Karakteristic fonsiyonlari kullanirsak:
$$1_{A\cap B}=1_A\cdot 1_B \qquad\mbox{and}\qquad 1_{A\Delta B}=1_A+1_B-1_{A\cap B}$$
O halde islemlerimiz:
$$1_{A\Delta(B\Delta C)}=1_A+1_{B\Delta C}-1_{A\cap (B\Delta C)}$$
$$=1_A+1_B+1_C-1_{B\cap C}-1_A\cdot(1_B+1_C-1_{B\cap C})$$
$$=1_A+1_B+1_C-1_{A\cap B}-1_{B\cap C}-1_{C\cap A}+1_{A\cap (B\cap C)}.$$
Burdan sonrasi bitti aslinda. Sunu da ispatlayalim: $A\cap (B\cap C)=A\cap B\cap C$:
$$1_{A\cap(B\cap C)}=1_A1_{B\cap C}=1_A1_B1_C=1_{A\cap B}1_C=1_{(A\cap B)\cap C}$$.