
theorem Th15:
  for L being complete Lattice holds ConceptLattice(Context(L)),L
  are_isomorphic
proof
  let L be complete Lattice;
  reconsider g = id the carrier of L as Function of the carrier of Context(L),
  the carrier of L;
  reconsider d = id the carrier of L as Function of the carrier' of Context(L)
  , the carrier of L;
A1: for o being Object of Context(L), a being Attribute of Context(L) holds
  o is-connected-with a iff g.o [= d.a
  proof
    let o be Object of Context(L), a be Attribute of Context(L);
    reconsider o9 = o, a9 = a as Element of L;
    o is-connected-with a iff [o,a] in the Information of Context(L) by
CONLAT_1:def 2;
    hence thesis by FILTER_1:31;
  end;
  for a being Element of L holds ex D9 being Subset of rng(d) st a = "/\"(
  D9,L)
  proof
    let a be Element of L;
    "/\"({a},L) = a & rng(d) = the carrier of L by LATTICE3:42,RELAT_1:45;
    hence thesis;
  end;
  then
A2: rng(d) is infimum-dense;
  rng(g) is supremum-dense
  proof
    let a be Element of L;
    "\/"({a},L) = a & rng(g) = the carrier of L by LATTICE3:42,RELAT_1:45;
    hence thesis;
  end;
  hence thesis by A2,A1,Th14;
end;
