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The proofs of (i) follows obviously from the definition of absolute value.
To prove (ii) we consider different cases. The required property is obvious if both x, y  0 .
In case  and  ) , we have
The other cases can be analyzed similarly.
To prove (iii) suppose  . If  , then
If  , then
Thus, 
Conversely, let  . If , then . If  , then  . Thus (iii) holds.
To prove (iv), note that
Adding the two we get
and hence by (iii),  .
Finally, to prove (v) , note that by (iv)
 .
Hence (v) follows from (iii).
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