reserve S,S9 for non void Signature,
  f,g for Function;

theorem
  for S9 being non void Signature for A being non-empty disjoint_valued
  MSAlgebra over S st A is Algebra of S9 holds S is Extension of S9
proof
  let S9 be non void Signature;
  let A be non-empty disjoint_valued MSAlgebra over S;
  assume A is Algebra of S9;
  then consider E being non void Extension of S9 such that
A1: A is feasible MSAlgebra over E by Def7;
A2: S9 is Subsignature of E by Def5;
A3: the ManySortedSign of S = the ManySortedSign of E by A1,Th62;
  then
A4: the ResultSort of S9 c= the ResultSort of S by A2,INSTALG1:11;
  the Arity of S9 c= the Arity of S by A2,A3,INSTALG1:11;
  hence S9 is Subsignature of S by A2,A3,A4,INSTALG1:10,13;
end;
