
theorem :: 1.12. COROLLARY, p. 183
  for S,T being complete LATTICE st T is algebraic
  for g being infs-preserving Function of S,T holds
  g is directed-sups-preserving iff
  (LowerAdj g)|the carrier of CompactSublatt T is
  finite-sups-preserving Function of CompactSublatt T, CompactSublatt S
proof
  let S,T be complete LATTICE such that
A1: T is algebraic;
  let g be infs-preserving Function of S,T;
  hereby
    assume g is directed-sups-preserving;
    then LowerAdj g is waybelow-preserving by Th22;
    then LowerAdj g is compact-preserving by Th56;
    hence (LowerAdj g)|the carrier of CompactSublatt T is
    finite-sups-preserving Function of CompactSublatt T, CompactSublatt S
    by Th66;
  end;
  assume (LowerAdj g)|the carrier of CompactSublatt T is
  finite-sups-preserving Function of CompactSublatt T, CompactSublatt S;
  then LowerAdj g is compact-preserving by Th66;
  then LowerAdj g is waybelow-preserving by A1,Th57;
  hence thesis by A1,Th23;
end;
