
theorem ::: see NUMBER02:24
  for n being Nat holds
    4 divides n or 4 divides n+1 or 4 divides n+2 or 4 divides n+3
  proof
    let n be Nat;
    assume
CC: not (4 divides n or 4 divides n+1 or 4 divides n+2 or 4 divides n+3);
    consider k being Nat such that
C1: n = 4*k or n = 4*k+1 or n = 4*k+2 or n = 4*k+3 by NUMBER02:24;
    per cases by C1;
    suppose n = 4 * k;
      hence thesis by CC;
    end;
    suppose
c2:   n = 4 * k + 1;
      4 divides 4 * (k + 1);
      hence thesis by c2,CC;
    end;
    suppose
c2:   n = 4 * k + 2;
      4 divides 4 * (k + 1);
      hence thesis by c2,CC;
    end;
    suppose
c2:   n = 4 * k + 3;
      4 divides 4 * (k + 1);
      hence thesis by c2,CC;
    end;
  end;
