Vtyjkyuk

the identity for the second Chebyshev function:
$$sum _j=1^pi left( x right) Biggllfloor frac ln
left( x right) ln left( p_j right) Biggrrfloor ln
left( p_j right)=ln(operatornamelcm(1,2,3,...,lfloor x rfloor))quadquadquadquadquadquadquadquadquadquadquadquadquadquadquadquadquad(0)
$$

Any natural number $N$ has a unique prime factorization product:
$$N=p_1,N^v_1,Np_2,N^v_2,Np_3,N^v_3,Ncdotcdotcdot p_omega(N),N^v_omega(N),Nquadquadquadquadquadquadquadquadquadquadquadquadquadquadquadquadquadquadquadquadquadquadquadquad(1)$$

Allows us to make the following assertion about the natural logarithm of any such natural:
$$ln left( N right) =sum _j=1^omega left( N right) v_j,N
ln left( p_j,N right)
quadquadquadquadquadquadquadquadquadquadquadquadquadquadquadquadquadquadquadquadquadquadquadquadquadquad(1')$$

My question is as to whether or not stating the above is sufficient justification for stating the following to be true:

$pi(x)=omega(operatornamelcm(1,2,3,...,lfloor xrfloor))quad quadquadquadquadquadquadquadquadquadquadquadquadquadquadquadquadquadquadquadquadquadquadquadquad(a)$

the largest multiplicity any factor of $operatornamelcm(1,2,3,...,lfloor xrfloor)$ may have is $ lfloorfracln left( x right) 2rfloor quadquadquadquadquadquadquad,,,(b)$

Both (a) and (b) I concluded from (0) & ('1), I just feel as if I am missing something that is necessary in a direct proof of these statements.