
theorem :: 1.8. COROLLARY, (3) => (1), p. 182
:: SHOULD BE:
:: for L being complete LATTICE, c being closure map of L,L
::  st
::    Image c is continuous & corestr c is waybelow-preserving
::  holds Image c is directed-sups-inheriting
:: ------------------------ a bug ???
  for L being continuous complete LATTICE, c being closure Function of L,L st
  corestr c is waybelow-preserving holds Image c is directed-sups-inheriting
proof
  let L be continuous complete LATTICE, c be closure Function of L,L;
  assume
A1: corestr c is waybelow-preserving;
A2: LowerAdj inclusion c = corestr c by Th35;
  inclusion c is infs-preserving by Th35;
  then inclusion c is directed-sups-preserving by A1,A2,Th23;
  hence thesis by Th36;
end;
