
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.