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property, hence all correspondences, postulated for the homogeneous 
set or, can be effected in such a way as to leave relations of order 
unaltered. 
We now formulate the following theorem: The linear continuum 
cannot be composed of two homogeneous sets of points, possessing 
the same geometric type. 
Our demonstration is to be an indirect one. Let the open line- 
segment AB consist of two sets of points a and z’ of the kind 
mentioned. These sets a and 2’ possess the same geometric type, 
that is there exists uniformly continuous one-one correspondence 
between them. Evidently « and a’ are both everywhere dense on AB. 
To begin with, we assume that this correspondence inverts rela- 
tions of order. Then a can be divided into two subsets 2, and zr, 
such that every point of a, is situated on the left, and every 
point of zr, on the right of the corresponding point of a’. Besides, 
every point of 2, lies on the left of every point of a,. Hence, as 
a, + 1, is everywhere dense, the DuprKIND axiom postulates the 
existence of a separating point A. This point & however can belong 
to neither a, nor z,. For instance let us assume A to be a point 
of a,, then it is situated on the left of the corresponding point of 
z' and the continuity of the correspondence makes that this is also 
the case for all points of a inside a certain finite neighbourhood 
of R, which means a contradiction. Hence A belongs to 7’, but 
this also leads to a contradiction as the fact that A is situated 
either on the left or on the right of its corresponding point cannot 
be made to agree with the circumstance that all points of 2’ on 
the left (right) of R are also situated on the left (right) of their 
corresponding points. 
We now come to the second possibility, namely that the corres- 
pondence between a and a’ leaves relations of order unaltered. 
We distinguish two cases: 
1. The set a contains both points situated on the left, and points 
situated on the right of the corresponding points or a’. 
2. All points of am lie on the same side of the corresponding 
points. 
First case. Let the point P, of z be situated on the left of its corres- 
ponding point P and P, on the right of Re The subset of a 
between P, and P,, including the endpoints shall be called 2,. Let 
‚mT, be the subset of mr, consisting of those points, which, together 
with all points of a, situated more to the left, precede their corres- 
