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 Th23:
  a = n1 mod p implies a|^n = n1|^n mod p
  proof
A1: p > 1 by INT_2:def 4;
    assume A2: a = n1 mod p;
    defpred P[Nat] means
    (power GF(p)).(a,$1) = n1|^($1) mod p;
    a|^0 = 1 by Th21;
    then A3: a|^0 = 1 mod p by A1,NAT_D:63;
A4: P[0] by A3,NEWTON:4;
A5: now let n be Nat;
      assume A6: P[n];
      reconsider b = (power GF(p)).(a,n) as Element of GF(p);
      (power GF(p)).(a,n+1) = b*a by GROUP_1:def 7
      .= (n1|^n*n1) mod p by A2,A6,Th18
      .= n1|^(n+1) mod p by NEWTON:6;
      hence P[n+1];
    end;
    for n be Nat holds P[n] from NAT_1:sch 2(A4,A5);
    hence thesis;
  end;
