
theorem Th22: :: THEOREM 4.19 (iii)
  for L be Semilattice holds Irr L c= IRR L
proof
  let L be Semilattice;
  let x be object;
  assume
A1: x in Irr L;
  then reconsider x1 = x as Element of L;
  x1 is completely-irreducible by A1,Def4;
  then consider q be Element of L such that
A2: x1 < q and
A3: for s be Element of L st x1 < s holds q <= s and
  uparrow x1 = {x1} \/ uparrow q by Th20;
  now
    let a,b be Element of L;
    assume that
A4: x1 = a "/\" b and
A5: a <> x1 and
A6: b <> x1;
    x1 <= b by A4,YELLOW_0:23;
    then x1 < b by A6,ORDERS_2:def 6;
    then
A7: q <= b by A3;
A8: x1 <= q by A2,ORDERS_2:def 6;
    x1 <= a by A4,YELLOW_0:23;
    then x1 < a by A5,ORDERS_2:def 6;
    then q <= a by A3;
    then q <= x1 by A4,A7,YELLOW_0:23;
    hence contradiction by A2,A8,ORDERS_2:2;
  end;
  then x1 is irreducible by WAYBEL_6:def 2;
  hence thesis by WAYBEL_6:def 4;
end;
