reserve a,b,c,d,a9,b9,c9,d9,y,x1,u,v for Real,
  s,t,h,z,z1,z2,z3,s1,s2,s3 for Complex;

theorem
  for n be non zero Nat, v be CRoot of n,0 holds v = 0
proof
  let n be non zero Nat, v be CRoot of n,0;
  defpred P[Nat] means v|^ $1 = 0;
A1: now
    let k be non zero Nat;
    assume that
A2: k <> 1 and
A3: P[k];
    consider t be Nat such that
A4: k = t+1 by NAT_1:6;
    t <> 0 by A2,A4;
    then reconsider t as non zero Nat;
    take t;
    thus t < k by A4,NAT_1:13;
    v|^ k = (v|^ t)*v by A4,NEWTON:6;
    then v|^ t = 0 or v = 0 by A3;
    hence P[t] by Lm2;
  end;
A5: ex n be non zero Nat st P[n]
  proof
    take n;
    thus thesis by COMPTRIG:def 2;
  end;
  P[1] from COMPTRIG:sch 1(A5,A1);
  hence thesis;
end;
