Tensor symmetry is an invariant property. The proof shows that if a tensor is symmetric in one coordinate system ( $T^{a b}=T^{b a}$ ), it will always be symmetric in any other transformed coordinate system ( $T^{\prime a^{\prime} b^{\prime}}=T^{\prime a^{\prime} b^{\prime}}$ ). This is demonstrated by applying the tensor transformation rule and using the initial symmetry to rearrange terms. The fact that the property holds true across all coordinate systems makes symmetry a fundamental characteristic of the tensor itself, rather than a coincidental feature of a specific coordinate representation.
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