<s>[INST] Logical statements: $A+B+C=A+B+C+AB+BC+AC$ 
Let $A,B, C$ be logical statements.
  Then: 
  $A+B+C=A+B+C+AB+BC+AC$

Prove it without a table.

My attempt:
$$A=A\cdot(1+B+C)$$
$$B=B\cdot(1+A+C)$$
$$C=C\cdot(1+A+B)$$
$$\implies A+B+C= A+AB+AC+B+AB+BC+C+BC+AC$$
$$\implies A+B+C=A+B+C+AB+AB+BC+BC+AC+AC$$
$$\implies A+B+C=A+B+C+AB+BC+AC$$
Is this correct?
 [/INST] Yes. it is completely correct.
You could have shortened it a bit by writing
$$A+B+C+AB+BC+AC = A(1+B+C) + B(1+C) + C $$ $$
\stackrel{1+x=1,1x=x}{=}A+B+C$$
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