
theorem Th31:
  for p being 2_greater Prime,
      a being Integer
  st a gcd p = 1
  holds a|^((p-1)/2), LegendreSymbol(a,p) are_congruent_mod p
proof
  let p be 2_greater Prime,
      a be Integer;
  p - 1 > 2 - 1 by Def1,XREAL_1:9;
  then A1: p -' 1 = p - 1 by NAT_D:39;
  assume a gcd p = 1;
  then Lege (a,p),a|^((p-'1) div 2) are_congruent_mod p by Def1,INT_5:28;
  then A2: Lege (a,p),a|^((p-1)/2) are_congruent_mod p by A1;
  Leg(a,p) = Lege(a,p) by Lm4;
  hence thesis by A2,INT_1:14;
end;
