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

theorem Th57:
  for S being Signature, E being non empty Signature, A being
MSAlgebra over E st A is Algebra of S holds the carrier of S c= the carrier of
  E & the carrier' of S c= the carrier' of E
proof
  let S be Signature, E be non empty Signature, A be MSAlgebra over E;
A1: dom the Charact of A = the carrier' of E by PARTFUN1:def 2;
  assume A is Algebra of S;
  then consider ES being non void Extension of S such that
A2: A is feasible MSAlgebra over ES by Def7;
  reconsider B = A as MSAlgebra over ES by A2;
A3: dom the Sorts of A = the carrier of E by PARTFUN1:def 2;
A4: S is Subsignature of ES by Def5;
  dom the Sorts of B = the carrier of ES by PARTFUN1:def 2;
  hence the carrier of S c= the carrier of E by A4,A3,INSTALG1:10;
  dom the Charact of B = the carrier' of ES by PARTFUN1:def 2;
  hence thesis by A4,A1,INSTALG1:10;
end;
