Nat
for A being MSAlgebra over S
for a, b6 being Function holds
( b6 = transl (o,i,a,A) iff ( dom b6 = the Sorts of A . (() /. i) & ( for x being object st x in the Sorts of A . (() /. i) holds
b6 . x = (Den (o,A)) . (a +* (i,x)) ) ) );

theorem Th10: :: MSUALG_6:10
for S being non empty non void ManySortedSign
for o being OperSymbol of S
for i being Element of NAT
for A being feasible MSAlgebra over S
for a being Function st a in Args (o,A) holds
transl (o,i,a,A) is Function of ( the Sorts of A . (() /. i)),( the Sorts of A . )
proof end;

definition
let S be non empty non void ManySortedSign ;
let s1, s2 be SortSymbol of S;
let A be MSAlgebra over S;
let f be Function;
pred f is_e.translation_of A,s1,s2 means :: MSUALG_6:def 5
ex o being OperSymbol of S st
( the_result_sort_of o = s2 & ex i being Element of NAT st
( i in dom () & () /. i = s1 & ex a being Function st
( a in Args (o,A) & f = transl (o,i,a,A) ) ) );
end;

:: deftheorem defines is_e.translation_of MSUALG_6:def 5 :
for S being non empty non void ManySortedSign
for s1, s2 being SortSymbol of S
for A being MSAlgebra over S
for f being Function holds
( f is_e.translation_of A,s1,s2 iff ex o being OperSymbol of S st
( the_result_sort_of o = s2 & ex i being Element of NAT st
( i in dom () & () /. i = s1 & ex a being Function st
( a in Args (o,A) & f = transl (o,i,a,A) ) ) ) );

theorem Th11: :: MSUALG_6:11
for S being non empty non void ManySortedSign
for s1, s2 being SortSymbol of S
for A being feasible MSAlgebra over S
for f being Function st f is_e.translation_of A,s1,s2 holds
( f is Function of ( the Sorts of A . s1),( the Sorts of A . s2) & the Sorts of A . s1 <> {} & the Sorts of A . s2 <> {} )
proof end;

theorem Th12: :: MSUALG_6:12
for S being non empty non void ManySortedSign
for s1, s2 being SortSymbol of S
for A being MSAlgebra over S
for f being Function st f is_e.translation_of A,s1,s2 holds
[s1,s2] in TranslationRel S by Def3;

theorem :: MSUALG_6:13
for S being non empty non void ManySortedSign
for s1, s2 being SortSymbol of S
for A being non-empty MSAlgebra over S st [s1,s2] in TranslationRel S holds
ex f being Function st f is_e.translation_of A,s1,s2
proof end;

theorem Th14: :: MSUALG_6:14
for S being non empty non void ManySortedSign
for A being feasible MSAlgebra over S
for s1, s2 being SortSymbol of S
for q being RedSequence of TranslationRel S
for p being Function-yielding FinSequence st len q = (len p) + 1 & s1 = q . 1 & s2 = q . (len q) & ( for i being Element of NAT
for f being Function
for s1, s2 being SortSymbol of S st i in dom p & f = p . i & s1 = q . i & s2 = q . (i + 1) holds
f is_e.translation_of A,s1,s2 ) holds
( compose (p,( the Sorts of A . s1)) is Function of ( the Sorts of A . s1),( the Sorts of A . s2) & ( p <> {} implies ( the Sorts of A . s1 <> {} & the Sorts of A . s2 <> {} ) ) )
proof end;

definition
let S be non empty non void ManySortedSign ;
let A be non-empty MSAlgebra over S;
let s1, s2 be SortSymbol of S;
assume A1: TranslationRel S reduces s1,s2 ;
mode Translation of A,s1,s2 -> Function of ( the Sorts of A . s1),( the Sorts of A . s2) means :Def6: :: MSUALG_6:def 6
ex q being RedSequence of TranslationRel S ex p being Function-yielding FinSequence st
( it = compose (p,( the Sorts of A . s1)) & len q = (len p) + 1 & s1 = q . 1 & s2 = q . (len q) & ( for i being Element of NAT
for f being Function
for s1, s2 being SortSymbol of S st i in dom p & f = p . i & s1 = q . i & s2 = q . (i + 1) holds
f is_e.translation_of A,s1,s2 ) );
existence
ex b1 being Function of ( the Sorts of A . s1),( the Sorts of A . s2) ex q being RedSequence of TranslationRel S ex p being Function-yielding FinSequence st
( b1 = compose (p,( the Sorts of A . s1)) & len q = (len p) + 1 & s1 = q . 1 & s2 = q . (len q) & ( for i being Element of NAT
for f being Function
for s1, s2 being SortSymbol of S st i in dom p & f = p . i & s1 = q . i & s2 = q . (i + 1) holds
f is_e.translation_of A,s1,s2 ) )
proof end;
end;

:: deftheorem Def6 defines Translation MSUALG_6:def 6 :
for S being non empty non void ManySortedSign
for A being non-empty MSAlgebra over S
for s1, s2 being SortSymbol of S st TranslationRel S reduces s1,s2 holds
for b5 being Function of ( the Sorts of A . s1),( the Sorts of A . s2) holds
( b5 is Translation of A,s1,s2 iff ex q being RedSequence of TranslationRel S ex p being Function-yielding FinSequence st
( b5 = compose (p,( the Sorts of A . s1)) & len q = (len p) + 1 & s1 = q . 1 & s2 = q . (len q) & ( for i being Element of NAT
for f being Function
for s1, s2 being SortSymbol of S st i in dom p & f = p . i & s1 = q . i & s2 = q . (i + 1) holds
f is_e.translation_of A,s1,s2 ) ) );

