
theorem
  for S,T being complete LATTICE for g being infs-preserving Function of S,T
  for t being Element of T holds (LowerAdj g).t = inf (g"uparrow t)
proof
  let S,T be complete LATTICE;
  let g be infs-preserving Function of S,T;
  let t be Element of T;
  [g, LowerAdj g] is Galois by Def1;
  then (LowerAdj g).t is_minimum_of g"(uparrow t) by WAYBEL_1:10;
  hence thesis;
end;
