reserve a,b,c for Integer;
reserve i,j,k,l for Nat;
reserve n for Nat;
reserve a,b,c,d,a1,b1,a2,b2,k,l for Integer;
reserve p,p1,q,l for Nat;

theorem
  for i,j being Integer st i >= 0 & j >= 0 holds
  |.i.| mod |.j.| = i mod j & |.i.| div |.j.| = i div j
proof
  let i,j be Integer;
  assume i >= 0 & j >= 0;
  then |.i.| = i & |.j.| = j by ABSVALUE:def 1;
  hence thesis;
end;
