
theorem Th20:
  for L being complete LATTICE holds (ClImageMap L)" is Function
  of (ClosureSystems L) opp, ClOpers L
proof
  let L be complete LATTICE;
  set f = ClImageMap L;
A1: rng (f") = dom f by FUNCT_1:33;
A2: dom f = the carrier of ClOpers L by FUNCT_2:def 1;
  the carrier of (ClosureSystems L) opp c= rng f
  proof
    let x be object;
    assume x in the carrier of (ClosureSystems L) opp;
    then reconsider x as Element of (ClosureSystems L) opp;
    reconsider x as infs-inheriting full strict SubRelStr of L by Th16;
A3: closure_op x is Element of ClOpers L by Th10;
    f.closure_op x = Image closure_op x by Def4
      .= x by Th18;
    hence thesis by A2,A3,FUNCT_1:def 3;
  end;
  then
A4: the carrier of (ClosureSystems L) opp = rng f;
  dom (f") = rng f by FUNCT_1:33;
  hence thesis by A1,A4,FUNCT_2:def 1,RELSET_1:4;
end;
