
theorem :: 1.7. COROLLARY, (1) => (3), p. 181
  for L being complete LATTICE, k being kernel Function of L,L
  st k is directed-sups-preserving
  for x,y being Element of L, a,b being Element of Image k st a = x & b = y
  holds x << y iff a << b
proof
  let L be complete LATTICE, k be kernel Function of L,L;
  set g = corestr k;
  assume k is directed-sups-preserving;
  then corestr k is directed-sups-preserving infs-preserving by Th29,Th30;
  then
A1: LowerAdj g is waybelow-preserving by Th22;
  let x,y be Element of L, a,b be Element of Image k;
A2: Image k is sups-inheriting by WAYBEL_1:53;
A3: inclusion k = LowerAdj g by Th29;
  then
A4: (LowerAdj g).a = a by FUNCT_1:18;
  (LowerAdj g).b = b by A3,FUNCT_1:18;
  hence thesis by A1,A2,A4,Th32;
end;
