
theorem Th5:
  for S being non empty non void ManySortedSign
  for A being non-empty MSAlgebra over S
  holds Image id the Sorts of A = the MSAlgebra of A
  proof
    let S be non empty non void ManySortedSign;
    let A be non-empty MSAlgebra over S;
    the MSAlgebra of A is strict non-empty MSSubAlgebra of A &
    id the Sorts of A is_homomorphism A,A &
    (id the Sorts of A).:.:the Sorts of A = the Sorts of A
    by EQUATION:15,MSUALG_2:5,MSUALG_3:9;
    hence Image id the Sorts of A = the MSAlgebra of A by MSUALG_3:def 12;
  end;
