reserve U0 for Universal_Algebra,
  U1 for SubAlgebra of U0,
  o for operation of U0;

theorem
  for U0 being with_const_op Universal_Algebra for l1,l2 being Element
of UnSubAlLattice(U0), U1,U2 being strict SubAlgebra of U0 st l1 = U1 & l2 = U2
  holds l1 [= l2 iff U1 is SubAlgebra of U2
proof
  let U0 be with_const_op Universal_Algebra;
  let l1,l2 be Element of UnSubAlLattice(U0);
  let U1,U2 be strict SubAlgebra of U0 such that
A1: l1 = U1 & l2 = U2;
  thus l1 [= l2 implies U1 is SubAlgebra of U2
  proof
    assume l1 [= l2;
    then the carrier of U1 c= the carrier of U2 by A1,Th15;
    hence thesis by UNIALG_2:11;
  end;
  thus U1 is SubAlgebra of U2 implies l1 [= l2
  proof
    assume U1 is SubAlgebra of U2;
    then the carrier of U1 is Subset of U2 by UNIALG_2:def 7;
    hence thesis by A1,Th15;
  end;
end;
