theorem
  i2 >= 0 implies i1 mod i2 >= 0
proof
  assume
A1: i2 >= 0;
  per cases by A1;
  suppose
A2: i2 > 0;
    [\ i1/i2 /] <= i1/i2 by Def6;
    then (i1 div i2)*i2 <= (i1/i2)*i2 by A2,XREAL_1:64;
    then (i1 div i2)*i2 <= i1 by A2,XCMPLX_1:87;
    then i1 - (i1 div i2)*i2 >= 0 by XREAL_1:48;
    hence thesis by Def10;
  end;
  suppose
    i2 = 0;
    hence thesis by Def10;
  end;
end;
