
theorem Th40:
  for L be Semilattice for x be Element of L holds waybelow x is meet-closed
proof
  let L be Semilattice;
  let x be Element of L;
  now
    let y,z be Element of L;
    assume that
A1: y in the carrier of subrelstr waybelow x and
    z in the carrier of subrelstr waybelow x and
    ex_inf_of {y,z},L;
    y in waybelow x by A1,YELLOW_0:def 15;
    then
A2: y << x by WAYBEL_3:7;
    y"/\"z <= y by YELLOW_0:23;
    then y"/\"z << x by A2,WAYBEL_3:2;
    then y"/\"z in waybelow x by WAYBEL_3:7;
    then inf {y,z} in waybelow x by YELLOW_0:40;
    hence inf {y,z} in the carrier of subrelstr waybelow x by YELLOW_0:def 15;
  end;
  then subrelstr waybelow x is meet-inheriting;
  hence thesis;
end;
