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 Th23:
  m is even implies 3|^m mod 4 = 1
  proof
    assume m is even;
    then consider n such that
A1: m = 2*n;
    ex k st 3|^(2*n) = 4*k+1 by Lm7;
    then 3|^(2*n) mod 4 = 1 mod 4 by NAT_D:21;
    hence thesis by A1,NAT_D:24;
  end;
