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 Th10:
  t divides ((t+z)|^n - z|^n) & z divides ((t+z)|^n - t|^n)
  proof
    defpred P[Nat] means t divides ((t+z)|^$1 - z|^$1)
    & z divides ((t+z)|^$1 - t|^$1);
    A1: P[0]
    proof
      (t+z)|^0 = 1 & z|^0 = 1 & t|^0 = 1 by NEWTON:4;
      hence thesis by INT_1:def 4,INT_1:11;
    end;
    A2: P[x] implies P[x+1]
    proof
      assume
      A2a: P[x]; then
      A4: t divides (t+z)|^x*z -z|^(x+1) by Lm2;
      A5: z divides (t+z)|^x*t -t|^(x+1) by Lm2,A2a;
      A6: t divides (t+z)|^x*t & z divides (t+z)|^x*z; then
      A7: t divides (t+z)|^x*t+((t+z)|^x*z - z|^(x+1)) by WSIERP_1:4,A4;
      A3: (t+z)|^(x+1) = (t+z)|^x*(t+z)|^1 by NEWTON:8
      .= (t+z)|^x*t+(t+z)|^x*z;
      z divides (t+z)|^x*z+((t+z)|^x*t - t|^(x+1)) by WSIERP_1:4,A5,A6;
      hence thesis by A3,A7;
    end;
    for m holds P[m] from NAT_1:sch 2(A1,A2);
    hence thesis;
  end;