theorem :: MSUALG_6:15
for S being non empty non void ManySortedSign
for A being non-empty MSAlgebra over S
for s1, s2 being SortSymbol of S st TranslationRel S reduces s1,s2 holds
for q being RedSequence of TranslationRel S
for p being Function-yielding FinSequence st len q = (len p) + 1 & s1 = q . 1 & s2 = q . (len q) & ( for i being Element of NAT
for f being Function
for s1, s2 being SortSymbol of S st i in dom p & f = p . i & s1 = q . i & s2 = q . (i + 1) holds
f is_e.translation_of A,s1,s2 ) holds
compose (p,( the Sorts of A . s1)) is Translation of A,s1,s2
proof end;

theorem Th16: :: MSUALG_6:16
for S being non empty non void ManySortedSign
for A being non-empty MSAlgebra over S
for s being SortSymbol of S holds id ( the Sorts of A . s) is Translation of A,s,s
proof end;

theorem Th17: :: MSUALG_6:17
for S being non empty non void ManySortedSign
for A being non-empty MSAlgebra over S
for s1, s2 being SortSymbol of S
for f being Function st f is_e.translation_of A,s1,s2 holds
( TranslationRel S reduces s1,s2 & f is Translation of A,s1,s2 )
proof end;

theorem Th18: :: MSUALG_6:18
for S being non empty non void ManySortedSign
for A being non-empty MSAlgebra over S
for s1, s2, s3 being SortSymbol of S st TranslationRel S reduces s1,s2 & TranslationRel S reduces s2,s3 holds
for t1 being Translation of A,s1,s2
for t2 being Translation of A,s2,s3 holds t2 * t1 is Translation of A,s1,s3
proof end;

theorem Th19: :: MSUALG_6:19
for S being non empty non void ManySortedSign
for A being non-empty MSAlgebra over S
for s1, s2, s3 being SortSymbol of S st TranslationRel S reduces s1,s2 holds
for t being Translation of A,s1,s2
for f being Function st f is_e.translation_of A,s2,s3 holds
f * t is Translation of A,s1,s3
proof end;

theorem :: MSUALG_6:20
for S being non empty non void ManySortedSign
for A being non-empty MSAlgebra over S
for s1, s2, s3 being SortSymbol of S st TranslationRel S reduces s2,s3 holds
for f being Function st f is_e.translation_of A,s1,s2 holds
for t being Translation of A,s2,s3 holds t * f is Translation of A,s1,s3
proof end;

scheme :: MSUALG_6:sch 1
TranslationInd{ F1() -> non empty non void ManySortedSign , F2() -> non-empty MSAlgebra over F1(), P1[ set , set , set ] } :
for s1, s2 being SortSymbol of F1() st TranslationRel F1() reduces s1,s2 holds
for t being Translation of F2(),s1,s2 holds P1[t,s1,s2]
provided
A1: for s being SortSymbol of F1() holds P1[ id ( the Sorts of F2() . s),s,s] and
A2: for s1, s2, s3 being SortSymbol of F1() st TranslationRel F1() reduces s1,s2 holds
for t being Translation of F2(),s1,s2 st P1[t,s1,s2] holds
for f being Function st f is_e.translation_of F2(),s2,s3 holds
P1[f * t,s1,s3]
proof end;

theorem Th21: :: MSUALG_6:21
for S being non empty non void ManySortedSign
for A1, A2 being non-empty MSAlgebra over S
for h being ManySortedFunction of A1,A2 st h is_homomorphism A1,A2 holds
for o being OperSymbol of S
for i being Element of NAT st i in dom () holds
for a being Element of Args (o,A1) holds (h . ) * (transl (o,i,a,A1)) = (transl (o,i,(h # a),A2)) * (h . (() /. i))
proof end;

theorem :: MSUALG_6:22
for S being non empty non void ManySortedSign
for A being non-empty MSAlgebra over S
for h being Endomorphism of A
for o being OperSymbol of S
for i being Element of NAT st i in dom () holds
for a being Element of Args (o,A) holds (h . ) * (transl (o,i,a,A)) = (transl (o,i,(h # a),A)) * (h . (() /. i)) by ;

theorem Th23: :: MSUALG_6:23
for S being non empty non void ManySortedSign
for A1, A2 being non-empty MSAlgebra over S
for h being ManySortedFunction of A1,A2 st h is_homomorphism A1,A2 holds
for s1, s2 being SortSymbol of S
for t being Function st t is_e.translation_of A1,s1,s2 holds
ex T being Function of ( the Sorts of A2 . s1),( the Sorts of A2 . s2) st
( T is_e.translation_of A2,s1,s2 & T * (h . s1) = (h . s2) * t )
proof end;

theorem :: MSUALG_6:24
for S being non empty non void ManySortedSign
for A being non-empty MSAlgebra over S
for h being Endomorphism of A
for s1, s2 being SortSymbol of S
for t being Function st t is_e.translation_of A,s1,s2 holds
ex T being Function of ( the Sorts of A . s1),( the Sorts of A . s2) st
( T is_e.translation_of A,s1,s2 & T * (h . s1) = (h . s2) * t ) by ;

theorem Th25: :: MSUALG_6:25
for S being non empty non void ManySortedSign
for A1, A2 being non-empty MSAlgebra over S
for h being ManySortedFunction of A1,A2 st h is_homomorphism A1,A2 holds
for s1, s2 being SortSymbol of S st TranslationRel S reduces s1,s2 holds
for t being Translation of A1,s1,s2 ex T being Translation of A2,s1,s2 st T * (h . s1) = (h . s2) * t
proof end;

theorem Th26: :: MSUALG_6:26
for S being non empty non void ManySortedSign
for A being non-empty MSAlgebra over S
for h being Endomorphism of A
for s1, s2 being SortSymbol of S st TranslationRel S reduces s1,s2 holds
for t being Translation of A,s1,s2 ex T being Translation of A,s1,s2 st T * (h . s1) = (h . s2) * t by ;

definition
let S be non empty non void ManySortedSign ;
let A be MSAlgebra over S;
let R be ManySortedRelation of A;
attr R is compatible means :: MSUALG_6:def 7
for o being OperSymbol of S
for a, b being Function st a in Args (o,A) & b in Args (o,A) & ( for n being Element of NAT st n in dom () holds
[(a . n),(b . n)] in R . (() /. n) ) holds
[((Den (o,A)) . a),((Den (o,A)) . b)] in R . ;
attr R is invariant means :Def8: :: MSUALG_6:def 8
for s1, s2 being SortSymbol of S
for t being Function st t is_e.translation_of A,s1,s2 holds
for a, b being set st [a,b] in R . s1 holds
[(t . a),(t . b)] in R . s2;
attr R is stable means :Def9: :: MSUALG_6:def 9
for h being Endomorphism of A
for s being SortSymbol of S
for a, b being set st [a,b] in R . s holds
[((h . s) . a),((h . s) . b)] in R . s;
end;

