let x be complex number ; :: thesis: x + 0 = x
x in COMPLEX by XCMPLX_0:def 2;
then consider x1, x2 being Element of REAL such that
A1: x = [*x1,x2*] by ARYTM_0:11;
0 = [*0,0*] by ARYTM_0:def 7;
then x + 0 = [*(+ (x1,0)),(+ (x2,0))*] by A1, XCMPLX_0:def 4
.= [*x1,(+ (x2,0))*] by ARYTM_0:13
.= x by A1, ARYTM_0:13 ;
hence x + 0 = x ; :: thesis: verum