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 = 3 or p mod 8 = 5) implies not 2 is_quadratic_residue_mod p
proof
  assume that
A1: p>2 and
A2: p mod 8 = 3 or p mod 8 = 5;
  set nn = p div 8;
  per cases by A2;
  suppose
    p mod 8 = 3;
    then p = 8*nn+3 by NAT_D:2;
    then (p^2 -'1) div 8 = (8*(8*nn^2)+8*(6*nn)+3*3-1) div 8 by NAT_1:12
,XREAL_1:233
      .= 8*(8*nn^2+6*nn+1) div 8
      .= 8*nn^2+6*nn+1;
    then Lege(2,p) = (-1)|^(8*nn^2+6*nn+1) by A1,Th42
      .= (-1)|^(2*(4*nn^2+3*nn))*(-1) by NEWTON:6
      .= ((-1)|^2)|^(4*nn^2+3*nn)*(-1) by NEWTON:9
      .= (1|^2)|^(4*nn^2+3*nn)*(-1) by WSIERP_1:1
      .= -1; then
    not (2 is_quadratic_residue_mod p & 2 mod p <> 0) &
    not (2 is_quadratic_residue_mod p & 2 mod p = 0) by Def3;
    hence thesis;
  end;
  suppose
    p mod 8 = 5;
    then p = 8*nn+5 by NAT_D:2;
    then (p^2 -'1) div 8=(8*(8*nn^2)+8*(10*nn)+25-1) div 8 by NAT_1:12
,XREAL_1:233
      .= 8*(8*nn^2+10*nn+3) div 8
      .= 8*nn^2+10*nn+3;
    then Lege(2,p) = (-1)|^(2*(4*nn^2)+2*(5*nn)+2*1+1) by A1,Th42
      .= (-1)|^(2*(4*nn^2+5*nn+1))*(-1) by NEWTON:6
      .= ((-1)|^2)|^(4*nn^2+5*nn+1)*(-1) by NEWTON:9
      .= (1|^2)|^(4*nn^2+5*nn+1)*(-1) by WSIERP_1:1
      .= -1; then
    not (2 is_quadratic_residue_mod p & 2 mod p <> 0) &
    not (2 is_quadratic_residue_mod p & 2 mod p = 0) by Def3;
    hence thesis;
  end;
end;
