let x be Complex; :: thesis: x * 0 = 0
x in COMPLEX by XCMPLX_0:def 2;
then consider x1, x2 being Element of REAL such that
A1: x = [*x1,x2*] by ARYTM_0:9;
0 = [*Z,Z*] by ARYTM_0:def 5;
then x * 0 = [*(+ ((* (x1,Z)),(opp (* (x2,Z))))),(+ ((* (x1,Z)),(* (x2,Z))))*] by A1, XCMPLX_0:def 5
.= [*(+ ((* (x1,Z)),(opp Z))),(+ ((* (x1,Z)),(* (x2,Z))))*] by ARYTM_0:12
.= [*(+ ((* (x1,Z)),(opp Z))),(+ ((* (x1,Z)),Z))*] by ARYTM_0:12
.= [*(+ (Z,(opp Z))),(+ ((* (x1,Z)),Z))*] by ARYTM_0:12
.= [*(+ (Z,(opp Z))),(+ (Z,Z))*] by ARYTM_0:12
.= [*(+ (Z,(opp Z))),Z*] by ARYTM_0:11
.= [*(opp Z),Z*] by ARYTM_0:11
.= 0 by Lm2, ARYTM_0:def 5 ;
hence x * 0 = 0 ; :: thesis: verum