
theorem
  for L being complete non empty Poset for D being non empty filtered
  Subset of [:L,L:] holds inf ((sup_op L).:D) = inf (proj1 D "\/" proj2 D)
proof
  let L be complete non empty Poset, D be non empty filtered Subset of [:L,L
  :];
  reconsider C = the carrier of L as non empty set;
  reconsider D9 = D as non empty Subset of [:C,C:] by YELLOW_3:def 2;
  set D1 = proj1 D, D2 = proj2 D, f = sup_op L;
A1: ex_inf_of D1 "\/" D2,L by YELLOW_0:17;
A2: ex_inf_of uparrow (f.:D),L & f.:[:D1,D2:] c= f.:(uparrow D) by RELAT_1:123
,YELLOW_0:17,YELLOW_3:49;
  f.:(uparrow D) c= uparrow (f.:D) & f.:[:D1,D2:] = D1 "\/" D2 by Th14,Th35;
  then inf (D1 "\/" D2) >= inf (uparrow (f.:D)) by A1,A2,XBOOLE_1:1,YELLOW_0:35
;
  then
A3: inf (D1 "\/" D2) >= inf(f.:D) by WAYBEL_0:38,YELLOW_0:17;
  f.:D9 c= f.:[:D1,D2:] by RELAT_1:123,YELLOW_3:1;
  then ex_inf_of f.:D9,L & f.:D9 c= D1 "\/" D2 by Th35,YELLOW_0:17;
  then inf (f.:D) >= inf (D1 "\/" D2) by A1,YELLOW_0:35;
  hence thesis by A3,ORDERS_2:2;
end;
