
theorem Th19:
  for L being complete LATTICE for c being closure Function of L,L
  holds closure_op Image c = c
proof
  let L be complete LATTICE, c be closure Function of L,L;
  now
    let x be Element of L;
A1: id L <= c by WAYBEL_1:def 14;
    x = (id L).x;
    then x <= c.x by A1,YELLOW_2:9;
    then
A2: c.x in uparrow x by WAYBEL_0:18;
    dom c = the carrier of L by FUNCT_2:def 1;
    then c.x in rng c by FUNCT_1:def 3;
    then c.x in (uparrow x) /\ rng c by A2,XBOOLE_0:def 4;
    then
A3: c.x >= "/\"((uparrow x) /\ rng c, L) by YELLOW_2:22;
    c.x is_<=_than (uparrow x) /\ rng c
    proof
      let z be Element of L;
      assume
A4:   z in (uparrow x) /\ rng c;
      then z in rng c by XBOOLE_0:def 4;
      then consider a being object such that
A5:   a in dom c and
A6:   z = c.a by FUNCT_1:def 3;
      reconsider a as Element of L by A5;
      z in uparrow x by A4,XBOOLE_0:def 4;
      then x <= c.a by A6,WAYBEL_0:18;
      then c.x <= c.(c.a) by WAYBEL_1:def 2;
      hence thesis by A6,YELLOW_2:18;
    end;
    then
A7: c.x <= "/\"((uparrow x) /\ rng c, L) by YELLOW_0:33;
    rng c = the carrier of Image c by YELLOW_0:def 15;
    hence (closure_op Image c).x = "/\"((uparrow x) /\ rng c, L) by Def5
      .= c.x by A3,A7,ORDERS_2:2;
  end;
  hence thesis by FUNCT_2:63;
end;
