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

theorem
  for S being non empty non void ManySortedSign, s1,s2 being SortSymbol
of S for A being non-empty MSAlgebra over S st [s1,s2] in TranslationRel S ex f
  being Function st f is_e.translation_of A,s1,s2
proof
  let S be non empty non void ManySortedSign, s1,s2 be SortSymbol of S;
  let A be non-empty MSAlgebra over S;
  assume [s1,s2] in TranslationRel S;
  then consider o being OperSymbol of S such that
A1: the_result_sort_of o = s2 and
A2: ex i being Element of NAT st i in dom the_arity_of o & (the_arity_of
  o)/.i = s1 by Def3;
  set a = the Element of Args(o,A);
  consider i being Element of NAT such that
A3: i in dom the_arity_of o and
A4: (the_arity_of o)/.i = s1 by A2;
  take transl(o,i,a,A), o;
  thus thesis by A1,A3,A4;
end;
