Goldbach Conjecture implies Weak Goldbach Conjecture

Theorem
The Goldbach Conjecture:
 * Every even integer greater than $$2$$ is the sum of two primes

implies Goldbach's Weak Conjecture:
 * Every odd integer greater than $$7$$ is the sum of three odd primes.

Proof
Take any odd integer $$n$$ such that $$n > 7$$.

Then $$m = n - 3$$ is an even integer $$n$$ such that $$m > 4$$.

If the Goldbach Conjecture holds, then $$m$$ is the sum of two primes: $$m = p_1 + p_2$$.

If one of them were $$2$$, then $$m - 2$$ would have to be even, which if it is prime it can not be.

So if $$m > 4$$, both $$p_1$$ and $$p_2$$ must be odd.

So then we have that $$n = p_1 + p_2 + 3$$, that is, the sum of three odd primes.

So, if the Goldbach Conjecture holds, then so does Goldbach's Weak Conjecture.