
theorem Th57: :: 1.11. PROPOSITION, (2) => (1) p.183
  for S,T being complete LATTICE, d being sups-preserving Function of T,S
  st T is algebraic & d is compact-preserving holds d is waybelow-preserving
proof
  let S,T be complete LATTICE, d be sups-preserving Function of T,S such that
A1: T is algebraic and
A2: for t being Element of T st t is compact holds d.t is compact;
  let t,t9 be Element of T;
  assume t << t9;
  then consider k being Element of T such that
A3: k in the carrier of CompactSublatt T and
A4: t <= k and
A5: k <= t9 by A1,WAYBEL_8:7;
  k is compact by A3,WAYBEL_8:def 1;
  then d.k is compact by A2;
  then
A6: d.k << d.k;
A7: d.t <= d.k by A4,WAYBEL_1:def 2;
  d.k <= d.t9 by A5,WAYBEL_1:def 2;
  hence thesis by A6,A7,WAYBEL_3:2;
end;
