
theorem Th22: :: 1.4. THEOREM, (1) => (3), p. 180
  for S,T being complete LATTICE, g being infs-preserving Function of S,T holds
  g is directed-sups-preserving implies LowerAdj g is waybelow-preserving
proof
  let S,T be complete LATTICE, g be infs-preserving Function of S,T such that
A1: g is directed-sups-preserving;
  set d = (LowerAdj g);
A2: [g,d] is Galois by Def1;
  let t,t9 be Element of T such that
A3: t << t9;
  let D be non empty directed Subset of S;
  assume d.t9 <= sup D;
  then
A4: t9 <= g.sup D by A2,WAYBEL_1:8;
A5: g preserves_sup_of D by A1;
  ex_sup_of D,S by YELLOW_0:17;
  then
A6: g.sup D = sup (g.:D) by A5;
  g.:D is directed non empty by YELLOW_2:15;
  then consider x being Element of T such that
A7: x in g.:D and
A8: t <= x by A3,A4,A6;
  consider s being object such that
A9: s in the carrier of S and
A10: s in D and
A11: x = g.s by A7,FUNCT_2:64;
  reconsider s as Element of S by A9;
  take s;
  thus thesis by A2,A8,A10,A11,WAYBEL_1:8;
end;
