
theorem Th15:
  for P being complete non empty Poset, V being non empty Subset
of P holds downarrow inf V = meet {downarrow u where u is Element of P : u in V
  }
proof
  let P be complete non empty Poset, V be non empty Subset of P;
  set F = {downarrow u where u is Element of P : u in V};
  consider u being object such that
A1: u in V by XBOOLE_0:def 1;
A2: F c= bool the carrier of P
  proof
    let X be object;
    assume X in F;
    then ex u being Element of P st X = downarrow u & u in V;
    hence thesis;
  end;
  reconsider u as Element of P by A1;
A3: downarrow u in F by A1;
  reconsider F as Subset-Family of P by A2;
  reconsider F as Subset-Family of P;
  now
    let x be object;
    hereby
      assume
A4:   x in downarrow inf V;
      then reconsider d = x as Element of P;
A5:   d <= inf V by A4,WAYBEL_0:17;
      now
        let Y be set;
        assume Y in F;
        then consider u being Element of P such that
A6:     Y = downarrow u and
A7:     u in V;
        inf V is_<=_than V by YELLOW_0:33;
        then inf V <= u by A7,LATTICE3:def 8;
        then d <= u by A5,ORDERS_2:3;
        hence x in Y by A6,WAYBEL_0:17;
      end;
      hence x in meet F by A3,SETFAM_1:def 1;
    end;
    assume
A8: x in meet F;
    then reconsider d = x as Element of P;
    now
      let b be Element of P;
      assume b in V;
      then downarrow b in F;
      then d in downarrow b by A8,SETFAM_1:def 1;
      hence d <= b by WAYBEL_0:17;
    end;
    then d is_<=_than V by LATTICE3:def 8;
    then d <= inf V by YELLOW_0:33;
    hence x in downarrow inf V by WAYBEL_0:17;
  end;
  hence thesis by TARSKI:2;
end;
