Abstract: Kim B.Bruce gave a system of axioms for second order logic L(Q).H.J.Keisler in 1977 published a book called 《Model Theory for Infinitary Logic》 to give a system of axioms for infinitary logic. This article combines the two ideas of the two systems of axioms, applying the consistency property introduced by the book 《Model Theory for Infinitary Logic》 to give a system of axioms and model theory for infinitary second order logic Lw₁w(Q). This part of the article is mainly the proofs of omitting type theorem for Lw₁w(Q) and prime model theorem for Lw₁w(Q).