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
  t|^n,t|^m are_congruent_mod t-1
  proof
    per cases;
    suppose n >= m;
      then consider k such that
      B2: n = m + k by NAT_1:10;
      t|^(m+k) = t|^m * t|^k by NEWTON:8; then
      t|^(m+k) - t|^m = t|^m*(t|^k-1|^k);
      hence thesis by B2,Th18,INT_2:2;
    end;
    suppose m >= n;
      then consider k such that
      B2: m = n + k by NAT_1:10;
      t|^(n+k) = t|^n * t|^k by NEWTON:8; then
      t|^(n+k) - t|^n = t|^n*(t|^k-1|^k); then
      (t-1) divides t|^m - t|^n by B2,Th18,INT_2:2; then
      (t-1) divides -(-t|^n + t|^m) by INT_2:10;
      hence thesis;
    end;
  end;
