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
  a is odd & b is odd implies 8 divides a|^2 - b|^2
  proof
    assume
    A0: a is odd & b is odd; then consider k such that
    A1: a = 2*k + 1 by ABIAN:9;
    consider n such that
    A1a: b = 2*n + 1 by A0,ABIAN:9;
    (k is even or (k+1) is even) & (n is even or (n+1) is even); then
    k*(k+1) - n*(n+1) is even; then
    A2: 2*4 divides 4*(k*(k+1) - n*(n+1)) by INT_6:8;
    a|^2 - b|^2  =  (2*k+1)|^2 -(4*n*(n+1)+1) by A1,A1a,Th62
    .= (4*k*(k+1)+1) - (4*n*(n+1)+1) by Th62
    .= 4*(k*(k+1) - n*(n+1));
    hence thesis by A2;
  end;
