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
  p is total implies dom PP_inversion(p) = {}
  proof
    set q = PP_inversion(p);
    assume
A1: dom p = D;
A2: dom q = {d where d is Element of D: not d in dom p} by Def19;
    thus dom q c= {}
    proof
      let x;
      assume x in dom q;
      then ex d being Element of D st x = d & not d in dom p by A2;
      hence thesis by A1;
    end;
    thus thesis by XBOOLE_1:2;
  end;
