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 Th30:
  for R being stable ManySortedRelation of A holds InvCl R is stable
proof
  let R be stable ManySortedRelation of A;
  let h be Endomorphism of A;
  let s be SortSymbol of S;
  let a,b be set;
  assume
A1: [a,b] in (InvCl R).s;
  then
A2: b in (the Sorts of A).s by ZFMISC_1:87;
  a in (the Sorts of A).s by A1,ZFMISC_1:87;
  then consider s9 being SortSymbol of S, x,y being Element of A,s9, t being
  Translation of A,s9,s such that
A3: TranslationRel S reduces s9,s and
A4: [x,y] in R.s9 and
A5: a = t.x and
A6: b = t.y by A1,A2,Th29;
  consider T being Translation of A,s9,s such that
A7: T*(h.s9) = (h.s)*t by A3,Th26;
  (T*(h.s9)).y = T.(h.s9.y) by FUNCT_2:15;
  then
A8: T.(h.s9.y) = h.s.b by A6,A7,FUNCT_2:15;
  (T*(h.s9)).x = T.(h.s9.x) by FUNCT_2:15;
  then
A9: T.(h.s9.x) = h.s.a by A5,A7,FUNCT_2:15;
  [h.s9.x,h.s9.y] in R.s9 by A4,Def9;
  hence thesis by A3,A9,A8,Th29;
end;
