reserve x for Real;

theorem
  for n be non zero Nat for v be CRoot of n,0 holds v = 0
proof
  let n be non zero Nat;
  let v be CRoot of n,0;
  defpred P[Nat] means v|^$1 = 0;
  assume
A1: v <> 0;
A2: now
    let k be non zero Nat;
    assume that
A3: k <> 1 and
A4: P[k];
    consider t be Nat such that
A5: k = t+1 by NAT_1:6;
    reconsider t as non zero Nat by A3,A5;
    take t;
    thus t < k by A5,NAT_1:13;
    v|^k = v|^t*v by A5,NEWTON:6;
    hence P[t] by A1,A4;
  end;
A6: ex n be non zero Nat st P[n]
  proof
    take n;
    thus thesis by Def2;
  end;
  P[1] from Regrwithout0(A6,A2);
  hence thesis by A1;
end;
