reserve i,i1,i2,i3,i4,i5,j,r,a,b,x,y for Integer,
  d,e,k,n for Nat,
  fp,fk for FinSequence of INT,
  f,f1,f2 for FinSequence of REAL,
  p for Prime;
reserve fr for FinSequence of REAL;
reserve fr,f for FinSequence of INT;
reserve b,m for Nat;
reserve b for Integer;
reserve m for Integer;
reserve fp for FinSequence of NAT;
reserve a,m for Nat;

theorem
  p>2 & (p mod 8 = 1 or p mod 8 = 7) implies 2 is_quadratic_residue_mod p
proof
  assume that
A1: p>2 and
A2: p mod 8 = 1 or p mod 8 = 7;
  set nn = p div 8;
  per cases by A2;
  suppose
    p mod 8 = 1;
    then p = 8*nn+1 by NAT_D:2;
    then (p^2 -'1) div 8 = (((8*nn)^2 + 2*(8*nn)) + 1-'1) div 8
      .= 8*(8*nn^2 + 2*nn) div 8 by NAT_D:34
      .= 2*(4*nn^2 + nn);
    then Lege(2,p) = (-1)|^(2*(4*nn^2 + nn)) by A1,Th42
      .= ((-1)|^2)|^(4*nn^2 + nn) by NEWTON:9
      .= (1|^2)|^(4*nn^2 + nn) by WSIERP_1:1
      .= 1;
    hence thesis by Def3;
  end;
  suppose
    p mod 8 = 7;
    then p = 8*nn+7 by NAT_D:2;
    then (p^2 -'1) div 8=(8*(8*nn^2)+8*(14*nn)+49-1) div 8 by NAT_1:12
,XREAL_1:233
      .= 8*(8*nn^2+14*nn+6) div 8
      .= 2*(4*nn^2+7*nn+3);
    then Lege(2,p) = (-1)|^(2*(4*nn^2+7*nn+3)) by A1,Th42
      .= ((-1)|^2)|^(4*nn^2+7*nn+3) by NEWTON:9
      .= (1|^2)|^(4*nn^2+7*nn+3) by WSIERP_1:1
      .= 1;
    hence thesis by Def3;
  end;
end;
