reserve I for set;
reserve S for non empty non void ManySortedSign,
  U0, U1 for non-empty MSAlgebra over S;

theorem Th19:
  for U0 being MSAlgebra over S, A being MSSubAlgebra of U0 for o
being OperSymbol of S, x being set st x in Args(o,A) holds Den(o,A).x = Den(o,
  U0).x
proof
  let U0 be MSAlgebra over S, A be MSSubAlgebra of U0, o be OperSymbol of S, x
  be set such that
A1: x in Args(o,A);
  reconsider B = the Sorts of A as MSSubset of U0 by MSUALG_2:def 9;
  B is opers_closed by MSUALG_2:def 9;
  then
A2: B is_closed_on o;
  thus Den(o,A).x = (Opers(U0,B).o).x by MSUALG_2:def 9
    .= (o/.B).x by MSUALG_2:def 8
    .= ((Den(o,U0)) | ((B# * the Arity of S).o)).x by A2,MSUALG_2:def 7
    .= Den(o,U0).x by A1,FUNCT_1:49;
end;
