reserve a,b,c,d,m,x,n,j,k,l for Nat,
  t,u,v,z for Integer,
  f,F for FinSequence of NAT;
reserve p,q,r,s for real number;

theorem
  |.t.| < a implies t mod a = |.t.| or t mod a = a -|.t.|
  proof
    assume |.t.| < a; then
    A2: t < a & t > -a by SEQ_2:1;
    per cases;
    suppose
      B1: t >= 0; then
      t mod a = t by A2,NAT_D:63;
      hence thesis by B1,ABSVALUE:def 1;
    end;
    suppose
      B1: t < 0; then
      t mod a = a + t by A2,NAT_D:63
      .= a - (-t)
      .= a -|.t.| by B1,ABSVALUE:def 1;
      hence thesis;
    end;
  end;
