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,c be positive Nat holds
  n*a*b divides c-b implies n*a*b divides (a|^n + b|^n) - (a+c)|^n
  proof
    let n be prime Nat, a,b,c be positive Nat;
    assume n*a*b divides c-b; then
    n*a*b divides -(b-c); then
    A1:  n*a*b divides b-c by INT_2:10;
    (a+b) - (a+c)  divides (a+b)|^n - (a+c)|^n by NEWTON01:33; then
    A3: n*a*b divides (a+b)|^n - (a+c)|^n by A1,INT_2:9;
    n*a*b divides -((a+b)|^n - (a|^n + b|^n)) by INT_2:10,Th55; then
    n*a*b divides (a+b)|^n - (a+c)|^n + -((a+b)|^n - (a|^n + b|^n))
      by A3,WSIERP_1:4;
    hence thesis;
  end;
