Proof: Subtract Move Over

Let's prove the following theorem:

if a - b = c, then a = c + b

Proof:

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Given
1 a - b = c
Proof Table
# Claim Reason
1 (a - b) + b = c + b if a - b = c, then (a - b) + b = c + b
2 a - b = a + (b ⋅ (-1)) a - b = a + (b ⋅ (-1))
3 (a - b) + b = (a + (b ⋅ (-1))) + b if a - b = a + (b ⋅ (-1)), then (a - b) + b = (a + (b ⋅ (-1))) + b
4 (a + (b ⋅ (-1))) + b = a + ((b ⋅ (-1)) + b) (a + (b ⋅ (-1))) + b = a + ((b ⋅ (-1)) + b)
5 (b ⋅ (-1)) + b = 0 (b ⋅ (-1)) + b = 0
6 (a + (b ⋅ (-1))) + b = a + 0 if (a + (b ⋅ (-1))) + b = a + ((b ⋅ (-1)) + b) and (b ⋅ (-1)) + b = 0, then (a + (b ⋅ (-1))) + b = a + 0
7 a + 0 = a a + 0 = a
8 (a + (b ⋅ (-1))) + b = a if (a + (b ⋅ (-1))) + b = a + 0 and a + 0 = a, then (a + (b ⋅ (-1))) + b = a
9 (a - b) + b = a if (a - b) + b = (a + (b ⋅ (-1))) + b and (a + (b ⋅ (-1))) + b = a, then (a - b) + b = a
10 a = c + b if (a - b) + b = a and (a - b) + b = c + b, then a = c + b

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