reserve x for set;
reserve i,j for Integer;
reserve n,n1,n2,n3 for Nat;
reserve K,K1,K2,K3 for Field;
reserve SK1,SK2 for Subfield of K;
reserve ek,ek1,ek2 for Element of K;
reserve p for Prime;
reserve a,b,c for Element of GF(p);
reserve F for FinSequence of GF(p);

theorem Th25:
  a <> 0 implies a |^ n <> 0
  proof
    assume A1: a <> 0;
    consider n1 be Nat such that A2: a = n1 mod p by Th13;
    not p divides n1 by A1,A2,INT_1:62;
    then not p divides n1|^n by NAT_3:5;
    then n1|^n mod p <> 0 by INT_1:62;
    hence thesis by A2,Th23;
  end;
