
theorem Th33:
  for L being up-complete Semilattice for D being non empty
directed Subset of [:L,L:] holds sup ((inf_op L).:D) = sup (proj1 D "/\" proj2
  D)
proof
  let L be up-complete Semilattice, D be non empty directed 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;
  reconsider D1 = proj1 D, D2 = proj2 D as non empty directed Subset of L by
YELLOW_3:21,22;
  set f = inf_op L;
A1: ex_sup_of D1 "/\" D2,L by WAYBEL_0:75;
A2: f.:[:D1,D2:] = D1 "/\" D2 by Th32;
A3: f.:[:D1,D2:] c= f.:(downarrow D) & f.:(downarrow D) c= downarrow (f.:D)
  by Th13,RELAT_1:123,YELLOW_3:48;
A4: f.:D is directed by YELLOW_2:15;
  then
A5: ex_sup_of f.:D,L by WAYBEL_0:75;
  ex_sup_of downarrow (f.:D),L by A4,WAYBEL_0:75;
  then sup (D1 "/\" D2) <= sup (downarrow (f.: D)) by A1,A3,A2,XBOOLE_1:1
,YELLOW_0:34;
  then
A6: sup (D1 "/\" D2) <= sup(f.:D) by A5,WAYBEL_0:33;
  f.:D9 c= f.:[:D1,D2:] by RELAT_1:123,YELLOW_3:1;
  then f.:D9 c= D1 "/\" D2 by Th32;
  then sup (f.:D) <= sup (D1 "/\" D2) by A5,A1,YELLOW_0:34;
  hence thesis by A6,ORDERS_2:2;
end;