:: deftheorem defines compatible MSUALG_6:def 7 :
for S being non empty non void ManySortedSign
for A being MSAlgebra over S
for R being ManySortedRelation of A holds
( R is compatible iff for o being OperSymbol of S
for a, b being Function st a in Args (o,A) & b in Args (o,A) & ( for n being Element of NAT st n in dom () holds
[(a . n),(b . n)] in R . (() /. n) ) holds
[((Den (o,A)) . a),((Den (o,A)) . b)] in R . );

:: deftheorem Def8 defines invariant MSUALG_6:def 8 :
for S being non empty non void ManySortedSign
for A being MSAlgebra over S
for R being ManySortedRelation of A holds
( R is invariant iff for s1, s2 being SortSymbol of S
for t being Function st t is_e.translation_of A,s1,s2 holds
for a, b being set st [a,b] in R . s1 holds
[(t . a),(t . b)] in R . s2 );

:: deftheorem Def9 defines stable MSUALG_6:def 9 :
for S being non empty non void ManySortedSign
for A being MSAlgebra over S
for R being ManySortedRelation of A holds
( R is stable iff for h being Endomorphism of A
for s being SortSymbol of S
for a, b being set st [a,b] in R . s holds
[((h . s) . a),((h . s) . b)] in R . s );

theorem :: MSUALG_6:27
for S being non empty non void ManySortedSign
for A being non-empty MSAlgebra over S
for R being MSEquivalence-like ManySortedRelation of A holds
( R is compatible iff R is MSCongruence of A )
proof end;

theorem Th28: :: MSUALG_6:28
for S being non empty non void ManySortedSign
for A being non-empty MSAlgebra over S
for R being ManySortedRelation of A holds
( R is invariant iff for s1, s2 being SortSymbol of S st TranslationRel S reduces s1,s2 holds
for f being Translation of A,s1,s2
for a, b being set st [a,b] in R . s1 holds
[(f . a),(f . b)] in R . s2 )
proof end;

registration
let S be non empty non void ManySortedSign ;
let A be non-empty MSAlgebra over S;
cluster MSEquivalence-like invariant -> MSEquivalence-like compatible for ManySortedRelation of the Sorts of A, the Sorts of A;
coherence
for b1 being MSEquivalence-like ManySortedRelation of A st b1 is invariant holds
b1 is compatible
proof end;
cluster MSEquivalence-like compatible -> MSEquivalence-like invariant for ManySortedRelation of the Sorts of A, the Sorts of A;
coherence
for b1 being MSEquivalence-like ManySortedRelation of A st b1 is compatible holds
b1 is invariant
proof end;
end;

registration
let X be non empty set ;
cluster id X -> non empty ;
coherence
not id X is empty
;
end;

scheme :: MSUALG_6:sch 2
MSRExistence{ F1() -> non empty set , F2() -> V2() ManySortedSet of F1(), P1[ object , object , object ] } :
ex R being ManySortedRelation of F2() st
for i being Element of F1()
for a, b being Element of F2() . i holds
( [a,b] in R . i iff P1[i,a,b] )
proof end;

scheme :: MSUALG_6:sch 3
MSRLambdaU{ F1() -> set , F2() -> ManySortedSet of F1(), F3( object ) -> set } :
( ex R being ManySortedRelation of F2() st
for i being object st i in F1() holds
R . i = F3(i) & ( for R1, R2 being ManySortedRelation of F2() st ( for i being object st i in F1() holds
R1 . i = F3(i) ) & ( for i being object st i in F1() holds
R2 . i = F3(i) ) holds
R1 = R2 ) )
provided
A1: for i being set st i in F1() holds
F3(i) is Relation of (F2() . i),(F2() . i)
proof end;

definition
let I be set ;
let A be ManySortedSet of I;
func id (I,A) -> ManySortedRelation of A means :Def10: :: MSUALG_6:def 10
for i being object st i in I holds
it . i = id (A . i);
correctness
existence
ex b1 being ManySortedRelation of A st
for i being object st i in I holds
b1 . i = id (A . i)
;
uniqueness
for b1, b2 being ManySortedRelation of A st ( for i being object st i in I holds
b1 . i = id (A . i) ) & ( for i being object st i in I holds
b2 . i = id (A . i) ) holds
b1 = b2
;
proof end;
end;

:: deftheorem Def10 defines id MSUALG_6:def 10 :
for I being set
for A being ManySortedSet of I
for b3 being ManySortedRelation of A holds
( b3 = id (I,A) iff for i being object st i in I holds
b3 . i = id (A . i) );

registration
let S be non empty non void ManySortedSign ;
let A be non-empty MSAlgebra over S;
cluster MSEquivalence-like -> V2() for ManySortedRelation of the Sorts of A, the Sorts of A;
coherence
for b1 being ManySortedRelation of A st b1 is MSEquivalence-like holds
b1 is V2()
proof end;
end;

registration
let S be non empty non void ManySortedSign ;
let A be non-empty MSAlgebra over S;
existence
ex b1 being ManySortedRelation of A st
( b1 is invariant & b1 is stable & b1 is MSEquivalence-like )
proof end;
end;

