
theorem
  for S, T being lower-bounded LATTICE st [:S,T:] is arithmetic holds S
  is arithmetic & T is arithmetic
proof
  let S, T be lower-bounded LATTICE such that
A1: [:S,T:] is algebraic and
A2: CompactSublatt [:S,T:] is meet-inheriting;
A3: S is up-complete & T is up-complete by A1,WAYBEL_2:11;
  hereby
    thus S is algebraic by A1,Th69;
    let x, y be Element of S such that
A4: x in the carrier of CompactSublatt S and
A5: y in the carrier of CompactSublatt S and
    ex_inf_of {x,y},S;
A6: [y,Bottom T]`1 = y & [y, Bottom T]`2 = Bottom T;
    y is compact by A5,WAYBEL_8:def 1;
    then [y,Bottom T] is compact by A3,A6,Th23,WAYBEL_3:15;
    then
A7: [y,Bottom T] in the carrier of CompactSublatt [:S,T:] by WAYBEL_8:def 1;
A8: [x,Bottom T]`1 = x & [x,Bottom T]`2 = Bottom T;
    x is compact by A4,WAYBEL_8:def 1;
    then [x,Bottom T] is compact by A3,A8,Th23,WAYBEL_3:15;
    then
    ex_inf_of {[x,Bottom T], [y,Bottom T]}, [:S,T:] & [x,Bottom T] in the
    carrier of CompactSublatt [:S,T:] by WAYBEL_8:def 1,YELLOW_0:21;
    then
    inf {[x,Bottom T], [y,Bottom T]} in the carrier of CompactSublatt [:S
    ,T:] by A2,A7;
    then
A9: inf {[x,Bottom T], [y,Bottom T]} is compact by WAYBEL_8:def 1;
    (inf {[x,Bottom T], [y,Bottom T]})`1 = ([x,Bottom T] "/\" [y,Bottom T
    ])`1 by YELLOW_0:40
      .= [x,Bottom T]`1 "/\" [y,Bottom T]`1 by Th13
      .= x "/\" [y,Bottom T]`1
      .= x "/\" y;
    then x "/\" y is compact by A3,A9,Th22;
    then inf {x,y} is compact by YELLOW_0:40;
    hence inf {x,y} in the carrier of CompactSublatt S by WAYBEL_8:def 1;
  end;
  thus T is algebraic by A1,Th69;
  let x, y be Element of T such that
A10: x in the carrier of CompactSublatt T and
A11: y in the carrier of CompactSublatt T and
  ex_inf_of {x,y},T;
A12: [Bottom S,y]`2 = y & [ Bottom S,y]`1 = Bottom S;
  y is compact by A11,WAYBEL_8:def 1;
  then [Bottom S,y] is compact by A3,A12,Th23,WAYBEL_3:15;
  then
A13: [Bottom S,y] in the carrier of CompactSublatt [:S,T:] by WAYBEL_8:def 1;
A14: [Bottom S,x]`2 = x & [Bottom S,x]`1 = Bottom S;
  x is compact by A10,WAYBEL_8:def 1;
  then [Bottom S,x] is compact by A3,A14,Th23,WAYBEL_3:15;
  then ex_inf_of {[Bottom S,x], [Bottom S,y]}, [:S,T:] & [Bottom S,x] in the
  carrier of CompactSublatt [:S,T:] by WAYBEL_8:def 1,YELLOW_0:21;
  then
  inf {[Bottom S,x], [Bottom S,y]} in the carrier of CompactSublatt [:S,
  T:] by A2,A13;
  then
A15: inf {[Bottom S,x], [Bottom S,y]} is compact by WAYBEL_8:def 1;
  (inf {[Bottom S,x], [Bottom S,y]})`2 = ([Bottom S,x] "/\" [Bottom S,y])
  `2 by YELLOW_0:40
    .= [Bottom S,x]`2 "/\" [Bottom S,y]`2 by Th13
    .= x "/\" [Bottom S,y]`2
    .= x "/\" y;
  then x "/\" y is compact by A3,A15,Th22;
  then inf {x,y} is compact by YELLOW_0:40;
  hence thesis by WAYBEL_8:def 1;
end;
