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
  for n be odd Nat holds 3 divides (a+b-c) iff 3 divides (a|^n+b|^n-c|^n)
  proof
    let n be odd Nat;
    thus 3 divides (a+b-c) implies 3 divides (a|^n+b|^n-c|^n)
    proof
      assume 3 divides a+b-c; then
      a+b-c mod 3 = 0 by INT_1:62; then
      (a|^n+b|^n-c|^n) mod 3 = 0 by Th89;
      hence thesis by INT_1:62;
    end;
    assume 3 divides a|^n+b|^n-c|^n; then
    a|^n+b|^n-c|^n mod 3 = 0 by INT_1:62; then
    (a+b-c) mod 3 = 0 by Th89;
    hence thesis by INT_1:62;
  end;
