
theorem :: 1.7. COROLLARY, (3) => (1), p. 181
  for L being complete LATTICE, k being kernel Function of L,L
  st Image k is continuous &
  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 holds k is directed-sups-preserving
proof
  let L be complete LATTICE, k be kernel Function of L,L such that
A1: Image k is continuous and
A2: 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;
  set g = corestr k;
A3: corestr k is infs-preserving by Th29;
  LowerAdj g is waybelow-preserving
  proof
    let t,t9 be Element of Image k;
A4: LowerAdj g = inclusion k by Th29;
    then
A5: (LowerAdj g).t = t by FUNCT_1:18;
    (LowerAdj g).t9 = t9 by A4,FUNCT_1:18;
    hence thesis by A2,A5;
  end;
  then corestr k is directed-sups-preserving by A1,A3,Th23;
  hence thesis by Th30;
end;
