reserve X for set, x,y,z for object,
  k,l,n for Nat,
  r for Real;
reserve i,i0,i1,i2,i3,i4,i5,i8,i9,j for Integer;
reserve r1,p,p1,g,g1,g2 for Real,
  Y for Subset of REAL;

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;
