reserve x for Complex;

theorem Th3:
  1 * x = x
proof
  0 in NAT & 1 in NAT;
then reconsider Z =0,J = 1 as Element of REAL by NUMBERS:19;
     +(Z,Z) = 0 by ARYTM_0:11;
  then Lm2: opp Z = 0 by ARYTM_0:def 3;     
  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;
  1 = [*J,Z*] by ARYTM_0:def 5;
  then x * 1 = [* +(*(x1,J),opp*(x2,Z)), +(*(x1,Z),*(x2,J)) *] by A1,
XCMPLX_0:def 5
    .= [* +(*(x1,J),opp Z), +(*(x1,Z),*(x2,J)) *] by ARYTM_0:12
    .= [* +(x1,opp Z), +(*(x1,Z),*(x2,J)) *] by ARYTM_0:19
    .= [* +(x1,opp Z), +(*(x1,Z),x2) *] by ARYTM_0:19
    .= [* +(x1,Z), +(Z,x2) *] by Lm2,ARYTM_0:12
    .= [* x1, +(Z,x2) *] by ARYTM_0:11
    .= x by A1,ARYTM_0:11;
  hence thesis;
end;
