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 a,b,c be odd Nat holds
    3 divides t|^a + t|^b + t|^c
  proof
    let a,b,c be odd Nat;
    A1: t|^a mod 3 = t mod 3 & t|^b mod 3 = t mod 3 & t|^c mod 3 = t mod 3
      by Th87;
    (t|^a + t|^b + t|^c) mod 3 = (t|^a + t|^b + (t|^c mod 3)) mod 3 by Th88
    .= ((t|^c mod 3) + t|^a + t|^b) mod 3
    .= ((t|^c mod 3) + t|^a + (t|^b mod 3))mod 3 by Th88
    .= ((t|^c mod 3) + (t|^b mod 3) + t|^a)mod 3
    .= ((t|^c mod 3) + (t|^b mod 3) + (t|^a mod 3))mod 3 by Th88
    .= (0+3*(t mod 3)) mod 3 by A1
    .= 0;
    hence thesis by INT_1:62;
  end;
