The cylindrical parts of a system of real algebraic equations and inequalities in variables 
are the terms
 |  |  |
(1)
|
 |  |  |
(2)
|
 |  |  |
(3)
|
 |  |  |
(4)
|
where '
'
is one of 
,
,
or 
,
and 
and 
are 
or algebraic expressions in variables 
that are real-valued for all 
-tuples of real numbers 
satisfying
 |  |  |
(5)
|
 |  |  |
(6)
|
 |  |  |
(7)
|
 |  |  |
(8)
|
The conjunction of a finite number of disjoint cylindrical parts is called a cylindrical algebraic decomposition.
