reserve x for object;
reserve D for set;
reserve p for PartialPredicate of D;

theorem
  x in dom PP_not(p) & PP_not(p).x = TRUE implies x in dom p & p.x = FALSE
  proof
    assume that
A1: x in dom PP_not(p) and
A2: PP_not(p).x = TRUE;
    thus
A3: x in dom p by A1,Def2;
    assume p.x <> FALSE;
    then p.x = TRUE by A3,Th3;
    hence thesis by A2,A3,Def2;
  end;
