reserve a,b,c,d,x,j,k,l,m,n,o,xi,xj for Nat,
  p,q,t,z,u,v for Integer,
  a1,b1,c1,d1 for Complex;

theorem NEWTON0258:
  for a,b be odd Integer holds
    4 divides a-b iff not 4 divides a+b
  proof
    let a,b be odd Integer;
    reconsider t = |.a.| as odd Nat;
    reconsider u = |.b.| as odd Nat;
A0: 4 divides t - u iff not 4 divides t + u by NEWTON02:58;
    4 divides u - t iff not 4 divides (u + t) by NEWTON02:58; then
A0a: 4 divides u - t iff not 4 divides -(u + t) by INT_2:10;
    per cases by ABSVALUE:1;
    suppose
  A1: |.a.| = a;
      then reconsider a as Nat;
      per cases by ABSVALUE:1;
      suppose
        |.b.| = b;
        hence thesis by NEWTON02:58,A1;
      end;
      suppose
        |.b.| = -b;
        hence thesis by A0,A1;
      end;
    end;
    suppose
  A1: |.a.| = -a;
      per cases by ABSVALUE:1;
      suppose |.b.| = b;
        hence thesis by A0a,A1;
      end;
      suppose
        |.b.| = -b;
        hence thesis by A0a,A1;
      end;
    end;
  end;
