
theorem Th21:
  for L being antisymmetric RelStr holds L is with_infima iff for
  a,b being Element of L holds ex_inf_of {a,b},L
proof
  let L be antisymmetric RelStr;
  hereby
    assume
A1: L is with_infima;
    let a,b be Element of L;
    ex z being Element of L st a >= z & b >= z & for z9 being Element of L
    st a >= z9 & b >= z9 holds z >= z9 by A1;
    hence ex_inf_of {a,b},L by Th19;
  end;
  assume
A2: for a,b being Element of L holds ex_inf_of {a,b},L;
  let x,y be Element of L;
  take x"/\"y;
  ex_inf_of {x,y},L by A2;
  hence thesis by Th19;
end;
