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

theorem
  for U0 being with_const_op strict Universal_Algebra holds
  UnSubAlLattice(U0) is complete
proof
  let U0 be with_const_op strict Universal_Algebra;
  let L be Subset of UnSubAlLattice(U0);
  per cases;
  suppose
A1: L = {};
    thus thesis
    proof
      take Top UnSubAlLattice(U0);
      thus Top UnSubAlLattice(U0) is_less_than L
      by A1;
      let l2 be Element of UnSubAlLattice(U0);
      assume l2 is_less_than L;
      thus thesis by LATTICES:19;
    end;
  end;
  suppose
    L <> {};
    then reconsider H = L as non empty Subset of Sub(U0);
    reconsider l1 = meet H as Element of UnSubAlLattice(U0) by UNIALG_2:def 14;
    take l1;
    set x = the Element of H;
    thus l1 is_less_than L
    proof
      let l2 be Element of UnSubAlLattice(U0);
      reconsider U1 = l2 as strict SubAlgebra of U0 by UNIALG_2:def 14;
      reconsider u = l2 as Element of Sub(U0);
      assume
A2:   l2 in L;
      (Carr U0).u = the carrier of U1 by Def4;
      then meet ((Carr U0).:H) c= the carrier of U1 by A2,FUNCT_2:35,SETFAM_1:3
;
      then the carrier of meet H c= the carrier of U1 by Def5;
      hence l1 [= l2 by Th15;
    end;
    let l3 be Element of UnSubAlLattice(U0);
    reconsider U1 = l3 as strict SubAlgebra of U0 by UNIALG_2:def 14;
    assume
A3: l3 is_less_than L;
A4: for A be set st A in (Carr U0).:H holds the carrier of U1 c= A
    proof
      let A be set;
      assume
A5:   A in (Carr U0).:H;
      then reconsider H1 = A as Subset of U0;
      consider l4 being Element of Sub(U0) such that
A6:   l4 in H & H1 = (Carr U0).l4 by A5,FUNCT_2:65;
      reconsider l4 as Element of UnSubAlLattice(U0);
      reconsider U2 = l4 as strict SubAlgebra of U0 by UNIALG_2:def 14;
      A = the carrier of U2 & l3 [= l4 by A3,A6,Def4;
      hence thesis by Th15;
    end;
    (Carr U0).x in (Carr U0).:L by FUNCT_2:35;
    then the carrier of U1 c= meet ((Carr U0).:H) by A4,SETFAM_1:5;
    then the carrier of U1 c= the carrier of meet H by Def5;
    hence l3 [= l1 by Th15;
  end;
end;