scheme :: MSUALG_6:sch 4
MSRelCl{ F1() -> non empty non void ManySortedSign , F2() -> non-empty MSAlgebra over F1(), P1[ set , set , set ], P2[ set ], F3() -> ManySortedRelation of F2(), F4() -> ManySortedRelation of F2() } :
( P2[F4()] & F3() c= F4() & ( for P being ManySortedRelation of F2() st P2[P] & F3() c= P holds
F4() c= P ) )
provided
A1: for R being ManySortedRelation of F2() holds
( P2[R] iff for s1, s2 being SortSymbol of F1()
for f being Function of ( the Sorts of F2() . s1),( the Sorts of F2() . s2) st P1[f,s1,s2] holds
for a, b being set st [a,b] in R . s1 holds
[(f . a),(f . b)] in R . s2 ) and
A2: for s1, s2, s3 being SortSymbol of F1()
for f1 being Function of ( the Sorts of F2() . s1),( the Sorts of F2() . s2)
for f2 being Function of ( the Sorts of F2() . s2),( the Sorts of F2() . s3) st P1[f1,s1,s2] & P1[f2,s2,s3] holds
P1[f2 * f1,s1,s3] and
A3: for s being SortSymbol of F1() holds P1[ id ( the Sorts of F2() . s),s,s] and
A4: for s being SortSymbol of F1()
for a, b being Element of F2(),s holds
( [a,b] in F4() . s iff ex s9 being SortSymbol of F1() ex f being Function of ( the Sorts of F2() . s9),( the Sorts of F2() . s) ex x, y being Element of F2(),s9 st
( P1[f,s9,s] & [x,y] in F3() . s9 & a = f . x & b = f . y ) )
proof end;

Lm1: now :: thesis: for S being non empty non void ManySortedSign
for A being non-empty MSAlgebra over S
for R, Q being ManySortedRelation of A st ( for s being SortSymbol of S
for a, b being Element of A,s holds
( [a,b] in Q . s iff ex s9 being SortSymbol of S ex f being Function of ( the Sorts of A . s9),( the Sorts of A . s) ex x, y being Element of A,s9 st
( S2[f,s9,s] & [x,y] in R . s9 & a = f . x & b = f . y ) ) ) holds
( S1[Q] & R c= Q & ( for P being ManySortedRelation of A st S1[P] & R c= P holds
Q c= P ) )
let S be non empty non void ManySortedSign ; :: thesis: for A being non-empty MSAlgebra over S
for R, Q being ManySortedRelation of A st ( for s being SortSymbol of S
for a, b being Element of A,s holds
( [a,b] in Q . s iff ex s9 being SortSymbol of S ex f being Function of ( the Sorts of A . s9),( the Sorts of A . s) ex x, y being Element of A,s9 st
( S2[f,s9,s] & [x,y] in R . s9 & a = f . x & b = f . y ) ) ) holds
( S1[Q] & R c= Q & ( for P being ManySortedRelation of A st S1[P] & R c= P holds
Q c= P ) )

let A be non-empty MSAlgebra over S; :: thesis: for R, Q being ManySortedRelation of A st ( for s being SortSymbol of S
for a, b being Element of A,s holds
( [a,b] in Q . s iff ex s9 being SortSymbol of S ex f being Function of ( the Sorts of A . s9),( the Sorts of A . s) ex x, y being Element of A,s9 st
( S2[f,s9,s] & [x,y] in R . s9 & a = f . x & b = f . y ) ) ) holds
( S1[Q] & R c= Q & ( for P being ManySortedRelation of A st S1[P] & R c= P holds
Q c= P ) )

let R, Q be ManySortedRelation of A; :: thesis: ( ( for s being SortSymbol of S
for a, b being Element of A,s holds
( [a,b] in Q . s iff ex s9 being SortSymbol of S ex f being Function of ( the Sorts of A . s9),( the Sorts of A . s) ex x, y being Element of A,s9 st
( S2[f,s9,s] & [x,y] in R . s9 & a = f . x & b = f . y ) ) ) implies ( S1[Q] & R c= Q & ( for P being ManySortedRelation of A st S1[P] & R c= P holds
Q c= P ) ) )

defpred S1[ ManySortedRelation of A] means $1 is invariant ; defpred S2[ Function, SortSymbol of S, SortSymbol of S] means ( TranslationRel S reduces$2,$3 &$1 is Translation of A,$2,$3 );
assume A1: for s being SortSymbol of S
for a, b being Element of A,s holds
( [a,b] in Q . s iff ex s9 being SortSymbol of S ex f being Function of ( the Sorts of A . s9),( the Sorts of A . s) ex x, y being Element of A,s9 st
( S2[f,s9,s] & [x,y] in R . s9 & a = f . x & b = f . y ) ) ; :: thesis: ( S1[Q] & R c= Q & ( for P being ManySortedRelation of A st S1[P] & R c= P holds
Q c= P ) )

A2: for s being SortSymbol of S holds S2[ id ( the Sorts of A . s),s,s] by ;
A3: now :: thesis: for R being ManySortedRelation of A holds
( ( S1[R] implies for s1, s2 being SortSymbol of S
for f being Function of ( the Sorts of A . s1),( the Sorts of A . s2) st S2[f,s1,s2] holds
for a, b being set st [a,b] in R . s1 holds
[(f . a),(f . b)] in R . s2 ) & ( ( for s1, s2 being SortSymbol of S
for f being Function of ( the Sorts of A . s1),( the Sorts of A . s2) st S2[f,s1,s2] holds
for a, b being set st [a,b] in R . s1 holds
[(f . a),(f . b)] in R . s2 ) implies S1[R] ) )
let R be ManySortedRelation of A; :: thesis: ( ( S1[R] implies for s1, s2 being SortSymbol of S
for f being Function of ( the Sorts of A . s1),( the Sorts of A . s2) st S2[f,s1,s2] holds
for a, b being set st [a,b] in R . s1 holds
[(f . a),(f . b)] in R . s2 ) & ( ( for s1, s2 being SortSymbol of S
for f being Function of ( the Sorts of A . s1),( the Sorts of A . s2) st S2[f,s1,s2] holds
for a, b being set st [a,b] in R . s1 holds
[(f . a),(f . b)] in R . s2 ) implies S1[R] ) )

