
theorem Th35:
  for L being complete LATTICE, c being closure Function of L,L holds
  corestr c is sups-preserving & inclusion c is infs-preserving &
  UpperAdj corestr c = inclusion c & LowerAdj inclusion c = corestr c
proof
  let L be complete LATTICE, c be closure Function of L,L;
A1: [inclusion c, corestr c] is Galois by WAYBEL_1:36;
  then
A2: inclusion c is upper_adjoint;
A3: corestr c is lower_adjoint by A1;
  hence corestr c is sups-preserving & inclusion c is infs-preserving by A2;
  thus thesis by A1,A2,A3,Def1,Def2;
end;
