let S be non empty non void ManySortedSign ; :: thesis: for A being non-empty MSAlgebra of 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 (InvCl R) . s iff ex s' being SortSymbol of S ex x, y being Element of A,s' ex t being Translation of A,s',s st
( TranslationRel S reduces s',s & [x,y] in R . s' & a = t . x & b = t . y ) )
let A be non-empty MSAlgebra of S; :: thesis: 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 (InvCl R) . s iff ex s' being SortSymbol of S ex x, y being Element of A,s' ex t being Translation of A,s',s st
( TranslationRel S reduces s',s & [x,y] in R . s' & a = t . x & b = t . y ) )
let P be ManySortedRelation of the Sorts of A; :: thesis: for s being SortSymbol of S
for a, b being Element of A,s holds
( [a,b] in (InvCl P) . s iff ex s' being SortSymbol of S ex x, y being Element of A,s' ex t being Translation of A,s',s st
( TranslationRel S reduces s',s & [x,y] in P . s' & a = t . x & b = t . y ) )
defpred S₁[ SortSymbol of S, set , set ] means ex s' being SortSymbol of S ex f being Function of (the Sorts of A . s'),(the Sorts of A . $1) ex x, y being Element of A,s' st
( TranslationRel S reduces s',$1 & f is Translation of A,s',$1 & [x,y] in P . s' & $2 = f . x & $3 = f . y );
consider Q being ManySortedRelation of the Sorts of A such that
A1: for s being SortSymbol of S
for a, b being Element of A,s holds
( [a,b] in Q . s iff S₁[s,a,b] ) from MSUALG_6:sch 2();
reconsider R = P, Q = Q as ManySortedRelation of A ;
A2: ( Q is invariant & R c= Q & ( for P being ManySortedRelation of A st P is invariant & R c= P holds
Q c= P ) ) by A1, Lm1;
reconsider Q = Q as invariant ManySortedRelation of A by A1, Lm1;
A3: InvCl R = Q

proof

R c= InvCl R by Def11;
hence ( InvCl R c= Q & Q c= InvCl R ) by A2, Def11; :: according to PBOOLE:def 13 :: thesis: verum

end;

let s be SortSymbol of S; :: thesis: for a, b being Element of A,s holds
( [a,b] in (InvCl P) . s iff ex s' being SortSymbol of S ex x, y being Element of A,s' ex t being Translation of A,s',s st
( TranslationRel S reduces s',s & [x,y] in P . s' & a = t . x & b = t . y ) )
let a, b be Element of A,s; :: thesis: ( [a,b] in (InvCl P) . s iff ex s' being SortSymbol of S ex x, y being Element of A,s' ex t being Translation of A,s',s st
( TranslationRel S reduces s',s & [x,y] in P . s' & a = t . x & b = t . y ) )

hereby :: thesis: ( ex s' being SortSymbol of S ex x, y being Element of A,s' ex t being Translation of A,s',s st
( TranslationRel S reduces s',s & [x,y] in P . s' & a = t . x & b = t . y ) implies [a,b] in (InvCl P) . s )

assume [a,b] in (InvCl P) . s ; :: thesis: ex s' being SortSymbol of S ex x, y being Element of A,s' ex t being Translation of A,s',s st
( TranslationRel S reduces s',s & [x,y] in P . s' & a = t . x & b = t . y )
then ex s' being SortSymbol of S ex f being Function of (the Sorts of A . s'),(the Sorts of A . s) ex x, y being Element of A,s' st
( TranslationRel S reduces s',s & f is Translation of A,s',s & [x,y] in P . s' & a = f . x & b = f . y ) by A1, A3;
hence ex s' being SortSymbol of S ex x, y being Element of A,s' ex t being Translation of A,s',s st
( TranslationRel S reduces s',s & [x,y] in P . s' & a = t . x & b = t . y ) ; :: thesis: verum

end;

thus ( ex s' being SortSymbol of S ex x, y being Element of A,s' ex t being Translation of A,s',s st
( TranslationRel S reduces s',s & [x,y] in P . s' & a = t . x & b = t . y ) implies [a,b] in (InvCl P) . s ) by A1, A3; :: thesis: verum