
theorem Th32:
  for L1,L2 be sup-Semilattice st L1,L2 are_isomorphic & L1 is
  lower-bounded algebraic holds L2 is algebraic
proof
  let L1,L2 be sup-Semilattice;
  assume that
A1: L1,L2 are_isomorphic and
A2: L1 is lower-bounded algebraic;
  consider f be Function of L1,L2 such that
A3: f is isomorphic by A1,WAYBEL_1:def 8;
  reconsider g = f" as Function of L2,L1 by A3,WAYBEL_0:67;
A4: g is isomorphic by A3,WAYBEL_0:68;
A5: now
    let y be Element of L2;
    y in the carrier of L2;
    then y in dom g by FUNCT_2:def 1;
    then
A6: y in rng f by A3,FUNCT_1:33;
A7: L2 is up-complete non empty Poset by A1,A2,Th30;
A8: compactbelow (g.y) is non empty directed by A2,WAYBEL_8:def 4;
    now
      let Y be finite Subset of compactbelow f.(g.y);
      Y c= the carrier of L2 by XBOOLE_1:1;
      then
A9:   Y c= rng f by A3,WAYBEL_0:66;
      now
        let z be object;
        assume z in g.:Y;
        then consider v be object such that
A10:    v in dom g and
A11:    v in Y and
A12:    z = g.v by FUNCT_1:def 6;
        reconsider v as Element of L2 by A10;
        v in compactbelow f.(g.y) by A11;
        then v in compactbelow y by A3,A6,FUNCT_1:35;
        then v <= y by WAYBEL_8:4;
        then
A13:    g.v <= g.y by A4,WAYBEL_0:66;
        v is compact by A11,WAYBEL_8:4;
        then g.v is compact by A2,A4,A7,Th28;
        hence z in compactbelow (g.y) by A12,A13,WAYBEL_8:4;
      end;
      then reconsider X = g.:Y as finite Subset of compactbelow (g.y) by
TARSKI:def 3;
      consider x be Element of L1 such that
A14:  x in compactbelow (g.y) and
A15:  x is_>=_than X by A8,WAYBEL_0:1;
      take fx = f.x;
      x <= g.y by A14,WAYBEL_8:4;
      then
A16:  f.x <= f.(g.y) by A3,WAYBEL_0:66;
      x is compact by A14,WAYBEL_8:4;
      then f.x is compact by A2,A3,A7,Th28;
      hence fx in compactbelow f.(g.y) by A16,WAYBEL_8:4;
      f.:X = f.:(f"Y) by A3,FUNCT_1:85
        .= Y by A9,FUNCT_1:77;
      hence fx is_>=_than Y by A3,A15,Th19;
    end;
    then compactbelow f.(g.y) is non empty directed by WAYBEL_0:1;
    hence compactbelow y is non empty directed by A3,A6,FUNCT_1:35;
  end;
  L2 is up-complete & L2 is satisfying_axiom_K by A1,A2,Th30,Th31;
  hence thesis by A5,WAYBEL_8:def 4;
end;
