reserve S for non empty non void ManySortedSign,
  A for MSAlgebra over S;
reserve A for non-empty MSAlgebra over S;
reserve S for non empty non void ManySortedSign,
  A for non-empty MSAlgebra over S,
  R for ManySortedRelation of the Sorts of A;

theorem Th29:
  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
  s9 being SortSymbol of S, x,y being Element of A,s9 st 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
  let P be ManySortedRelation of the Sorts of A;
  defpred Z[SortSymbol of S,set,set] means ex s9 being SortSymbol of S, f
being Function of (the Sorts of A).s9,(the Sorts of A).$1, x,y being Element of
A,s9 st TranslationRel S reduces s9,$1 & f is Translation of A,s9,$1 & [x,y] in
  P.s9 & $2 = f.x & $3 = f.y;
  let s be SortSymbol of S;
  let a,b be Element of A,s;
  consider Q being ManySortedRelation of the Sorts of A such that
A1: for s being SortSymbol of S, a,b being Element of A,s holds [a,b] in
  Q.s iff Z[s,a,b] from MSRExistence;
  reconsider R = P,Q as ManySortedRelation of A;
A2: R c= Q by A1,Lm1;
  reconsider Q as invariant ManySortedRelation of A by A1,Lm1;
  R c= InvCl R by Def11;
  then
A3: Q c= InvCl R by A1,Lm1;
  InvCl R c= Q by A2,Def11;
  then
A4: InvCl R = Q by A3,PBOOLE:146;
  hereby
    assume [a,b] in (InvCl P).s;
    then
    ex s9 being SortSymbol of S, f being Function of (the Sorts of A).s9,(
the Sorts of A).s, x,y being Element of A,s9 st TranslationRel S reduces s9,s &
    f is Translation of A,s9,s & [x,y] in P.s9 & a = f.x & b = f.y by A1,A4;
    hence ex s9 being SortSymbol of S, x,y being Element of A,s9 st ex t being
Translation of A,s9,s st TranslationRel S reduces s9,s & [x,y] in P.s9 & a = t.
    x & b = t.y;
  end;
  thus thesis by A1,A4;
end;
