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Fifith Postulate in Affine Spaces

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Fifith Postulate in Affine Spaces

If possible, could anyone criticize the following sequence of sentences ?
1) Affine subspaces are parallel iff they have the same associated linear spaces.
2) Two different (non-equal) parallel affine subspaces are disjoint sets.
3) If two n-dimensional affine subspaces ( 'A' and 'B' ) are parallel to a third n-dimensional subspace then
they are parallel to each other. (A || B )
4) Euclid's Fifth Postulate: Based on item 3), if two n-dimensional subspaces ( 'A' and 'B' ) parallel to a third n-dimensional subspace have
a point in common, then they are equal (A = B).
5) Based on item 4), every affine space obeys the parallel postulate.

My question is : What item is wrong ?
thanx in advance


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