thus ( S1[R] implies for s1, s2 being SortSymbol of S
for f being Function of ( the Sorts of A . s1),( the Sorts of A . s2) st S2[f,s1,s2] holds
for a, b being set st [a,b] in R . s1 holds
[(f . a),(f . b)] in R . s2 ) by Th28; :: thesis: ( ( for s1, s2 being SortSymbol of S
for f being Function of ( the Sorts of A . s1),( the Sorts of A . s2) st S2[f,s1,s2] holds
for a, b being set st [a,b] in R . s1 holds
[(f . a),(f . b)] in R . s2 ) implies S1[R] )

assume for s1, s2 being SortSymbol of S
for f being Function of ( the Sorts of A . s1),( the Sorts of A . s2) st S2[f,s1,s2] holds
for a, b being set st [a,b] in R . s1 holds
[(f . a),(f . b)] in R . s2 ; :: thesis: S1[R]
then for s1, s2 being SortSymbol of S st TranslationRel S reduces s1,s2 holds
for f being Translation of A,s1,s2
for a, b being set st [a,b] in R . s1 holds
[(f . a),(f . b)] in R . s2 ;
hence S1[R] by Th28; :: thesis: verum
end;
A4: for s1, s2, s3 being SortSymbol of S
for f1 being Function of ( the Sorts of A . s1),( the Sorts of A . s2)
for f2 being Function of ( the Sorts of A . s2),( the Sorts of A . s3) st S2[f1,s1,s2] & S2[f2,s2,s3] holds
S2[f2 * f1,s1,s3] by ;
thus ( S1[Q] & R c= Q & ( for P being ManySortedRelation of A st S1[P] & R c= P holds
Q c= P ) ) from MSUALG_6:sch 4(A3, A4, A2, A1); :: thesis: verum
end;

definition
let S be non empty non void ManySortedSign ;
let A be non-empty MSAlgebra over S;
let R be ManySortedRelation of the Sorts of A;
func InvCl R -> invariant ManySortedRelation of A means :Def11: :: MSUALG_6:def 11
( R c= it & ( for Q being invariant ManySortedRelation of A st R c= Q holds
it c= Q ) );
uniqueness
for b1, b2 being invariant ManySortedRelation of A st R c= b1 & ( for Q being invariant ManySortedRelation of A st R c= Q holds
b1 c= Q ) & R c= b2 & ( for Q being invariant ManySortedRelation of A st R c= Q holds
b2 c= Q ) holds
b1 = b2
proof end;
existence
ex b1 being invariant ManySortedRelation of A st
( R c= b1 & ( for Q being invariant ManySortedRelation of A st R c= Q holds
b1 c= Q ) )
proof end;
end;

:: deftheorem Def11 defines InvCl MSUALG_6:def 11 :
for S being non empty non void ManySortedSign
for A being non-empty MSAlgebra over S
for R being ManySortedRelation of the Sorts of A
for b4 being invariant ManySortedRelation of A holds
( b4 = InvCl R iff ( R c= b4 & ( for Q being invariant ManySortedRelation of A st R c= Q holds
b4 c= Q ) ) );

theorem Th29: :: MSUALG_6:29
for S being non empty non void ManySortedSign
for A being non-empty MSAlgebra over S
for R being ManySortedRelation of the Sorts of A
for s being SortSymbol of S
for a, b being Element of A,s holds
( [a,b] in () . s iff ex s9 being SortSymbol of S ex x, y being Element of A,s9 ex t being Translation of A,s9,s st
( TranslationRel S reduces s9,s & [x,y] in R . s9 & a = t . x & b = t . y ) )
proof end;

theorem Th30: :: MSUALG_6:30
for S being non empty non void ManySortedSign
for A being non-empty MSAlgebra over S
for R being stable ManySortedRelation of A holds InvCl R is stable
proof end;

Lm2: now :: thesis: for S being non empty non void ManySortedSign
for A being non-empty MSAlgebra over S
for R, Q being ManySortedRelation of A st ( for s being SortSymbol of S
for a, b being Element of A,s holds
( [a,b] in Q . s iff ex s9 being SortSymbol of S ex f being Function of ( the Sorts of A . s9),( the Sorts of A . s) ex x, y being Element of A,s9 st
( S2[f,s9,s] & [x,y] in R . s9 & a = f . x & b = f . y ) ) ) holds
( S1[Q] & R c= Q & ( for P being ManySortedRelation of A st S1[P] & R c= P holds
Q c= P ) )
let S be non empty non void ManySortedSign ; :: thesis: for A being non-empty MSAlgebra over S
for R, Q being ManySortedRelation of A st ( for s being SortSymbol of S
for a, b being Element of A,s holds
( [a,b] in Q . s iff ex s9 being SortSymbol of S ex f being Function of ( the Sorts of A . s9),( the Sorts of A . s) ex x, y being Element of A,s9 st
( S2[f,s9,s] & [x,y] in R . s9 & a = f . x & b = f . y ) ) ) holds
( S1[Q] & R c= Q & ( for P being ManySortedRelation of A st S1[P] & R c= P holds
Q c= P ) )

let A be non-empty MSAlgebra over S; :: thesis: for R, Q being ManySortedRelation of A st ( for s being SortSymbol of S
for a, b being Element of A,s holds
( [a,b] in Q . s iff ex s9 being SortSymbol of S ex f being Function of ( the Sorts of A . s9),( the Sorts of A . s) ex x, y being Element of A,s9 st
( S2[f,s9,s] & [x,y] in R . s9 & a = f . x & b = f . y ) ) ) holds
( S1[Q] & R c= Q & ( for P being ManySortedRelation of A st S1[P] & R c= P holds
Q c= P ) )

