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 Th87:
  for n be odd Nat holds t|^n mod 3 = t mod 3
  proof
    let n be odd Nat;
    per cases by Lm3;
    suppose
      A1: t mod 3 = 0; then
      0 = (t mod 3)|^n mod 3;
      hence thesis by A1,GR_CY_3:30;
    end;
    suppose
      A1: t mod 3 = 1; then
      1 = (t mod 3)|^n mod 3;
      hence thesis by A1,GR_CY_3:30;
    end;
    suppose
      A1: t mod 3 = 2; then
      2 = (t mod 3)|^n mod 3 by LmMod;
      hence thesis by A1,GR_CY_3:30;
    end;
  end;
