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 Th17:
  m>0 implies t*z divides (t+z)|^m - (t|^m + z|^m)
  proof
    assume m>0; then
    consider n such that
    A0: m = 1+n by NAT_1:14,10;
    t*z divides (t+z)|^(n+1) - (t|^(n+1) + z|^(n+1))
    proof
      defpred P[Nat] means
        t*z divides ((t+z)|^($1+1) - (t|^($1+1) + z|^($1+1)));
      A1: P[0] by INT_2:12;
      A2: P[x] implies P[x+1]
      proof
        assume
        P[x]; then
    B2: t*z divides ((t+z)|^(x+1) - (t|^(x+1) + z|^(x+1)))*(t+z) by INT_2:2;
        B3: ((t+z)|^(x+1) - (t|^(x+1) + z|^(x+1)))*(t+z) =
        (t+z)|^(x+1)*(t+z) - t|^(x+1)*t - z*t|^(x+1) - t*z|^(x+1) - z|^(x+1)*z
        .= (t+z)|^(x+1+1) - t|^(x+1)*t- z*t|^(x+1) - t*z|^(x+1) - z|^(x+1)*z
          by NEWTON:6
        .= (t+z)|^(x+1+1) - t|^(x+1+1)- z*t|^(x+1) - t*z|^(x+1) - z|^(x+1)*z
          by NEWTON:6
        .= (t+z)|^(x+1+1) - t|^(x+1+1)- z*(t|^x*t) - t*z|^(x+1) - z|^(x+1)*z
          by NEWTON:6
        .= (t+z)|^(x+1+1) - t|^(x+1+1)- z*t|^x*t - t*(z|^x*z) - z|^(x+1)*z
          by NEWTON:6
        .= (t+z)|^(x+1+1) - t|^(x+1+1) - z*(t|^x*t) - t*(z|^x*z) - z|^(x+1+1)
          by NEWTON:6
        .= (t+z)|^(x+1+1) -t|^(x+1+1)- z|^(x+1+1) + t*z*(-t|^x - z|^x);
        t*z divides t*z*(-t|^x - z|^x);
        hence thesis by B2,B3,INT_2:1;
      end;
      for m holds P[m] from NAT_1:sch 2(A1,A2);
      hence thesis;
    end;
    hence thesis by A0;
  end;
