
theorem Th40:
  for L being with_infima Poset, a,b being Element of L holds inf
  {a,b} = a"/\"b
proof
  let L be with_infima Poset, a,b be Element of L;
A1: now
    let c be Element of L;
    assume c is_<=_than {a,b};
    then c <= a & c <= b by Th8;
    hence c <= a"/\"b by Th23;
  end;
  a"/\"b <= a & a"/\"b <= b by Th23;
  then
A2: a"/\"b is_<=_than {a,b} by Th8;
  then ex_inf_of {a,b}, L by A1,Th16;
  hence thesis by A2,A1,Def10;
end;
