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 Th29:
  for x be Element of MultGroup GF(p), x1 be Element of GF(p),
  n be Nat st x = x1 holds x|^n = x1 |^n
  proof
    let x be Element of MultGroup GF(p), x1 be Element of GF(p), n be Nat;
    assume
    A1: x = x1;
    defpred P[Nat] means x|^$1 = x1 |^$1;
    A2: for n be Nat st P[n] holds P[n+1]
    proof
      let n be Nat;
      A3: x|^(n+1) = (x|^n)*x by GROUP_1:34;
      A4: x1 |^(n+1) = (x1 |^n)*x1 by Th24;
      assume x|^n = x1 |^n;
      hence thesis by A1,A3,A4,UNIROOTS:16;
    end;
    x|^0 = 1_(MultGroup GF(p)) by GROUP_1:25
    .= 1_GF(p) by UNIROOTS:17
    .= x1 |^0 by Th21; then
A5: P[0];
    for n be Nat holds P[n] from NAT_1:sch 2(A5,A2);
    hence thesis;
  end;
