
theorem Th29:
  for L being complete LATTICE, k being kernel Function of L,L holds
  corestr k is infs-preserving & inclusion k is sups-preserving &
  LowerAdj corestr k = inclusion k & UpperAdj inclusion k = corestr k
proof
  let L be complete LATTICE, k be kernel Function of L,L;
A1: [corestr k, inclusion k] is Galois by WAYBEL_1:39;
  then
A2: inclusion k is lower_adjoint;
A3: corestr k is upper_adjoint by A1;
  hence corestr k is infs-preserving & inclusion k is sups-preserving by A2;
  thus thesis by A1,A2,A3,Def1,Def2;
end;
