
theorem
  for L being with_infima transitive antisymmetric RelStr for a, b, c, d
  being Element of L st a <= c & b <= d holds a "/\" b <= c "/\" d
proof
  let L be with_infima transitive antisymmetric RelStr, a, b, c, d be Element
  of L such that
A1: a <= c and
A2: b <= d;
A3: ex_inf_of {a,b},L by YELLOW_0:21;
  then a "/\" b <= b by YELLOW_0:19;
  then
A4: a "/\" b <= d by A2,ORDERS_2:3;
  a "/\" b <= a by A3,YELLOW_0:19;
  then
  ( ex x being Element of L st c >= x & d >= x & for z being Element of L
  st c >= z & d >= z holds x >= z)& a "/\" b <= c by A1,LATTICE3:def 11
,ORDERS_2:3;
  hence thesis by A4,LATTICE3:def 14;
end;
