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 Th89:
  for n be odd Nat holds (a+b-c) mod 3 = (a|^n+b|^n-c|^n) mod 3
  proof
    let n be odd Nat;
A1: (a|^n mod 3) = (a mod 3) & (b|^n mod 3) = (b mod 3) & (c|^n mod 3) =
      (c mod 3) by Th87;
    (a+b-c) mod 3 = ((a mod 3) + (b mod 3) + 2*(c mod 3)) mod 3 by LmSum
    .=  (a|^n+b|^n-c|^n) mod 3 by A1,LmSum;
    hence thesis;
  end;
