A binary operation "\\cdot" is associative when "(x \\cdot y) \\cdot z = x \\cdot (y \\cdot z)" for any "x, y, z" .
Suppose we have
"(x \\cdot y) \\cdot z"
We can replace the binary operation with addition using its definition:
"(x \\cdot y) \\cdot z = (1 + x + y) \\cdot z = 1 + (1+x+y) +z"
Now we can remove parentheses (using associativity of addition), regroup terms (using commutativity of addition) and put parentheses where we want (using associativity of addition once again):
"1 + (1 + x + y) + z = \\\\\n=1 + 1 + x + y + z = \\\\\n= 1 + x + 1+y +z = \\\\\n=1 + x + (1+y+z)"
"1+y+z = y \\cdot z" by definition of this binary operation.
"1+ x + (1+y+ z) = 1+x+(y \\cdot z)"
"1+x+(y \\cdot z) = x \\cdot (y \\cdot z)" by definition of our binary operation.
We proved that "(x \\cdot y) \\cdot z = x\\cdot (y \\cdot z)" which means we proved that the binary operation is associative algebraically. What was to be shown.
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