
theorem Th38:
  for L be Semilattice for x be Element of L holds uparrow x is meet-closed
proof
  let L be Semilattice;
  let x be Element of L;
  reconsider x1 = x as Element of L;
  now
    let y,z be Element of L;
    assume that
A1: y in the carrier of subrelstr uparrow x and
A2: z in the carrier of subrelstr uparrow x and
    ex_inf_of {y,z},L;
    z in uparrow x by A2,YELLOW_0:def 15;
    then
A3: z >= x1 by WAYBEL_0:18;
    y in uparrow x by A1,YELLOW_0:def 15;
    then y >= x1 by WAYBEL_0:18;
    then y"/\"z >= x1"/\"x1 by A3,YELLOW_3:2;
    then y"/\"z >= x1 by YELLOW_5:2;
    then y"/\"z in uparrow x by WAYBEL_0:18;
    then inf {y,z} in uparrow x by YELLOW_0:40;
    hence inf {y,z} in the carrier of subrelstr uparrow x by YELLOW_0:def 15;
  end;
  then subrelstr uparrow x is meet-inheriting;
  hence thesis;
end;
