 reserve a,Z1,Z2,Z3 for set,
         x,y,z for object,
         k for Nat;

theorem ThMin02:
  for P being lower-bounded non empty Poset,
      p being Element of P
        st p is_<=_than the carrier of P
            holds p = Bottom P
  proof
    let P be lower-bounded non empty Poset;
    let p be Element of P;
    assume
A1: p is_<=_than the carrier of P;
A2: ex_sup_of {},P by YELLOW_0:42;
A3: {} is_<=_than p;
    for a being Element of P st {} is_<=_than a holds p <= a by A1;
    hence thesis by A2,A3,YELLOW_0:def 9;
  end;
