by (4).
This gives the theorem in question.
In the same manner every one of the first ten propositions is proved.
It will be seen that the operations performed are exactly the same as if the letters denoted numbers.
Props. 5 and 6 may also be written thus—
(a + b)(a − b) = a² − b².
Prop. 7, which is an easy consequence of Prop. 4, may be transformed. If we denote by c the line a + b, so that
c = a + b, a = c − b,
we get
| c² + (c − b)² | = 2c(c − b) + b² |
| = 2c² − 2bc + b². |