let R, Q be ManySortedRelation of A; :: thesis: ( ( for s being SortSymbol of S
for a, b being Element of A,s holds
( [a,b] in Q . s iff ex s9 being SortSymbol of S ex f being Function of ( the Sorts of A . s9),( the Sorts of A . s) ex x, y being Element of A,s9 st
( S2[f,s9,s] & [x,y] in R . s9 & a = f . x & b = f . y ) ) ) implies ( S1[Q] & R c= Q & ( for P being ManySortedRelation of A st S1[P] & R c= P holds
Q c= P ) ) )

defpred S1[ ManySortedRelation of A] means $1 is stable ; defpred S2[ Function, SortSymbol of S, SortSymbol of S] means ($2 = $3 & ex h being Endomorphism of A st$1 = h . \$2 );
A1: for s1, s2, s3 being SortSymbol of S
for f1 being Function of ( the Sorts of A . s1),( the Sorts of A . s2)
for f2 being Function of ( the Sorts of A . s2),( the Sorts of A . s3) st S2[f1,s1,s2] & S2[f2,s2,s3] holds
S2[f2 * f1,s1,s3]
proof
let s1, s2, s3 be SortSymbol of S; :: thesis: for f1 being Function of ( the Sorts of A . s1),( the Sorts of A . s2)
for f2 being Function of ( the Sorts of A . s2),( the Sorts of A . s3) st S2[f1,s1,s2] & S2[f2,s2,s3] holds
S2[f2 * f1,s1,s3]

let f1 be Function of ( the Sorts of A . s1),( the Sorts of A . s2); :: thesis: for f2 being Function of ( the Sorts of A . s2),( the Sorts of A . s3) st S2[f1,s1,s2] & S2[f2,s2,s3] holds
S2[f2 * f1,s1,s3]

let f2 be Function of ( the Sorts of A . s2),( the Sorts of A . s3); :: thesis: ( S2[f1,s1,s2] & S2[f2,s2,s3] implies S2[f2 * f1,s1,s3] )
assume A2: s1 = s2 ; :: thesis: ( for h being Endomorphism of A holds not f1 = h . s1 or not S2[f2,s2,s3] or S2[f2 * f1,s1,s3] )
given h1 being Endomorphism of A such that A3: f1 = h1 . s1 ; :: thesis: ( not S2[f2,s2,s3] or S2[f2 * f1,s1,s3] )
assume A4: s2 = s3 ; :: thesis: ( for h being Endomorphism of A holds not f2 = h . s2 or S2[f2 * f1,s1,s3] )
given h2 being Endomorphism of A such that A5: f2 = h2 . s2 ; :: thesis: S2[f2 * f1,s1,s3]
thus s1 = s3 by A2, A4; :: thesis: ex h being Endomorphism of A st f2 * f1 = h . s1
reconsider h = h2 ** h1 as Endomorphism of A ;
take h ; :: thesis: f2 * f1 = h . s1
thus f2 * f1 = h . s1 by ; :: thesis: verum
end;
A6: for R being ManySortedRelation of A holds
( S1[R] iff for s1, s2 being SortSymbol of S
for f being Function of ( the Sorts of A . s1),( the Sorts of A . s2) st S2[f,s1,s2] holds
for a, b being set st [a,b] in R . s1 holds
[(f . a),(f . b)] in R . s2 )
proof
let R be ManySortedRelation of A; :: thesis: ( S1[R] iff for s1, s2 being SortSymbol of S
for f being Function of ( the Sorts of A . s1),( the Sorts of A . s2) st S2[f,s1,s2] holds
for a, b being set st [a,b] in R . s1 holds
[(f . a),(f . b)] in R . s2 )

thus ( R is stable implies for s1, s2 being SortSymbol of S
for f being Function of ( the Sorts of A . s1),( the Sorts of A . s2) st s1 = s2 & ex h being Endomorphism of A st f = h . s1 holds
for a, b being set st [a,b] in R . s1 holds
[(f . a),(f . b)] in R . s2 ) ; :: thesis: ( ( for s1, s2 being SortSymbol of S
for f being Function of ( the Sorts of A . s1),( the Sorts of A . s2) st S2[f,s1,s2] holds
for a, b being set st [a,b] in R . s1 holds
[(f . a),(f . b)] in R . s2 ) implies S1[R] )

assume A7: for s1, s2 being SortSymbol of S
for f being Function of ( the Sorts of A . s1),( the Sorts of A . s2) st S2[f,s1,s2] holds
for a, b being set st [a,b] in R . s1 holds
[(f . a),(f . b)] in R . s2 ; :: thesis: S1[R]
thus R is stable by A7; :: thesis: verum
end;
A8: for s being SortSymbol of S holds S2[ id ( the Sorts of A . s),s,s]
proof
reconsider h = id the Sorts of A as Endomorphism of A by Th4;
let s be SortSymbol of S; :: thesis: S2[ id ( the Sorts of A . s),s,s]
thus s = s ; :: thesis: ex h being Endomorphism of A st id ( the Sorts of A . s) = h . s
take h ; :: thesis: id ( the Sorts of A . s) = h . s
thus id ( the Sorts of A . s) = h . s by MSUALG_3:def 1; :: thesis: verum
end;
assume A9: for s being SortSymbol of S
for a, b being Element of A,s holds
( [a,b] in Q . s iff ex s9 being SortSymbol of S ex f being Function of ( the Sorts of A . s9),( the Sorts of A . s) ex x, y being Element of A,s9 st
( S2[f,s9,s] & [x,y] in R . s9 & a = f . x & b = f . y ) ) ; :: thesis: ( S1[Q] & R c= Q & ( for P being ManySortedRelation of A st S1[P] & R c= P holds
Q c= P ) )

thus ( S1[Q] & R c= Q & ( for P being ManySortedRelation of A st S1[P] & R c= P holds
Q c= P ) ) from MSUALG_6:sch 4(A6, A1, A8, A9); :: thesis: verum
end;

definition
let S be non empty non void ManySortedSign ;
let A be non-empty MSAlgebra over S;
let R be ManySortedRelation of the Sorts of A;
func StabCl R -> stable ManySortedRelation of A means :Def12: :: MSUALG_6:def 12
( R c= it & ( for Q being stable ManySortedRelation of A st R c= Q holds
it c= Q ) );
uniqueness
for b1, b2 being stable ManySortedRelation of A st R c= b1 & ( for Q being stable ManySortedRelation of A st R c= Q holds
b1 c= Q ) & R c= b2 & ( for Q being stable ManySortedRelation of A st R c= Q holds
b2 c= Q ) holds
b1 = b2
proof end;
existence
ex b1 being stable ManySortedRelation of A st
( R c= b1 & ( for Q being stable ManySortedRelation of A st R c= Q holds
b1 c= Q ) )
proof end;
end;

