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ponding points*), and let A be the last limiting point of ‚a, on 
the right hand side. Then the assumption that A precedes its cor- 
responding point, as well as the assumption according to which R 
follows on its corresponding point, leads immediately to a contra- 
diction (we here consider the transformation of the entire segment 
AB in itself, which is determined by the correspondence between 
a and 2’). Hence the point R must correspond to itself, but this is 
out of the question, both if A belongs to 2 or to a’. 
Second case. All points of a lie on the left of the corresponding 
points. Let the points P. and P. of 2’ correspond to P, and P, of 
x respectively and let the order from left to right be P,, P., 12, P.. 
Of course, such a system of points can always be found. 
We choose a point C of a’ on the left of P, and we subject a! 
to a uniformly continuous one-one transformation in itself, such that 
P passes into C and P. remains in its place. A transformation of 
this kind can certainly be found as a! is homogeneous. Let 2" be 
the transformed set, then a uniformly continuous one-one correspond- 
ence exists between 2” and a, such that 2" contains both points 
preceding and points coming after the corresponding points, and the 
reasoning used for the jist case can now be applied. 
To Prof. L. E J. Brouwer I am indebted for several remarks 
turned to advantage in the preceding note. 
bj “Precede” here stands for “are situated on the left of”. 
