
theorem Th10:
  for L1,L2 be LATTICE st L1,L2 are_isomorphic & L1 is
  lower-bounded arithmetic holds L2 is arithmetic
proof
  let L1,L2 be LATTICE;
  assume that
A1: L1,L2 are_isomorphic and
A2: L1 is lower-bounded arithmetic;
  consider f be Function of L1,L2 such that
A3: f is isomorphic by A1;
  reconsider g = (f qua Function)" as Function of L2,L1 by A3,WAYBEL_0:67;
A4: g is isomorphic by A3,WAYBEL_0:68;
A5: L2 is up-complete LATTICE by A1,A2,WAYBEL13:30;
  now
    let x2,y2 be Element of L2;
    assume that
A6: x2 in the carrier of CompactSublatt L2 and
A7: y2 in the carrier of CompactSublatt L2 and
A8: ex_inf_of {x2,y2},L2;
    x2 is compact by A6,WAYBEL_8:def 1;
    then g.x2 is compact by A2,A4,A5,WAYBEL13:28;
    then
A9: g.x2 in the carrier of CompactSublatt L1 by WAYBEL_8:def 1;
    y2 is compact by A7,WAYBEL_8:def 1;
    then g.y2 is compact by A2,A4,A5,WAYBEL13:28;
    then
A10: g.y2 in the carrier of CompactSublatt L1 by WAYBEL_8:def 1;
    x2 in the carrier of L2;
    then
A11: x2 in dom g by FUNCT_2:def 1;
A12: CompactSublatt L1 is meet-inheriting by A2,WAYBEL_8:def 5;
    y2 in the carrier of L2;
    then
A13: y2 in dom g by FUNCT_2:def 1;
    g is infs-preserving by A4,WAYBEL13:20;
    then
A14: g preserves_inf_of {x2,y2} by WAYBEL_0:def 32;
    then ex_inf_of g.:{x2,y2},L1 by A8,WAYBEL_0:def 30;
    then ex_inf_of {g.x2,g.y2},L1 by A11,A13,FUNCT_1:60;
    then inf {g.x2,g.y2} in the carrier of CompactSublatt L1 by A9,A10,A12,
YELLOW_0:def 16;
    then
A15: inf {g.x2,g.y2} is compact by WAYBEL_8:def 1;
    g.(inf {x2,y2}) = inf (g.:{x2,y2}) by A8,A14,WAYBEL_0:def 30
      .= inf {g.x2,g.y2} by A11,A13,FUNCT_1:60;
    then inf {x2,y2} is compact by A2,A4,A5,A15,WAYBEL13:28;
    hence inf {x2,y2} in the carrier of CompactSublatt L2 by WAYBEL_8:def 1;
  end;
  then
A16: CompactSublatt L2 is meet-inheriting by YELLOW_0:def 16;
  L2 is algebraic by A1,A2,WAYBEL13:32;
  hence thesis by A16,WAYBEL_8:def 5;
end;
