
theorem
  for L being complete non empty Poset for A, B being non empty Subset
  of L holds inf (A "/\" B) = inf A "/\" inf B
proof
  let L be complete non empty Poset, A, B be non empty Subset of L;
  B is_>=_than inf (A "/\" B) by Th54;
  then
A1: inf B >= inf (A "/\" B) by YELLOW_0:33;
  A is_>=_than inf (A "/\" B) by Th54;
  then inf A >= inf (A "/\" B) by YELLOW_0:33;
  then
A2: inf A "/\" inf B >= inf (A "/\" B) "/\" inf (A "/\" B) by A1,YELLOW_3:2;
  A is_>=_than inf A & B is_>=_than inf B by YELLOW_0:33;
  then ex_inf_of A "/\" B,L & A "/\" B is_>=_than inf A "/\" inf B by Th55,
YELLOW_0:17;
  then
A3: inf (A "/\" B) >= inf A "/\" inf B by YELLOW_0:def 10;
  inf (A "/\" B) "/\" inf (A "/\" B) = inf (A "/\" B) by YELLOW_0:25;
  hence thesis by A3,A2,ORDERS_2:2;
end;
