reserve x for object;
reserve n for Nat;
reserve D for non empty set;
reserve p,q for PartialPredicate of D;
reserve D for set;
reserve p,q for PartialPredicate of D;
reserve f,g for BinominativeFunction of D;
reserve D for non empty set;
reserve p,q for PartialPredicate of D;
reserve f,g,h for BinominativeFunction of D;

theorem
  for d being Element of D holds
  d in dom p iff not d in dom(PP_inversion(p))
  proof
    let d be Element of D;
A1: dom(PP_inversion(p)) = {d where d is Element of D: not d in dom p}
    by Def19;
    thus d in dom p implies not d in dom(PP_inversion(p))
    proof
      assume
A2:   d in dom p;
      assume d in dom(PP_inversion(p));
      then ex d1 being Element of D st d = d1 & not d1 in dom p by A1;
      hence thesis by A2;
    end;
    thus thesis by A1;
  end;
