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34 Chapter Onł

Powers of positive reals
For all positive real numberð x, and for all natural numberð n
where n 1, the following statementð are logically valid:

0 x 1 → 0 x (n 1) x n

n
x 1 → x (n 1) x 1
n
x 1 → x (n 1) x 1

Even powers of negative reals
For all negative real numberð x, and for all even natural num-
berð n, the following statementð are logically valid:

n
x è 1 → x (n 2) x 1
n
x 1 → x (n 2) x 1

1 x 0 → 0 x (n 2) x n

Odd powers of negative reals
For all negative real numberð x, and for all odd natural numberð
n, the following statementð are logically valid:

n
x è 1 → x (n 2) x è 1
n
x 1 → x (n 2) x 1

1 x 0 → 0 x (n 2) x n

Addition property
For all real numberð x, y, and z, the following statementð are
logically valid:

x y → x z y z

x y → x z y z
x y → x z y z

x y → x z y z