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Addition Axiom
If equals are added to equals, their sums are equal. If unequals are added to equals, their sums are unequal.
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Division Axiom
If equals are divided by equals, their quotients is equal. If unequals are divided by equals, their quotients is unequal.
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Multiplication Axiom
If equals are multiplied by equals, their products are equal. If unequals are multiplied by equals, their products are unequal.
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Partition Axiom
A quantity is equal to the sum of its parts. A quantity is greater than any one of its parts.
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Reflexive Property
A quantity is equal to itself.
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Substitution Axiom
Equals can be substituted for each other in any equality or inequality.
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Subtraction Axiom
If equals are subtracted from equals, their differences are equal. If unequals are subtracted from equals, their differences are unequal.
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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.