
theorem Th18:
  for L1,L2 be LATTICE st the RelStr of L1 = the RelStr of L2 & L1
  is arithmetic holds L2 is arithmetic
proof
  let L1,L2 be LATTICE;
  assume that
A1: the RelStr of L1 = the RelStr of L2 and
A2: L1 is arithmetic;
A3: L2 is algebraic by A1,A2,Th17;
A4: CompactSublatt L1 is meet-inheriting by A2;
  now
    let x2,y2 be Element of L2;
    reconsider x1 = x2, y1 = y2 as Element of L1 by A1;
    assume that
A5: x2 in the carrier of CompactSublatt L2 and
A6: y2 in the carrier of CompactSublatt L2 and
A7: ex_inf_of {x2,y2},L2;
    x2 is compact by A5,Def1;
    then x1 is compact by A1,A3,Th9;
    then
A8: x1 in the carrier of CompactSublatt L1 by Def1;
    y2 is compact by A6,Def1;
    then y1 is compact by A1,A3,Th9;
    then
A9: y1 in the carrier of CompactSublatt L1 by Def1;
    ex_inf_of {x1,y1},L1 by A1,A7,YELLOW_0:14;
    then inf {x1,y1} in the carrier of CompactSublatt L1 by A4,A8,A9,
YELLOW_0:def 16;
    then
A10: inf {x1,y1} is compact by Def1;
    inf {x1,y1} = inf {x2,y2} by A1,A7,YELLOW_0:27;
    then inf {x2,y2} is compact by A1,A2,A10,Th9;
    hence inf {x2,y2} in the carrier of CompactSublatt L2 by Def1;
  end;
  then CompactSublatt L2 is meet-inheriting by YELLOW_0:def 16;
  hence thesis by A3;
end;
