reserve x for Complex;

theorem Th2:
  x * 0 = 0
proof
  0 in NAT;
  then reconsider Z =0 as Element of REAL by NUMBERS:19;

  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;

     +(Z,Z) = 0 by ARYTM_0:11;
  then Lm2: opp Z = 0 by ARYTM_0:def 3;
  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 thesis;
end;
