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 Th5:
  n mod 4 = 0 or n mod 4 = 2 implies (n * m) mod 4 = 0 or (n * m) mod 4 = 2
  proof
    m = 2 * (m div 2) + (m mod 2) by NAT_D:2;
    then m = 2 * (m div 2) + 0 or m = 2 * (m div 2) + 1 by NAT_D:12;
    then
A1: 2*m = 4 * (m div 2) + 0 or 2*m = 4 * (m div 2) + 2*1;
    assume n mod 4 = 0 or n mod 4 = 2;
    then (m * n) mod 4 = (m *0) mod 4 or (m * n) mod 4 = (m * 2) mod 4
    by RADIX_2:3;
    hence thesis by A1,NUMBER02:16;
  end;
