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;
reserve a,b,c,d,m,x,n,k,l for Nat,
  t,z for Integer,
  f,F,G for FinSequence of REAL;
reserve q,r,s for real number;
reserve D for set;

theorem
  for n be prime Nat, a,b be positive Nat holds
    n*a*b divides (a+b)|^(k*n) - (a|^n + b|^n)|^k
proof
  let n be prime Nat, a,b be positive Nat;
  (a+b)|^n - (a|^n + b|^n) divides ((a+b)|^n)|^k - (a|^n + b|^n)|^k
    by NEWTON01:33; then
  A1: (a+b)|^n - (a|^n + b|^n) divides (a+b)|^(k*n) - (a|^n + b|^n)|^k
    by NEWTON:9;
  n*a*b divides (a+b)|^n - (a|^n + b|^n) by Th55;
  hence thesis by A1,INT_2:9;
end;
