Constraints :

1. Basis and dimensions are not introduced yet
2. Matrix and system of equations can't be used

I am aware that this might possibly be a duplicate of the [same question here](https://math.stackexchange.com/questions/2234481/if-a-subspace-s-is-spanned-by-n-vectors-any-subset-of-s-with-mn-vectors-is), but I wanted to take this question further and see if any of you have come across an alternative proof than the one mentioned in the first answer. Btw, I am new to this platform, so in case I am making a mistake by posting this, please let me know.