reserve X for set;
reserve a,b,c,k,m,n for Nat;
reserve i for Integer;
reserve r for Real;
reserve p for Prime;

theorem Th41:
  3 divides 2|^645-2
  proof
    30 = 3*10;
    then
A1: 32,2 are_congruent_mod 3;
    1023 = 3*341;
    then 2|^10,1 are_congruent_mod 3 by Lm10;
    then 2|^10|^64,1|^64 are_congruent_mod 3 by GR_CY_3:34;
    then 2|^640*2|^5,1*2|^5 are_congruent_mod 3 by Lm1122,INT_4:11;
    then 2|^645,2 are_congruent_mod 3 by Lm5,Lm1123,A1,INT_1:15;
    hence thesis;
  end;
