
theorem
  for L being non empty Poset, k being Function of L,L st k is kernel
  holds (corestr k) is infs-preserving & for X being Subset of L st X c= the
carrier of Image k & ex_inf_of X,L holds ex_inf_of X,Image k & "/\"(X,Image k)
  = k.("/\"(X,L))
proof
  let L be non empty Poset, k be Function of L,L;
A1: (corestr k) = k by Th30;
  assume
A2: k is kernel;
  then
A3: k is idempotent by Def13;
  [corestr k,inclusion k] is Galois by A2,Th39;
  then
A4: corestr k is upper_adjoint;
  hence (corestr k) is infs-preserving;
  let X be Subset of L such that
A5: X c= the carrier of Image k and
A6: ex_inf_of X,L;
  X c= rng k by A5,YELLOW_0:def 15;
  then
A7: k.:X = X by A3,YELLOW_2:20;
  corestr k preserves_inf_of X by A4,WAYBEL_0:def 32;
  hence thesis by A6,A1,A7;
end;
