This line brings sorrow to the poem. This is important because it's sending a message. What he's saying is that he wants her to stop grieving. Afterwords she'll hear the church bell stop ringing. It builds the theme because  we don't know who the reader is speaking to but you'd have an idea. 
no longer mourn for me when I am dead. Then you shall hear the surly sullen bell. 
This line builds up theme because he's speaking to his lover. This is important because now we know what it's about. These lines mean that his departure will be announced when the ringing stops. If his lover reads his note, he rather be forgotten then to cause them more sadness through his absence. 
Give warning to the world that I am fled. From this vile world with vilest worms to dwell. Nay, if you read this line, remember not The hand that writ it; for I love you so, That I in your sweet thoughts would be forgot, If thinking on me then should make you woe.
A device used in this poem is Apostrophe. In the poem, the narrator is speaking to someone but doesn't give a name or explains who they're talking to. He uses words like "you" and lover because the poem is a note. The poem is message to they're lover in death, so it adds more emotion to the poem. The narrator loves their significant other so much, he rather be forgotten then to cause them more pain. 
Line 13 then marks a turning point in the poem, indeed, the final couplet of an English sonnet is called the "turn" when the speaker says that the world will register the lover's sorrow if the lover doesn't manage to sufficiently move on from mourning the speaker's death.
Let the wise world should look into your moan 
"Let the wise world should look into your moan" means the world will notice his lover's sorrow. It's important because  it shows how the world reacts when someone's loved one has died. Sometimes they don't know what to say or mention it when they don't need to. 
And Mock you with me while i am gone
What this line means is that people will taunt his lover with the reminder that he is gone. They'll constantly send their condolences reminding the lover. It's important to the story because it builds the theme. The theme is "No longer mourn those you lost". Sometimes the mourn you mourn, you'll never be able to move on. 
The Theme "No lounger mourn for me when I am dead"
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