:: deftheorem Def12 defines StabCl MSUALG_6:def 12 :
for S being non empty non void ManySortedSign
for A being non-empty MSAlgebra over S
for R being ManySortedRelation of the Sorts of A
for b4 being stable ManySortedRelation of A holds
( b4 = StabCl R iff ( R c= b4 & ( for Q being stable ManySortedRelation of A st R c= Q holds
b4 c= Q ) ) );

theorem Th31: :: MSUALG_6:31
for S being non empty non void ManySortedSign
for A being non-empty MSAlgebra over S
for R being ManySortedRelation of the Sorts of A
for s being SortSymbol of S
for a, b being Element of A,s holds
( [a,b] in () . s iff ex x, y being Element of A,s ex h being Endomorphism of A st
( [x,y] in R . s & a = (h . s) . x & b = (h . s) . y ) )
proof end;

theorem :: MSUALG_6:32
for S being non empty non void ManySortedSign
for A being non-empty MSAlgebra over S
for R being ManySortedRelation of the Sorts of A holds InvCl () is stable by Th30;

definition
let S be non empty non void ManySortedSign ;
let A be non-empty MSAlgebra over S;
let R be ManySortedRelation of the Sorts of A;
func TRS R -> invariant stable ManySortedRelation of A means :Def13: :: MSUALG_6:def 13
( R c= it & ( for Q being invariant stable ManySortedRelation of A st R c= Q holds
it c= Q ) );
uniqueness
for b1, b2 being invariant stable ManySortedRelation of A st R c= b1 & ( for Q being invariant stable ManySortedRelation of A st R c= Q holds
b1 c= Q ) & R c= b2 & ( for Q being invariant stable ManySortedRelation of A st R c= Q holds
b2 c= Q ) holds
b1 = b2
proof end;
existence
ex b1 being invariant stable ManySortedRelation of A st
( R c= b1 & ( for Q being invariant stable ManySortedRelation of A st R c= Q holds
b1 c= Q ) )
proof end;
end;

:: deftheorem Def13 defines TRS MSUALG_6:def 13 :
for S being non empty non void ManySortedSign
for A being non-empty MSAlgebra over S
for R being ManySortedRelation of the Sorts of A
for b4 being invariant stable ManySortedRelation of A holds
( b4 = TRS R iff ( R c= b4 & ( for Q being invariant stable ManySortedRelation of A st R c= Q holds
b4 c= Q ) ) );

registration
let S be non empty non void ManySortedSign ;
let A be non-empty MSAlgebra over S;
let R be V2() ManySortedRelation of A;
cluster InvCl R -> V2() invariant ;
coherence
proof end;
cluster StabCl R -> V2() stable ;
coherence
proof end;
coherence
TRS R is non-empty
proof end;
end;

theorem :: MSUALG_6:33
for S being non empty non void ManySortedSign
for A being non-empty MSAlgebra over S
for R being invariant ManySortedRelation of A holds InvCl R = R
proof end;

theorem :: MSUALG_6:34
for S being non empty non void ManySortedSign
for A being non-empty MSAlgebra over S
for R being stable ManySortedRelation of A holds StabCl R = R
proof end;

theorem :: MSUALG_6:35
for S being non empty non void ManySortedSign
for A being non-empty MSAlgebra over S
for R being invariant stable ManySortedRelation of A holds TRS R = R
proof end;

theorem :: MSUALG_6:36
for S being non empty non void ManySortedSign
for A being non-empty MSAlgebra over S
for R being ManySortedRelation of the Sorts of A holds
( StabCl R c= TRS R & InvCl R c= TRS R & StabCl () c= TRS R )
proof end;

theorem Th37: :: MSUALG_6:37
for S being non empty non void ManySortedSign
for A being non-empty MSAlgebra over S
for R being ManySortedRelation of the Sorts of A holds InvCl () = TRS R
proof end;

theorem :: MSUALG_6:38
for S being non empty non void ManySortedSign
for A being non-empty MSAlgebra over S
for R being ManySortedRelation of the Sorts of A
for s being SortSymbol of S
for a, b being Element of A,s holds
( [a,b] in (TRS R) . s iff ex s9 being SortSymbol of S st
( TranslationRel S reduces s9,s & ex l, r being Element of A,s9 ex h being Endomorphism of A ex t being Translation of A,s9,s st
( [l,r] in R . s9 & a = t . ((h . s9) . l) & b = t . ((h . s9) . r) ) ) )
proof end;

theorem Th39: :: MSUALG_6:39
for A being set
for R, E being Relation of A st ( for a, b being set st a in A & b in A holds
( [a,b] in E iff a,b are_convertible_wrt R ) ) holds
( E is total & E is symmetric & E is transitive )
proof end;

theorem Th40: :: MSUALG_6:40
for A being set
for R being Relation of A
for E being Equivalence_Relation of A st R c= E holds
for a, b being object st a in A & a,b are_convertible_wrt R holds
[a,b] in E
proof end;

theorem Th41: :: MSUALG_6:41
for A being non empty set
for R being Relation of A
for a, b being Element of A holds
( [a,b] in EqCl R iff a,b are_convertible_wrt R )
proof end;

theorem Th42: :: MSUALG_6:42
for S being non empty set
for A being V2() ManySortedSet of S
for R being ManySortedRelation of A
for s being Element of S
for a, b being Element of A . s holds
( [a,b] in (EqCl R) . s iff a,b are_convertible_wrt R . s )
proof end;

definition
let S be non empty non void ManySortedSign ;
let A be non-empty MSAlgebra over S;
mode EquationalTheory of A is MSEquivalence-like invariant stable ManySortedRelation of A;
let R be ManySortedRelation of A;
func EqCl (R,A) -> MSEquivalence-like ManySortedRelation of A equals :: MSUALG_6:def 14
EqCl R;
coherence by MSUALG_4:def 3;
end;

