
theorem :: PROPOSITION 4.12.(iii)
  for S be lower-bounded LATTICE holds InclPoset Ids S is arithmetic
proof
  let S be lower-bounded LATTICE;
  now
    let x,y be Element of CompactSublatt InclPoset Ids S;
    reconsider x1 = x, y1 = y as Element of InclPoset Ids S by YELLOW_0:58;
    x1 is compact by WAYBEL_8:def 1;
    then consider a be Element of S such that
A1: x1 = downarrow a by Th12;
    y1 is compact by WAYBEL_8:def 1;
    then consider b be Element of S such that
A2: y1 = downarrow b by Th12;
    Bottom S <= b by YELLOW_0:44;
    then
A3: Bottom S in downarrow b by WAYBEL_0:17;
    Bottom S <= a by YELLOW_0:44;
    then Bottom S in downarrow a by WAYBEL_0:17;
    then reconsider
    xy = downarrow a /\ downarrow b as non empty Subset of S by A3,
XBOOLE_0:def 4;
    reconsider xy as lower non empty Subset of S by WAYBEL_0:27;
    reconsider xy as lower directed non empty Subset of S by WAYBEL_0:44;
    xy is Ideal of S;
    then downarrow a /\ downarrow b in the set of all X where X is Ideal of S;
    then downarrow a /\ downarrow b in Ids S by WAYBEL_0:def 23;
    then x1 "/\" y1 = downarrow a /\ downarrow b by A1,A2,YELLOW_1:9;
    then reconsider
    z1 = downarrow a /\ downarrow b as Element of InclPoset Ids S;
    z1 c= y1 by A2,XBOOLE_1:17;
    then
A4: z1 <= y1 by YELLOW_1:3;
A5: downarrow (a "/\" b) c= downarrow a /\ downarrow b
    proof
      let v be object;
      assume
A6:   v in downarrow (a "/\" b);
      then reconsider v1 = v as Element of S;
A7:   v1 <= a "/\" b by A6,WAYBEL_0:17;
      a "/\" b <= b by YELLOW_0:23;
      then v1 <= b by A7,ORDERS_2:3;
      then
A8:   v in downarrow b by WAYBEL_0:17;
      a "/\" b <= a by YELLOW_0:23;
      then v1 <= a by A7,ORDERS_2:3;
      then v in downarrow a by WAYBEL_0:17;
      hence thesis by A8,XBOOLE_0:def 4;
    end;
    downarrow a /\ downarrow b c= downarrow (a "/\" b)
    proof
      let v be object;
      assume
A9:   v in downarrow a /\ downarrow b;
      then reconsider v1 = v as Element of S;
      v in downarrow b by A9,XBOOLE_0:def 4;
      then
A10:  v1 <= b by WAYBEL_0:17;
      v in downarrow a by A9,XBOOLE_0:def 4;
      then v1 <= a by WAYBEL_0:17;
      then v1 <= a "/\" b by A10,YELLOW_0:23;
      hence thesis by WAYBEL_0:17;
    end;
    then z1 = downarrow (a "/\" b) by A5,XBOOLE_0:def 10;
    then z1 is compact by Th12;
    then reconsider z = z1 as Element of CompactSublatt InclPoset Ids S by
WAYBEL_8:def 1;
    take z;
    z1 c= x1 by A1,XBOOLE_1:17;
    then z1 <= x1 by YELLOW_1:3;
    hence z <= x & z <= y by A4,YELLOW_0:60;
    let v be Element of CompactSublatt InclPoset Ids S;
    reconsider v1 = v as Element of InclPoset Ids S by YELLOW_0:58;
    assume that
A11: v <= x and
A12: v <= y;
    v1 <= y1 by A12,YELLOW_0:59;
    then
A13: v1 c= y1 by YELLOW_1:3;
    v1 <= x1 by A11,YELLOW_0:59;
    then v1 c= x1 by YELLOW_1:3;
    then v1 c= z1 by A1,A2,A13,XBOOLE_1:19;
    then v1 <= z1 by YELLOW_1:3;
    hence v <= z by YELLOW_0:60;
  end;
  then
A14: CompactSublatt InclPoset Ids S is with_infima by LATTICE3:def 11;
  InclPoset Ids S is algebraic by Th10;
  hence thesis by A14,WAYBEL_8:19;
end;
