reserve S for non empty non void ManySortedSign,
  A for MSAlgebra over S;

theorem
  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 for q being RedSequence of TranslationRel S, p being Function-yielding
FinSequence st len q = len p+1 & s1 = q.1 & s2 = q.len q & for i being (Element
of NAT), f being Function, 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
  let S be non empty non void ManySortedSign;
  let A be non-empty MSAlgebra over S;
  let s1,s2 be SortSymbol of S such that
A1: TranslationRel S reduces s1,s2;
  let q be RedSequence of TranslationRel S, p be Function-yielding FinSequence
  such that
A2: len q = len p+1 and
A3: s1 = q.1 and
A4: s2 = q.len q and
A5: for i being (Element of NAT), f being Function, 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;
  compose(p, (the Sorts of A).s1) is Function of (the Sorts of A).s1, (the
  Sorts of A).s2 by A2,A3,A4,A5,Th14;
  hence thesis by A1,A2,A3,A4,A5,Def6;
end;
