theorem
  i2 <> 0 implies i1 = (i1 div i2) * i2 + (i1 mod i2)
proof
  assume i2 <> 0;
  then
  (i1 div i2) * i2 +(i1 mod i2) = (i1 div i2 )*i2 + (i1 - ( i1 div i2 )*i2
  ) by Def10
    .= i1;
  hence thesis;
end;
