
theorem Th36:
  for L be Semilattice for x be Element of L holds downarrow 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 downarrow x and
    z in the carrier of subrelstr downarrow x and
    ex_inf_of {y,z},L;
    y in downarrow x by A1,YELLOW_0:def 15;
    then
A2: y <= x1 by WAYBEL_0:17;
    y"/\"z <= y by YELLOW_0:23;
    then y"/\"z <= x1 by A2,YELLOW_0:def 2;
    then y"/\"z in downarrow x by WAYBEL_0:17;
    then inf {y,z} in downarrow x by YELLOW_0:40;
    hence inf {y,z} in the carrier of subrelstr downarrow x by YELLOW_0:def 15;
  end;
  then subrelstr downarrow x is meet-inheriting;
  hence thesis;
end;