:: deftheorem defines EqCl MSUALG_6:def 14 :
for S being non empty non void ManySortedSign
for A being non-empty MSAlgebra over S
for R being ManySortedRelation of A holds EqCl (R,A) = EqCl R;

theorem Th43: :: MSUALG_6:43
for S being non empty non void ManySortedSign
for A being non-empty MSAlgebra over S
for R being ManySortedRelation of A holds R c= EqCl (R,A)
proof end;

theorem Th44: :: MSUALG_6:44
for S being non empty non void ManySortedSign
for A being non-empty MSAlgebra over S
for R being ManySortedRelation of A
for E being MSEquivalence-like ManySortedRelation of A st R c= E holds
EqCl (R,A) c= E
proof end;

theorem Th45: :: MSUALG_6:45
for S being non empty non void ManySortedSign
for A being non-empty MSAlgebra over S
for R being stable ManySortedRelation of A
for s being SortSymbol of S
for a, b being Element of A,s st a,b are_convertible_wrt R . s holds
for h being Endomorphism of A holds (h . s) . a,(h . s) . b are_convertible_wrt R . s
proof end;

theorem Th46: :: MSUALG_6:46
for S being non empty non void ManySortedSign
for A being non-empty MSAlgebra over S
for R being stable ManySortedRelation of A holds EqCl (R,A) is stable
proof end;

registration
let S be non empty non void ManySortedSign ;
let A be non-empty MSAlgebra over S;
let R be stable ManySortedRelation of A;
coherence
EqCl (R,A) is stable
by Th46;
end;

theorem Th47: :: MSUALG_6:47
for S being non empty non void ManySortedSign
for A being non-empty MSAlgebra over S
for R being invariant ManySortedRelation of A
for s1, s2 being SortSymbol of S
for a, b being Element of A,s1 st a,b are_convertible_wrt R . s1 holds
for t being Function st t is_e.translation_of A,s1,s2 holds
t . a,t . b are_convertible_wrt R . s2
proof end;

theorem Th48: :: MSUALG_6:48
for S being non empty non void ManySortedSign
for A being non-empty MSAlgebra over S
for R being invariant ManySortedRelation of A holds EqCl (R,A) is invariant
proof end;

registration
let S be non empty non void ManySortedSign ;
let A be non-empty MSAlgebra over S;
let R be invariant ManySortedRelation of A;
coherence
EqCl (R,A) is invariant
by Th48;
end;

theorem Th49: :: MSUALG_6:49
for S being non empty set
for A being V2() ManySortedSet of S
for R, E being ManySortedRelation of A st ( for s being Element of S
for a, b being Element of A . s holds
( [a,b] in E . s iff a,b are_convertible_wrt R . s ) ) holds
E is MSEquivalence_Relation-like
proof end;

theorem Th50: :: MSUALG_6:50
for S being non empty non void ManySortedSign
for A being non-empty MSAlgebra over S
for R, E being ManySortedRelation of A st ( for s being SortSymbol of S
for a, b being Element of A,s holds
( [a,b] in E . s iff a,b are_convertible_wrt (TRS R) . s ) ) holds
E is EquationalTheory of A
proof end;

theorem Th51: :: MSUALG_6:51
for S being non empty set
for A being V2() ManySortedSet of S
for R being ManySortedRelation of A
for E being MSEquivalence_Relation-like ManySortedRelation of A st R c= E holds
for s being Element of S
for a, b being Element of A . s st a,b are_convertible_wrt R . s holds
[a,b] in E . s
proof end;

definition
let S be non empty non void ManySortedSign ;
let A be non-empty MSAlgebra over S;
let R be ManySortedRelation of the Sorts of A;
func EqTh R -> EquationalTheory of A means :Def15: :: MSUALG_6:def 15
( R c= it & ( for Q being EquationalTheory of A st R c= Q holds
it c= Q ) );
uniqueness
for b1, b2 being EquationalTheory of A st R c= b1 & ( for Q being EquationalTheory of A st R c= Q holds
b1 c= Q ) & R c= b2 & ( for Q being EquationalTheory of A st R c= Q holds
b2 c= Q ) holds
b1 = b2
proof end;
existence
ex b1 being EquationalTheory of A st
( R c= b1 & ( for Q being EquationalTheory of A st R c= Q holds
b1 c= Q ) )
proof end;
end;

:: deftheorem Def15 defines EqTh MSUALG_6:def 15 :
for S being non empty non void ManySortedSign
for A being non-empty MSAlgebra over S
for R being ManySortedRelation of the Sorts of A
for b4 being EquationalTheory of A holds
( b4 = EqTh R iff ( R c= b4 & ( for Q being EquationalTheory of A st R c= Q holds
b4 c= Q ) ) );

theorem :: MSUALG_6:52
for S being non empty non void ManySortedSign
for A being non-empty MSAlgebra over S
for R being ManySortedRelation of A holds
( EqCl (R,A) c= EqTh R & InvCl R c= EqTh R & StabCl R c= EqTh R & TRS R c= EqTh R )
proof end;

theorem :: MSUALG_6:53
for S being non empty non void ManySortedSign
for A being non-empty MSAlgebra over S
for R being ManySortedRelation of A
for s being SortSymbol of S
for a, b being Element of A,s holds
( [a,b] in (EqTh R) . s iff a,b are_convertible_wrt (TRS R) . s )
proof end;

theorem :: MSUALG_6:54
for S being non empty non void ManySortedSign
for A being non-empty MSAlgebra over S
for R being ManySortedRelation of A holds EqTh R = EqCl ((TRS R),A)
proof end;