
theorem Th28:
  for L being complete LATTICE holds Image ((ClImageMap L)|
  DsupClOpers L) = (Subalgebras L) opp
proof
  let L be complete LATTICE;
  defpred P[object] means
   $1 is directed-sups-inheriting closure strict System of L;
A1: now
    let a be object;
    hereby
      assume a is Element of Image ((ClImageMap L)|DsupClOpers L);
      then consider x being Element of DsupClOpers L such that
A2:   ((ClImageMap L)|DsupClOpers L).x = a by YELLOW_2:10;
      reconsider x as directed-sups-preserving closure Function of L,L by Th26;
      a = (ClImageMap L).x by A2,Th6
        .= Image x by Def4;
      hence P[a];
    end;
    assume P[a];
    then reconsider
    S = a as directed-sups-inheriting closure strict System of L;
    reconsider x = closure_op S as Element of DsupClOpers L by Th26;
    S = Image closure_op S by Th18
      .= (ClImageMap L).closure_op S by Def4
      .= ((ClImageMap L)|DsupClOpers L).x by Th6;
    then S in rng ((ClImageMap L)|DsupClOpers L) by FUNCT_2:4;
    hence a is Element of Image ((ClImageMap L)|DsupClOpers L) by
YELLOW_0:def 15;
  end;
A3: for a be object
   holds a is Element of (Subalgebras L) opp iff P[a] by Th27;
  the RelStr of Image ((ClImageMap L)|DsupClOpers L) = the RelStr of (
  Subalgebras L) opp from SubrelstrEq1(A1,A3);
  hence thesis;
end;
