reserve a,b,k,m,n,s for Nat;
reserve c,c1,c2,c3 for Complex;
reserve i,j,z for Integer;
reserve p for Prime;
reserve x for object;

theorem Th24:
  m is odd implies 3|^m mod 4 = 3
  proof
    assume m is odd;
    then consider n such that
A1: m = 2*n+1 by ABIAN:9;
    ex k st 3|^(2*n+1) = 4*k+3 by Lm8;
    then 3|^(2*n+1) mod 4 = 3 mod 4 by NAT_D:21;
    hence thesis by A1,NAT_D:24;
  end;
