We know that from BR10 that one of the following is true:

    (i)
    b-a > 0
    (ii)
    b-a = 0
    (iii)
    -(b-a) > 0

Now we use process of elimination.

If b-a = 0 then this implies that b = a and then b2.=.a2 which contradicts the given statement
a2.<.b2.

If -(b-a) > 0 then a-b > 0 which implies that b.<.a. Therefore b2.<.a2 .(from 8) which also contradict the statement that a2.<.b2.

Hence b-a > 0 which implies that a.<.b.

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