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 Th37:
  a <> 0 & n mod 2 = 0 implies Lege_p(a|^n) = 1
  proof
    assume A1: a <> 0 & n mod 2 = 0;
A2: n = (n div 2) * 2 + (n mod 2) by INT_1:59
    .= (n div 2) * 2 by A1;
    reconsider n1 = n div 2 as Nat;
    (a|^n1) |^ 2 is quadratic_residue by A1,Th31,Th25;
    then a|^n is quadratic_residue by A2,Lm4;
    hence thesis by Def5;
  end;
