reserve n,k for Element of NAT;

theorem Th12:
  for L be distributive finite LATTICE for x be Element of L holds
  sup (downarrow x /\ Join-IRR L) = x
proof
  let L be distributive finite LATTICE;
  let x be Element of L;
A1: x <= sup( downarrow x /\ Join-IRR L)
  proof
    defpred X[Element of L] means sup(downarrow $1 /\ Join-IRR L) = $1;
A2: for x being Element of L st for b be Element of L st b < x holds X[b]
    holds X[x] by Lm1;
    for x being Element of L holds X[x] from LattInd(A2);
    hence thesis;
  end;
  ex_sup_of downarrow x /\ Join-IRR L,L & ex_sup_of downarrow x,L by
YELLOW_0:17;
  then sup(downarrow x /\ Join-IRR L) <= sup(downarrow x) by XBOOLE_1:17
,YELLOW_0:34;
  then sup(downarrow x /\ Join-IRR L) <= x by WAYBEL_0:34;
  hence thesis by A1,ORDERS_2:2;
end;
