
theorem Th14: :: THEOREM 2.10.
  for L be continuous complete LATTICE for p be kernel Function of
  L,L st p is directed-sups-preserving holds Image p is continuous LATTICE
proof
  let L be continuous complete LATTICE;
  let p be kernel Function of L,L such that
A1: p is directed-sups-preserving;
  now
    let X be Subset of L;
    assume X is non empty directed;
    then
A2: p preserves_sup_of X by A1,WAYBEL_0:def 37;
A3: Image p is sups-inheriting by WAYBEL_1:53;
    now
      assume
A4:   ex_sup_of X,L;
      Image p is complete by WAYBEL_1:54;
      hence ex_sup_of (corestr(p)).:X,Image p by YELLOW_0:17;
A5:   (corestr(p)).:X = p.:X by WAYBEL_1:30;
      ex_sup_of (p).:X,L by YELLOW_0:17;
      then "\/"((corestr p).:X,L) in the carrier of Image p by A3,A5,
YELLOW_0:def 19;
      hence sup ((corestr(p)).:X) = sup (p.:X) by A5,YELLOW_0:68
        .= p.sup X by A2,A4,WAYBEL_0:def 31
        .= (corestr(p)).sup X by WAYBEL_1:30;
    end;
    hence corestr(p) preserves_sup_of X by WAYBEL_0:def 31;
  end;
  then corestr(p) is directed-sups-preserving by WAYBEL_0:def 37;
  hence thesis by WAYBEL_1:56,WAYBEL_5:33;
end;
