reserve X for set;

theorem
  for X be non empty set st for x,y be set st x in X & y in X holds x /\
  y in X holds InclPoset X is with_infima
proof
  let X be non empty set;
  set L = InclPoset X;
  assume
A1: for x,y be set st x in X & y in X holds x /\ y in X;
  now
    let a,b be Element of L;
    ex c be Element of L st {a,b} is_>=_than c & for d be Element of L st
    {a,b} is_>=_than d holds c >= d
    proof
      take c = a "/\" b;
A2:   a /\ b = c by A1,Th9;
      then c c= b by XBOOLE_1:17;
      then
A3:   c <= b by Th3;
      c c= a by A2,XBOOLE_1:17;
      then c <= a by Th3;
      hence {a,b} is_>=_than c by A3,YELLOW_0:8;
      let d be Element of L;
      assume
A4:   {a,b} is_>=_than d;
      b in {a,b} by TARSKI:def 2;
      then d <= b by A4;
      then
A5:   d c= b by Th3;
      a in {a,b} by TARSKI:def 2;
      then d <= a by A4;
      then d c= a by Th3;
      then d c= a /\ b by A5,XBOOLE_1:19;
      then d c= c by A1,Th9;
      hence thesis by Th3;
    end;
    hence ex_inf_of {a,b},L by YELLOW_0:16;
  end;
  hence thesis by YELLOW_0:21;
end;
