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of sp, and w’ we can effectuate that any base triangle of sp, covers 
in w’ either a single base triangle, or a single base side. 
By choosing one of the base sides of sp, represented by a) in 
a single point, and considering it as a single point and accordingly 
the two base triangles adjacent to it as line segments, sp, passes 
into an other sphere sp,’ represented likewise simplicially by a»). 
In the same way we deduce from sp,’ an other sphere sp," if this 
be possible, and we continue this process until after a finite number 
of steps we obtain a sphere sp,” no more possessing for a” any 
singular image triangle. 
Let us denote by B and D the two base points of sp,0—V 
identified for sp and by a and c the two base triangles of 
sp eontracted into line segments for sp”. Then the triangles 
a and c have either only the side 4) in common, or moreover a 
second side, which we may assume to contain the vertex B. 
In the first case we represent the third vertex of a, resp. c, by 
A, resp. C, and the domain covered by a and c together, by d. 
At least one of the base points B and D, say D, does not coincide 
with a point P,,. We then connect in sp,™—") outside d the points 
A and C by an are of simple curve 8 situated in the vicinity of 
the broken line ADC, and we represent the domain included 
between 8 and the broken line ADC, by d’. By means of a 
continuous series of continuous one-one transformations leaving the 
points of @ invariant and transforming each point of AB and BC 
into points coinciding with it on sp”, we can reduce the domain 
dtd’ with its boundary continuously into the domain d’ with its 
boundary. If we represent by «,”) an arbitrary univalent continuous 
representation of sp” on a’, then to the continuous reduction of 
d+d’ to d’ corresponds a continuous series of univalent continuous 
representations of sp,@"— on sp”) transforming the representation 
obtained by the identification of B and D, into a continuous one-one 
correspondence en in which the points P,- correspond to them- 
selves, thus also a continuous series of univalent continuous repre- 
sentations of sp.” on w’, leaving invariant the images of the 
points P,., and transforming «,@) considered as a representation of 
spr: on p’, into that representation mot of sp," ona. 
which follows from @,° by means of mem —- 
In the second case we represent the third vertex of aande by #, 
choose on the side DF of a, the side DF of c, and the common 
side 5/” successively three such points A, C, and G, as in passing 
from sp.0"—) to sp.” are brought to coincidence, connect A within a 
rectilinearly with B and G, C within c rectilinearly with 5 and G, 
