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
  for R being ManySortedRelation of the Sorts of A for s being
  SortSymbol of S, 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, h being Endomorphism of A, t being Translation of A, s9, s st [l,r] in
  R.s9 & a = t.(h.s9.l) & b = t.(h.s9.r)
proof
  let R be ManySortedRelation of the Sorts of A;
  let s be SortSymbol of S, a,b be Element of A,s;
A1: InvCl StabCl R = TRS R by Th37;
  hereby
    assume [a,b] in (TRS R).s;
    then consider s9 being SortSymbol of S, x,y being Element of A,s9, t being
    Translation of A,s9,s such that
A2: TranslationRel S reduces s9,s and
A3: [x,y] in (StabCl R).s9 and
A4: a = t.x and
A5: b = t.y by A1,Th29;
    take s9;
    thus TranslationRel S reduces s9,s by A2;
    reconsider t as Translation of A,s9,s;
    consider u,v being Element of A,s9, h being Endomorphism of A such that
A6: [u,v] in R.s9 and
A7: x = h.s9.u and
A8: y = h.s9.v by A3,Th31;
    take u,v,h;
    take t;
    thus [u,v] in R.s9 & a = t.((h.s9).u) & b = t.((h.s9).v) by A4,A5,A6,A7,A8;
  end;
  given s9 being SortSymbol of S such that
A9: TranslationRel S reduces s9, s and
A10: ex l,r being Element of A,s9, h being Endomorphism of A, t being
Translation of A, s9, s st [l,r] in R.s9 & a = t.((h.s9).l) & b = t.((h.s9).r);
  consider l,r being Element of A,s9, h being Endomorphism of A, t being
  Translation of A, s9, s such that
A11: [l,r] in R.s9 and
A12: a = t.((h.s9).l) and
A13: b = t.((h.s9).r) by A10;
  [h.s9.l,h.s9.r] in (StabCl R).s9 by A11,Th31;
  hence thesis by A1,A9,A12,A13,Th29;
end;
