The following is assumed, in addition to what is assumed in the preceding section:


Lp) [x~y]=[x~y], where ~ defines applicable relation sign.


Lp (the Equal distribution principle) defines that a relation (between x and y) not change if the arguments/variables change equally (defi-ned by ). Lp is just only assumed, thus without analysing Lp's validity per se, to see the implications of this assumption, thus of Lp.


Assume (for example, even if this particular assumption is highly deliberately made):


E≠∞*; E=[the World], ∞*=min[∞]; ∞=infinity:


E+E+∞*≠∞*+E+∞*; Lp.


To be able to continue it is just only assumed that, thus without deeper analysis (if symmetry (around +: x+y=y+x (symmetrically valid), commutativity) had been valid, that could have been used here, but symmetry is thus not generally valid, so the following is the easiest to assume):


Terms can when Up is used be unified to places (in which the unifiable is located) in sentences at will:


E+∞*≠E+∞*; Up:


E=∞*; Kp.


Further assume the intuitive that:


d(x,x)≠d(x,x]; d(x,x),d(x,x]E; (x=[excluding x], [x=[including x]; d(x,x)=[distance between x and x]:


d(x,x)+x≠d(x,x]+x; Lp:


d(x,x]≠d(x,x]; Up:


t1) x)=x]; Kp.


Excluding x is thus identically including x (and vice versa), which at least for volumes is contradictory (or weaker an absurd p-super-positionality, but finds it rationally to call it a contradiction, because it is so patently absurd), for p (points, non-extended positions) maybe not. Anyway, it defines continuity rule in E, that there not exists a distance between p (p]) and a next (to p) closest p (p)).


Assume further that only finite distances exists in E:




d(x,∞*]E; t1.


E is thus boundless, which means that Nothing not exists beyond E, nor within E, thus given t1 (the continuity), which altogether simpli-citer defines that Nothing not exists, which also can be shown more directly ‒ for Nothing it is valid by definition that:


xNothing; x=[at least one property]:


x+xNothing+x; Lp:


xx; Up; Nothing+x=x or Nothingx:


T1 is valid; Kp (the definition of Nothing leads to a contradiction (xx; Ip)*, which in accordance with Kp means that Nothing cannot exist).


The assumption of Lp thus leads to the conclusion that Nothing not exists, which in no way can be realized before that this result is present, but it can only be trusted in Lp (if Lp is trusted, which Lp not can be, which the further analysis will show), if there is no suppor-ting evidence, such as then T1 (T1 which then is supporting evidence in the previous proof, and also to t1, see further T2 below, even if t1 (obviously) also can be seen as a proof of where Lp leads wrong; In conventional formalism there is an, must be said, idiotic axiom, which defines that true derives true, no, naturally it is the (rational) reason (mind) that decides that, thus what is true (or false)).


The preceding just only implicates that E=∞*, that E not is >∞*, it just only follows from Lp, so to more explicitly show that it is valid (in context of Lp), the following is assumed:


d(x,x)=d(x,x)+d(x,x)=∞*; d(x,x),d(x,x)<∞*.


Given this, there exists an x before which d(x,x) is finite, after which d(x,x) is infinite:


d(x,x)<∞*; d(x,x)<∞*.


d(x,x)=∞*; d(x,x]<∞*.


Which given t1 defines d(x,x)=d(x,x] to be both finite and infinite in x, an absurd/contradictory p-superpositionality, so at least one sub-distance must be infinite, say d(x,x), which defines:




It may seem that d(x,x)+∞*>∞* (or at least ≥), but given the continuous perspective in accordance with t1, two sub-distances must be able to define (exactly) ∞*, this which defines that d(x,x)=0, or more commonly that:


T2) ∞*0=∞*; 0=d(x,x)<∞*.


Thus, finite distances are 0 in relation to infinite distances (≥∞*).


Are there distances longer than ∞*? Not finitely added in accordance with T2, but in that case infinitely added:


∞*+d; d≥∞*.


Which given the continuous perspective in accordance with t1 (and T1 also, see further the following) defines the existence of distances between ∞* and ∞*+d which not exists, which is absurd:


T2) E=∞*:


x<∞*; x≠E; xE.


T2 which also can be stated without the help of Lp (important for the E-section, that it not needs to rely on Lp, especially given what is coming in this section), but given T1, because given T1 there exists no limits in E after which Nothing takes on:


E is homogeneously continuous, infinitely ongoing in all directions.


Especially a smallest E=∞* continues with this infinitely in all directions, E>E need ex ante not necessarily continue infinitely in all directions, but in the directions that E continues infinitely, so does E, with which it is established that E=E (since E<E cannot be valid; It defines E to be finite (contrary to that E>E), given that E is a smallest infinity):


T2 is valid.


Also Up can be proved given Lp, assume Up not valid:




x+x+{x}≠{x}+x+{x}; Lp:


x+{x}≠x+{x}; Up:


Up valid; Kp.


More specific proof of that all xE are finite ‒ assume not:




x+x>E+x; Lp:


x>x+x; Up, where x is part of E which maybe not belong to x:


x≤E; Kp, which given that x≠E:




Assume more commonly:




x+E≠E+E; Lp:


E≠E; Up(; xE):


x=E; Kp.


Which may seem contradictory, but actually is intuitive, given T1, in accordance with which all (existing) x(≠E) at least must exist as (eternal) possibilities (not even a possibility can (rationally) arise out of(/in) the non-existing Nothing, or for that matter disappear in, pass over in to being the non-existing Nothing), with which the question especially is how x, which not are only possibilities in E, can be crea-ted, which will be returned to in the next section.


Assume further:




x+E≠x+E; Lp:


E≠E; Up(; x,xE):


x=x; Kp.


Which especially is valid for rational superpositionalities, but not in general, so here Lp categorically leads wrong (and no small error either); The same conclusion is obtained if x+x in accordance with Lp, instead of E, is added to either side of ≠.


Another case where Lp fails:


Assume that y and z can be different even though they have a cluster ({x}) of joint properties:




{x}+{x}+{x}+{x}≠{x}+{x}+{x}+{x}; Lp:


{x}+{x}+{x}≠{x}+{x}+{x}; Up.


Which given Kp then defines that y and z not are different, but well can x according to experience be different despite having a cluster of joint properties, thinks especially of Siamese twins.


Especially the two latter examples destroy the confidence in Lp, perhaps Lp will do in certain context, in which it consequently, before Lp is used (ex ante), rigorously must be investigated if Lp can be used, is rational and gives rational results (in its context), generally Lp definitely not can be assumed valid (ex ante), and the same must (ex ante) be stated to be valid for any principle other than those in the previous section, thus the principles of the Rational basis:


Principles beyond the Rational basis cannot be taken for granted (but perhaps be assumed after ex ante analysis in certain context).