What's wrong with this proof for the set C={a1}?

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The proof for the set C={a1} is invalid because it incorrectly assumes that C and B can be equal while excluding an element from set A, specifically a2. The argument fails when it states that set A consists of k+1 numbers, which can be identical or distinct. This oversight leads to a contradiction, as it implies that not all elements of A are accounted for in B or C. Therefore, the proof collapses at the point of defining set A. The logical inconsistency renders the entire argument unsound.
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Why can't C={a1}?
 
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Which part of
The reason is that C = {a1} = B, so an element of A, namely a2, is not in either B or C.
don't you understand?
 
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The proof clearly falls apart when it is stated let Set ##A## be a set of ##k+1## numbers. These number can either be all the same, or different. Hence the proof is invalid from that point on.
 

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