When we calculate xor$($S$,$K$),$ it equals xor$($S$,$K$),$ so the input to the S$-$boxes and the rearrangement remains the same. After the modified output is xored with the other half of S$,$ we get C$$ for that round of DES, and this process repeats for all $16$ rounds. We can observe that the input to the S$-$boxes is determined by the XOR operation between the key and the plaintext. Since XOR operation is commutative and associative, changing all the $1$s to $0$s and all the $0$s to $1$s in both the key and the plaintext will result in the same input to the S$-$boxes. Therefore, if the original key encrypts the original plaintext to the original ciphertext, then the complementary key will encrypt the complementary plaintext to the complementary ciphertext in DES encryption.