reserve r, r1, r2, x, y, z,
        x1, x2, x3, y1, y2, y3 for Real;
reserve R, R1, R2, R3 for Element of 3-tuples_on REAL;
reserve p, q, p1, p2, p3, q1, q2 for Element of REAL 3;
reserve f1, f2, f3, g1, g2, g3, h1, h2, h3 for PartFunc of REAL,REAL;
reserve t, t0, t1, t2 for Real;

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)
  implies VFuncdiff(r(#)(f1-g1),r(#)(f2-g2),r(#)(f3-g3),t0) =
  r*VFuncdiff(f1,f2,f3,t0)-r*VFuncdiff(g1,g2,g3,t0)
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;
 f1-g1 is_differentiable_in t0 & f2-g2 is_differentiable_in t0 &
    f3-g3 is_differentiable_in t0 by A1,A2,FDIFF_1:14;
then VFuncdiff(r(#)(f1-g1),r(#)(f2-g2),r(#)(f3-g3),t0)
 = r*VFuncdiff(f1-g1,f2-g2,f3-g3,t0) by Th90
.= r*(VFuncdiff(f1,f2,f3,t0)-VFuncdiff(g1,g2,g3,t0)) by A1,A2,Th89
.= r*VFuncdiff(f1,f2,f3,t0)-r*VFuncdiff(g1,g2,g3,t0) by EUCLIDLP:12;
    hence thesis;
end;
