Statement:

  Let \(\pmb{(V,+,\cdot)} \) be a vector space over a field \(\pmb{(F,+,\cdot)} \) and \(\pmb{S} \) and \(\pmb{T} \) be two subspaces of \(\pmb{V} \). Then the linear sum \(\pmb{S+T} \) is a subspace of \(\pmb{V} \).

  Proof:
  Given that \(\pmb{(V,+,\cdot)} \) is a vector space over a field \(\pmb{(F,+,\cdot)} \) and \(\pmb{S} \) and \(\pmb{T} \) are two subspaces of \(\pmb{V} \).
  To prove \(\pmb{S+T} \) is a subspace of \(\pmb{V} \).

1. To prove \(\pmb{S+T\ne \Phi} \)
   Since \(\pmb{S} \) and \(\pmb{T} \) are subspaces
   \(\implies \pmb{\theta\in S} \) and \(\pmb{\theta\in T} \)
   \(\implies \pmb{\theta+\theta\in S+T} \)
   \(\implies \pmb{\theta\in S+T} \)
   
2. To prove \(\pmb{\alpha+\beta\in S+T~\forall~\alpha,\beta\in S+T} \)
   Let \(\pmb{\alpha,\beta\in S+T} \)
   Then \(\pmb{\exists~ s_{1},s_{2}\in S} \) and \(\pmb{t_{1},t_{2}\in T} \) such that \(\pmb{\alpha=s_{1}+t_{1}} \) and \(\pmb{\beta=s_{2}+t_{2}} \)
   Now \(\pmb{\alpha+\beta= (s_{1}+t_{1})+(s_{2}+t_{2})} \)
   \(\implies \pmb{\alpha+\beta= (s_{1}+s_{2})+(t_{1}+t_{2}) } \)
   \(\implies \pmb{\alpha+\beta= s_{3}+t_{3} }\) where \(\pmb{s_{1}+s_{2}=s_{3}\in S}\) and \(\pmb{t_{1}+t_{2}=t_{3}\in T}\) since \(\pmb{S} \) and \(\pmb{T} \) are subspaces.
   \(\implies \pmb{\alpha+\beta\in S+T }\)
3. To prove \(\pmb{c\cdot\gamma\in S+T~\forall~\gamma\in S+T,~c\in F} \)
   Let \(\pmb{\gamma\in S+T,~c\in F} \)
   Then \(\pmb{\exists~ s_{3}\in S} \) and \(\pmb{t_{3}\in T} \) such that \(\pmb{\gamma=s_{3}+t_{3}} \).
   Now \(\pmb{c\cdot\gamma=c\cdot(s_{3}+t_{3} )} \)
   \(\implies \pmb{c\cdot\gamma=c\cdot s_{3} +c\cdot t_{3}} \)
   \(\implies \pmb{c\cdot\gamma=s_{4} + t_{4}} \) where \(\pmb{c\cdot s_{3}=s_{4}\in S}\) and \(\pmb{c\cdot t_{3}=t_{4}\in T}\) since \(\pmb{S} \) and \(\pmb{T} \) are subspaces.
   \(\implies \pmb{c\cdot\gamma\in S+T }\)