Hello to everyone that's reading this.

(Theory of Computation) proof problem about proving that L = L(A) by proving that L ⊆ L(A) and that L(A) ⊆ L and its solution (not by me):

My question is: does the "δ̂(p, w) = p ⇒ w has even number of 1's" from second part (L ⊆ L(A)) need the first part (L(A) ⊆ L) to already be shown, or is it completely independent, such that the second part (L ⊆ L(A)) could have been cut and pasted, verbatim, before the first part (L(A) ⊆ L)?

Any input would be GREATLY appreciated!