Proof: Removing the Common Term

Let's prove the following theorem:

if the following are true:
  • a ⋅ x = a ⋅ y
  • not (a = 0)

then x = y

The Multiplicative Property of Equality allows us to multiply both the left and right sides by (1/a). This is the same as dividing both sides by a

Since 1/a ⋅ a ⋅ x = x, the equation becomes:

x = y

Proof:

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Given
1 a ⋅ x = a ⋅ y
2 not (a = 0)
Proof Table
# Claim Reason
1 (1 / a) ⋅ (a ⋅ x) = (1 / a) ⋅ (a ⋅ y) if a ⋅ x = a ⋅ y, then (1 / a) ⋅ (a ⋅ x) = (1 / a) ⋅ (a ⋅ y)
2 (1 / a) ⋅ (a ⋅ x) = x if not (a = 0), then (1 / a) ⋅ (a ⋅ x) = x
3 (1 / a) ⋅ (a ⋅ y) = y if not (a = 0), then (1 / a) ⋅ (a ⋅ y) = y
4 x = y if (1 / a) ⋅ (a ⋅ y) = y and (1 / a) ⋅ (a ⋅ x) = x and (1 / a) ⋅ (a ⋅ x) = (1 / a) ⋅ (a ⋅ y), then x = y

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