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 Th28:
  a <> 0 implies (a") |^n = (a|^n)"
  proof
    assume A1: a <> 0;
A2: p > 1 by INT_2:def 4;
    consider n1 be Nat such that A3: a = n1 mod p by Th13;
    consider n2 be Nat such that A4: a" = n2 mod p by Th13;
A5: a|^n = n1|^n mod p by A3,Th23;
A6: (a") |^ n = (n2|^n) mod p by A4,Th23;
A7: (n1 * n2) |^ n mod p = (n1|^n * n2|^n) mod p by NEWTON:7
    .= (a|^n) * ((a") |^ n) by A5,A6,Th18;
    a <> 0.GF(p) by A1,Th11;
    then a" * a = 1.GF(p) by VECTSP_1:def 10
    .= 1 by Th12;
    then (n1*n2) mod p = 1 by A3,A4,Th18;
    then (n1 * n2) |^ n mod p = 1 by A2,PEPIN:35;
    then A8: ((a") |^ n) * (a|^n) = 1.GF(p) by A7;
    then a|^n <> 0.GF(p);
    hence thesis by A8,VECTSP_1:def 10;
  end;
