
theorem Th34:
  for L be complete non empty Poset for X be Subset of L for p
  be Element of L st p is completely-irreducible & p = inf X holds p in X
proof
  let L be complete non empty Poset;
  let X be Subset of L;
  let p be Element of L;
  assume that
A1: p is completely-irreducible and
A2: p = inf X;
  assume
A3: not p in X;
  consider q be Element of L such that
A4: p < q and
A5: for s be Element of L st p < s holds q <= s and
  uparrow p = {p} \/ uparrow q by A1,Th20;
A6: p is_<=_than X by A2,YELLOW_0:33;
  now
    let b be Element of L;
    assume
A7: b in X;
    then p <= b by A6,LATTICE3:def 8;
    then p < b by A3,A7,ORDERS_2:def 6;
    hence q <= b by A5;
  end;
  then
A8: q is_<=_than X by LATTICE3:def 8;
A9: p <= q by A4,ORDERS_2:def 6;
  now
    let b be Element of L;
    assume b is_<=_than X;
    then p >= b by A2,YELLOW_0:33;
    hence q >= b by A9,ORDERS_2:3;
  end;
  hence contradiction by A2,A4,A8,YELLOW_0:33;
end;
