
theorem Th18:
  for L being complete LATTICE for S being closure System of L
  holds Image closure_op S = the RelStr of S
proof
  let L be complete LATTICE;
  let S be infs-inheriting full non empty SubRelStr of L;
  the carrier of Image closure_op S = the carrier of S
  proof
    hereby
      let x be object;
      assume x in the carrier of Image closure_op S;
      then reconsider a = x as Element of Image closure_op S;
      consider b being Element of L such that
A1:   a = (closure_op S).b by YELLOW_2:10;
      set X = (uparrow b) /\ the carrier of S;
      reconsider X as Subset of S by XBOOLE_1:17;
A2:   ex_inf_of X,L by YELLOW_0:17;
      a = "/\"(X,L) by A1,Def5;
      hence x in the carrier of S by A2,YELLOW_0:def 18;
    end;
    set c = closure_op S;
    let x be object;
    assume x in the carrier of S;
    then reconsider a = x as Element of S;
    reconsider a as Element of L by YELLOW_0:58;
    set X = (uparrow a) /\ the carrier of S;
A3: (id L).a = a;
    a <= a;
    then a in uparrow a by WAYBEL_0:18;
    then
A4: a in X by XBOOLE_0:def 4;
    c.a = "/\"(X,L) by Def5;
    then
A5: c.a <= a by A4,YELLOW_2:22;
    id L <= c by WAYBEL_1:def 14;
    then a <= c.a by A3,YELLOW_2:9;
    then
A6: a = c.a by A5,ORDERS_2:2;
    dom c = the carrier of L by FUNCT_2:def 1;
    then a in rng closure_op S by A6,FUNCT_1:def 3;
    hence thesis by YELLOW_0:def 15;
  end;
  hence thesis by YELLOW_0:57;
end;
