reserve x, y, i for object,
  L for up-complete Semilattice;
reserve L for complete LATTICE,
  a, b, c for Element of L,
  J for non empty set,
  K for non-empty ManySortedSet of J;
reserve J, K, D for non empty set,
  j for Element of J,
  k for Element of K;
reserve J for non empty set,
  K for non-empty ManySortedSet of J;

theorem Th29:
  for S being non empty Poset st ex g being Function of L, S st g
  is infs-preserving onto holds S is complete LATTICE
proof
  let S be non empty Poset;
  given g being Function of L, S such that
A1: g is infs-preserving and
A2: g is onto;
  for A being Subset of S holds ex_inf_of A,S
  proof
    let A be Subset of S;
    set Y = g"A;
    rng g = the carrier of S by A2,FUNCT_2:def 3;
    then
A3: A = g.:Y by FUNCT_1:77;
    ex_inf_of Y,L & g preserves_inf_of Y by A1,WAYBEL_0:def 32,YELLOW_0:17;
    hence thesis by A3,WAYBEL_0:def 30;
  end;
  then S is complete non empty Poset by YELLOW_2:28;
  hence thesis;
end;
