theorem
  L WHERE NOT NOT O = L WHERE O
  proof
    thus L WHERE NOT NOT O c= L WHERE O
    proof
      let z be object; assume
A1:   z in L WHERE NOT NOT O; then
      reconsider x = z as Element of X;
A2:   x in L & x.NOT NOT O <> {} by A1,Th3; then
      x in dom NOT NOT O by RELAT_1:170; then
      x in dom O by Th38; then
      x.O <> {} by RELAT_1:170;
      hence thesis by A2,Th3;
    end;
    let z be object;
    assume
A3: z in L WHERE O; then
    reconsider x = z as Element of X;
A4: z in L & x.O <> {} by A3,Th3; then
    x in dom O by RELAT_1:170; then
    x in dom NOT NOT O by Th38; then
    x.NOT NOT O <> {} by RELAT_1:170;
    hence thesis by A4,Th3;
  end;
