
theorem
  for p be Prime st p > 2 holds p mod 4 = 1 or p mod 4 = 3
proof
  let p be Prime;
  set k = p mod 4;
  assume
A1: p > 2;
A2: now
    assume
A3: k = 0 or k = 2;
    now
      per cases by A3;
      suppose
        k = 0;
        then 4 divides p by INT_1:62;
        hence contradiction by INT_2:29,def 4;
      end;
      suppose
        k = 2;
        then p = (p div 4)*4+2 by INT_1:59
          .= ((p div 4)*2+1)*2;
        then 2 divides p by INT_1:def 3;
        hence contradiction by A1,INT_2:def 4;
      end;
    end;
    hence contradiction;
  end;
  k < 3+1 by NAT_D:62;
  then k <= 0+3 by NAT_1:13;
  then k = 0 or ... or k = 3;
  hence thesis by A2;
end;
