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 Th55:
  for n be prime Nat, a,b be positive Nat holds
    n*a*b divides (a+b)|^n - (a|^n + b|^n)
  proof
    let n be prime Nat, a,b be positive Nat;
    reconsider g = ((a,b) In_Power n) as FinSequence of NAT by Th1;
    reconsider h = (((a,b) In_Power n)|n)/^1 as FinSequence of NAT;
    A1: for k st k in dom h holds n*a*b divides h.k by Th54;
    Sum g = a|^n + b|^n + Sum h by Th48; then
    (a+b)|^n = Sum h + a|^n + b|^n by NEWTON:30;
    hence thesis by A1,INT_4:36;
  end;
