How we cite our quotes: (Act.Scene.Line). Line numbers correspond to the 2008 Norton edition.
Quote #1
ANTONIO
Let's all sink wi' th' king.
SEBASTIAN
Let's take leave of him. (1.1.66-67)
From the first sign that things might not be rosy, Sebastian shows he has no loyalty to his brother or his king (Alonso is both). He has a partner in callousness in Antonio—this foreshadows their later treachery on the island.
Quote #2
PROSPERO
Both, both, my girl.
By foul play, as thou sayst, were we heaved thence,
But blessedly holp hither. (1.2.77-79)
Prospero has the distance and perspective of wisdom when thinking about how they ended up on the island. Antonio's treachery put them there, but the help of the natural elements, and Gonzalo's loyalty, allowed them to survive and prosper.
Quote #3
PROSPERO
Being once perfected how to grant suits,
How to deny them, who t' advance and who
To trash for over-topping, new created
The creatures that were mine, I say, or changed 'em,
Or else new formed 'em, having both the key
Of officer and office, set all hearts i' th' state
To what tune pleased his ear, that now he was
The ivy which had hid my princely trunk
And sucked my verdure out on 't. (1.2.98-106)
Prospero values the brotherly bond more than Antonio; Prospero assumed his brother would be loyal to him. Instead, Antonio learned all the tricks of political treachery while serving in the place of Prospero, and used them to betray his brother.