
theorem Th12:
  for L1 be finite LATTICE holds L1 is arithmetic
proof
  let L1 be finite LATTICE;
  thus for x be Element of L1 holds compactbelow x is non empty directed;
  thus L1 is up-complete;
  thus L1 is satisfying_axiom_K
  proof
    let x be Element of L1;
A1: for y be Element of L1 st y is_>=_than compactbelow x holds x <= y
       by Th11,LATTICE3:def 9;
    for y be Element of L1 st y in compactbelow x holds y <= x by WAYBEL_8:4;
    then x is_>=_than compactbelow x by LATTICE3:def 9;
    hence thesis by A1,YELLOW_0:30;
  end;
  thus CompactSublatt L1 is meet-inheriting
  proof
    let x,y be Element of L1;
    assume that
    x in the carrier of CompactSublatt L1 and
    y in the carrier of CompactSublatt L1 and
    ex_inf_of {x,y},L1;
    x "/\" y is compact by WAYBEL_3:17;
    then x "/\" y in the carrier of CompactSublatt L1 by WAYBEL_8:def 1;
    hence thesis by YELLOW_0:40;
  end;
end;
