theorem
  (f1 is_differentiable_in t0 & f2 is_differentiable_in t0 &
  f3 is_differentiable_in t0) & (g1 is_differentiable_in t0 &
  g2 is_differentiable_in t0 & g3 is_differentiable_in t0) &
  (g1.t0 <> 0 & g2.t0 <> 0 & g3.t0 <> 0) implies
  VFuncdiff(f1/g1,f2/g2,f3/g3,t0) =
  |[ (diff(f1,t0)*(g1.t0)-diff(g1,t0)*(f1.t0))/(g1.t0)^2,
  (diff(f2,t0)*(g2.t0)-diff(g2,t0)*(f2.t0))/(g2.t0)^2,
  (diff(f3,t0)*(g3.t0)-diff(g3,t0)*(f3.t0))/(g3.t0)^2  ]|
proof
    assume that
A1: f1 is_differentiable_in t0 & f2 is_differentiable_in t0 &
    f3 is_differentiable_in t0 and
A2: g1 is_differentiable_in t0 & g2 is_differentiable_in t0 &
    g3 is_differentiable_in t0 and
A3: g1.t0 <> 0 & g2.t0 <> 0 & g3.t0 <> 0;
    VFuncdiff(f1/g1,f2/g2,f3/g3,t0)
    = |[ (diff(f1,t0)*(g1.t0)-diff(g1,t0)*(f1.t0))/(g1.t0)^2,
      diff(f2/g2,t0),diff(f3/g3,t0) ]| by A1,A2,A3,FDIFF_2:14
   .= |[ (diff(f1,t0)*(g1.t0)-diff(g1,t0)*(f1.t0))/(g1.t0)^2,
      (diff(f2,t0)*(g2.t0)-diff(g2,t0)*(f2.t0))/(g2.t0)^2,
      diff(f3/g3,t0) ]| by A1,A2,A3,FDIFF_2:14
   .= |[ (diff(f1,t0)*(g1.t0)-diff(g1,t0)*(f1.t0))/(g1.t0)^2,
      (diff(f2,t0)*(g2.t0)-diff(g2,t0)*(f2.t0))/(g2.t0)^2,
      (diff(f3,t0)*(g3.t0)-diff(g3,t0)*(f3.t0))/(g3.t0)^2 ]|
      by A1,A2,A3,FDIFF_2:14;
    hence thesis;
end;
