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

theorem Th15:
  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 the carrier of U1 c= the carrier 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;
  reconsider l1 = U1 as Element of UnSubAlLattice(U0) by UNIALG_2:def 14;
  reconsider l2 = U2 as Element of UnSubAlLattice(U0) by UNIALG_2:def 14;
A1: l1 [= l2 implies the carrier of U1 c= the carrier of U2
  proof
    reconsider U21 = the carrier of U2 as Subset of U0 by UNIALG_2:def 7;
    reconsider U11 = the carrier of U1 as Subset of U0 by UNIALG_2:def 7;
    reconsider U3 = U11 \/ U21 as non empty Subset of U0;
    assume l1 [= l2;
    then l1 "\/" l2 = l2;
    then U1 "\/" U2 = U2 by UNIALG_2:def 15;
    then GenUnivAlg (U3) = U2 by UNIALG_2:def 13;
    then
A2: (the carrier of U1) \/ the carrier of U2 c= the carrier of U2 by
UNIALG_2:def 12;
    the carrier of U2 c= (the carrier of U1) \/ the carrier of U2 by XBOOLE_1:7
;
    then (the carrier of U1) \/ the carrier of U2 = the carrier of U2 by A2;
    hence thesis by XBOOLE_1:7;
  end;
  the carrier of U1 c= the carrier of U2 implies l1 [= l2
  proof
    reconsider U21 = the carrier of U2 as Subset of U0 by UNIALG_2:def 7;
    reconsider U11 = the carrier of U1 as Subset of U0 by UNIALG_2:def 7;
    reconsider U3 = U11 \/ U21 as non empty Subset of U0;
    assume the carrier of U1 c= the carrier of U2;
    then GenUnivAlg (U3) = U2 by UNIALG_2:19,XBOOLE_1:12;
    then U1 "\/" U2 = U2 by UNIALG_2:def 13;
    then l1 "\/" l2 = l2 by UNIALG_2:def 15;
    hence thesis;
  end;
  hence thesis by A1;
end;
