Addition Axiom  -  If equals are added to equals, their sums are equal. If unequals are added to equals, their sums are unequal.

Division Axiom  -  If equals are divided by equals, their quotients is equal. If unequals are divided by equals, their quotients is unequal.

Multiplication Axiom  -  If equals are multiplied by equals, their products are equal. If unequals are multiplied by equals, their products are unequal.

Partition Axiom  -  A quantity is equal to the sum of its parts. A quantity is greater than any one of its parts.

Reflexive Property  -  A quantity is equal to itself.

Substitution Axiom  -  Equals can be substituted for each other in any equality or inequality.

Subtraction Axiom  -  If equals are subtracted from equals, their differences are equal. If unequals are subtracted from equals, their differences are unequal.

Transitive Property  -  If two quantities are equal to a third quantity, they are equal to each other. If a quantity is greater than another quantity, which is greater than a third quantity, then the first quantity is greater than the third quantity.