Prove the following results about the theory \( L_1 \).

a. \( B \Rightarrow \vdash L_1 \lor DOV \)

b. \( \vdash L_1 (\Rightarrow) \Rightarrow ((\Rightarrow \Rightarrow) \Rightarrow (g \Rightarrow \lambda)) \)

c. \( \Rightarrow \Rightarrow B, \Rightarrow \vdash L_1 \Rightarrow \)

d. \( \vdash L_1 B \Rightarrow \) (i.e., \( \vdash L_1 \lnot V \lor D \))

The question is about mathematical or formal logic, but the symbolic expressions provided do not use standard logical notation, making interpretation and response difficult.

Explanation:

The provided problem set is about logical statements, symbolic logic, or symbolic expressions in the field of Mathematics, specifically in an area called Mathematical Logic or Formal Logic.

However, the notations used in the current form of the problem (e.g., B⇒⊢ L1 ,∨DOV etc.) are not standard and thus difficult to interpret. It is crucial to state logical expressions using standard notation to understand their correct interpretation and consequently solve or prove them. For instance, if '⇒' represents 'implies', '⊢' asserts that a statement is a theorem, and '∧', '∨', '¬' denote 'and', 'or', 'not' respectively, the meaning of the expressions becomes clearer.

Without the adequate context or proper notation, it is not possible to provide the proofs or solutions requested. Please resubmit your question with standard logical notation and sufficient context to enable a more accurate response.

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