
theorem Th54:
  for L being complete transitive antisymmetric non empty RelStr
for A being Subset of L, B being non empty Subset of L holds A is_>=_than inf (
  A "/\" B)
proof
  let L be complete transitive antisymmetric non empty RelStr, A be Subset
  of L, B be non empty Subset of L;
  set b = the Element of B;
  let x be Element of L;
  assume x in A;
  then
A1: x "/\" b in A "/\" B;
  ex xx being Element of L st x >= xx & b >= xx & for c being Element of L
  st x >= c & b >= c holds xx >= c by LATTICE3:def 11;
  then
A2: x >= x "/\" b by LATTICE3:def 14;
  ex_inf_of A "/\" B,L by YELLOW_0:17;
  then A "/\" B is_>=_than inf (A "/\" B) by YELLOW_0:def 10;
  then x "/\" b >= inf (A "/\" B) by A1;
  hence thesis by A2,YELLOW_0:def 2;
end;
