reserve X for set;

theorem
  for X be non empty set holds InclPoset X is lower-bounded implies meet X in X
proof
  let X be non empty set;
  assume InclPoset X is lower-bounded;
  then consider x be Element of InclPoset X such that
A1: x is_<=_than the carrier of InclPoset X by YELLOW_0:def 4;
  now
    let y be object;
    assume
A2: y in x;
    now
      let Y be set;
      assume Y in X;
      then reconsider Y9 = Y as Element of InclPoset X;
      x <= Y9 by A1;
      then x c= Y9 by Th3;
      hence y in Y by A2;
    end;
    hence y in meet X by SETFAM_1:def 1;
  end;
  then
A3: x c= meet X;
  x in X & meet X c= x by SETFAM_1:3;
  hence thesis by A3,XBOOLE_0:def 10;
end;
