reserve z,z1,z2,z3,z4 for Element of F_Complex;

theorem
  for n be non zero Element of NAT for v be CRoot of n,0.F_Complex
  holds v = 0.F_Complex
proof
  let n be non zero Element of NAT;
  let v be CRoot of n,0.F_Complex;
  defpred P[Element of omega] means (power F_Complex).(v,$1) = 0.F_Complex;
A1: now
    let k be non zero Element of NAT;
    assume that
A2: k <> 1 and
A3: P[k];
    consider t be Nat such that
A4: k = t+1 by NAT_1:6;
    reconsider t as Element of NAT by ORDINAL1:def 12;
    reconsider t as non zero Element of NAT by A2,A4;
    take t;
    thus t < k by A4,NAT_1:13;
    (power F_Complex).(v,k) = (power F_Complex).(v,t)*v by A4,GROUP_1:def 7;
    then (power F_Complex).(v,t) = 0.F_Complex or v = 0.F_Complex by A3,
VECTSP_1:12;
    hence P[t] by NAT_1:3,VECTSP_1:36;
  end;
A5: ex n be non zero Element of NAT st P[n]
  proof
    take n;
    thus thesis by Def2;
  end;
  P[1] from Regrwithout0(A5,A1);
  hence thesis by GROUP_1:50;
end;
