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
  n mod 2 = 1 implies Lege_p(a|^n) = Lege_p(a)
  proof
    assume A1: n mod 2 = 1;
    A2: n = (n div 2) * 2 + 1 by A1,INT_1:59;
    reconsider n1 = n - 1 as Nat by A2;
    a |^ (n1+1) = a |^ n1 * a by Th24;
    then A3: Lege_p(a|^n) = Lege_p(a |^ n1) * Lege_p(a) by Th36;
    per cases;
    suppose a = 0;
      then Lege_p(a) = 0 by Def5;
      hence thesis by A3;
    end;
    suppose A4: a <> 0;
      n-1 mod 2 = 0 + ((n div 2) * 2) mod 2 by A2
      .= 0 mod 2 by NAT_D:61
      .= 0 by NAT_D:26;
      then Lege_p(a |^ n1) = 1 by A4,Th37;
      hence thesis by A3;
    end;
  end;
