reserve X for set;
reserve a,b,c,k,m,n for Nat;
reserve i,j for Integer;
reserve r for Real;
reserve p,p1,p2 for Prime;

theorem
  4*n mod 8 = 0 or 4*n mod 8 = 4
  proof
A1: n mod (7+1) = 0 or ... or n mod (7+1) = 7 by NUMBER03:11;
A2: 1*8+0 mod 8 = 0 & 1*8+4 mod 8 = 4 & 2*8+0 mod 8 = 0 & 2*8+4 mod 8 = 4
    & 3*8+0 mod 8 = 0 & 3*8+4 mod 8 = 4 by NAT_D:def 2;
    4 mod 8 = 4 by NAT_D:24;
    then 4*n mod (7+1) = 4*0 mod 8 or ... or 4*n mod (7+1) = 4*7 mod 8
    by A1,NAT_D:67;
    hence 4*n mod 8 = 0 or 4*n mod 8 = 4 by A2;
  end;
