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

theorem
  q is total implies dom p c= dom PP_imp(p,q)
  proof
    assume
A1: q is total;
    set a = PP_imp(p,q);
    let x;
    assume
A2: x in dom p;
A3: dom a = {d where d is Element of D:
    d in dom p & p.d = FALSE or d in dom q & q.d = TRUE
    or d in dom p & p.d = TRUE & d in dom q & q.d = FALSE} by PARTPR_1:31;
    per cases by A2,PARTPR_1:3;
    suppose
A4:   p.x = TRUE;
      q.x = TRUE or q.x = FALSE by A1,A2,PARTPR_1:3;
      hence thesis by A1,A2,A3,A4;
    end;
    suppose p.x = FALSE;
      hence thesis by A2,A3;
    end;
  end;
