My problem is how can I state my proof more clearly (if it is correct). I wanted to prove Steinitz's exchange lemma, which, according to my book states that:
Steinitz exchange lemma: Let $\vec{c_1},...,\vec{c_p}$ be linearly independent vectors which can be written as a linear combination of $\vec{v_1},...,\vec{v_l}$. Then you can substitute $p$ of the vectors from $V=(\vec{v_1},...,\vec{v_l})$ with $\vec{c_1},...,\vec{c_p}$ and obtain a list of vectors equivalent to V.
Here a list of vectors is a $n$-tuple of vectors, and two lists of vectors $A,B$ are said to be equivalent iff all vectors $\vec{a}\in A$ can be written as a linear combination of vectors from $B$ and all vectors $\vec{b}\in B$ can be written as a linear combination of vectors from $A$. We represent that by $A\approx B$.
My proof: I proved for induction. We want to prove for all $n\in N$: $C(n) \iff $ Steinitz exchange lemma for $p=n$.
Base case: $C(1)\iff ((\vec{c}\neq\vec{0}$ and $\vec{c}$ is a linear combination of $V$$) \implies (\vec{v_1},...,\vec{v_i}...,\vec{v_l})\approx(\vec{v_1},...,\vec{c}...,\vec{v_l}))$. This can be easily proved.
Induction hypothesis: $C(k)\iff ((\vec{c_1},...,\vec{c_k} $ are linearly independent and linear combination of $V)\implies $ for some $i_1,...,i_k: (\vec{v_1},...,\vec{v_{i_1}},...,\vec{v_{i_k}},...,\vec{v_l})\approx(\vec{v_1},...,\vec{c_1},...,\vec{c_k},...,\vec{v_l}))$.
Assuming $C(k)$, we choose $\vec{c_{k+1}}$ that is linear combination of $V$ and such that $\vec{c_1},...,\vec{c_k},\vec{c_{k+1}}$. Since $V\approx(\vec{v_1},...,\vec{c_1},...,\vec{c_k},...,\vec{v_l})$, we can easily prove that $\vec{c_{k+1}}$ is linear combination of the second list of vectors. But in it's linear combination it must have at least a coeficient $\alpha_j\neq 0$ for which $j\neq i_1,...,i_k$, else $\vec{c_1},...,\vec{c_k},\vec{c_{k+1}}$ would not be linearly independent. Therefore by the same lemma that we could use to prove the base case, we will have by the transitive property ($A\approx B$ and $B\approx C \implies A\approx C)$: $(\vec{v_1},...,\vec{v_l})\approx (\vec{v_1},...,\vec{c_1},...,\vec{c_k},...,\vec{v_j},...,\vec{v_l})\approx(\vec{v_1},...,\vec{c_1},...,\vec{c_k},...,\vec{c_{k+1}},...,\vec{v_l})$. Q.E.D
Problem is that this is very confusing to look and read, and also possibly kind of ambiguous. How could I write this down nicely, more clear and succint? And is my proof correct?