reserve a,b,c,d,m,x,n,j,k,l for Nat,
  t,u,v,z for Integer,
  f,F for FinSequence of NAT;
reserve p,q,r,s for real number;

theorem Th58:
  for a,b be odd Nat holds (4 divides a + b iff not 4 divides a - b)
  proof
    let a,b be odd Nat;
    a+b is even & a-b is even; then
    2 divides a+b & 2 divides a - b; then
    consider t,z such that
    A3: a + b = 2*t & a - b = 2*z;
    A5: t is odd iff z is even
    proof
      a = t + z & b = t-z by A3;
      hence thesis;
    end;
    A6: 2*(2 gcd z) = |.2.|*(2 gcd z)
      .= 2*2 gcd 2*z by INT_6:16;
    A7: 2*2 divides (a+b) implies not 2*2 divides (a-b)
    proof
      assume (2*2) divides a+b; then
      |.(2*2).| = 2*2 gcd (2*t) by A3,Th3
      .=|.2.|*(t gcd 2) by INT_6:16; then
      2*2 gcd 2*z <> |.4.| by A5,A6,Th3;
      hence thesis by A3,Th3;
    end;
    (not 2*2 divides (a+b)) implies 2*2 divides (a-b)
    proof
      assume not (2*2) divides a+b; then
      not |.(2*2).| = 2*2 gcd (2*t) by A3,Th3; then
      not |.(2*2).| =|.2.|*(t gcd 2) by INT_6:16; then
      2*2 gcd 2*z = |.4.| by A5,A6,Th3;
      hence thesis by A3,Th3;
    end;
    hence thesis by A7;
  end;
