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) & (f1.t0 <> 0 & f2.t0 <> 0 & f3.t0 <> 0) &
  r <> 0 implies VFuncdiff(r(#)f1^,r(#)f2^,r(#)f3^,t0) =
  - 1/r*|[ diff(f1,t0)/(f1.t0)^2,diff(f2,t0)/(f2.t0)^2,
  diff(f3,t0)/(f3.t0)^2 ]|
proof
    assume that
A1: f1 is_differentiable_in t0 & f2 is_differentiable_in t0 &
    f3 is_differentiable_in t0 and
A2: f1.t0 <> 0 & f2.t0 <> 0 & f3.t0 <> 0 and
A3: r <> 0;
A4: r(#)f1 is_differentiable_in t0 & r(#)f2 is_differentiable_in t0 &
    r(#)f3 is_differentiable_in t0 by A1,FDIFF_1:15;
A5: (r(#)f1).t0 = r*(f1.t0) by VALUED_1:6;
A6: (r(#)f2).t0 = r*(f2.t0) by VALUED_1:6;
A7: (r(#)f3).t0 = r*(f3.t0) by VALUED_1:6;
  then VFuncdiff(r(#)f1^,r(#)f2^,r(#)f3^,t0)
 = - |[ diff(r(#)f1,t0)/((r(#)f1).t0)^2,diff(r(#)f2,t0)/((r(#)f2).t0)^2,
         diff(r(#)f3,t0)/((r(#)f3).t0)^2 ]|
         by A4,A5,A6,A2,A3,Th94
 .= - |[ r*diff(f1,t0)/((r(#)f1).t0)^2,diff(r(#)f2,t0)/((r(#)f2).t0)^2,
         diff(r(#)f3,t0)/((r(#)f3).t0)^2 ]| by A1,FDIFF_1:15
 .= - |[ r*diff(f1,t0)/((r(#)f1).t0)^2,r*diff(f2,t0)/((r(#)f2).t0)^2,
         diff(r(#)f3,t0)/((r(#)f3).t0)^2 ]| by A1,FDIFF_1:15
 .= - |[ (r*diff(f1,t0))/((r(#)f1).t0)^2,(r*diff(f2,t0))/((r(#)f2).t0)^2,
         (r*diff(f3,t0))/((r(#)f3).t0)^2 ]| by A1,FDIFF_1:15
 .= - |[ r*(diff(f1,t0)/((r(#)f1).t0)^2),(r*diff(f2,t0))/((r(#)f2).t0)^2,
         (r*diff(f3,t0))/((r(#)f3).t0)^2 ]| by XCMPLX_1:74
 .= - |[ r*(diff(f1,t0)/((r(#)f1).t0)^2),r*(diff(f2,t0)/((r(#)f2).t0)^2),
         (r*diff(f3,t0))/((r(#)f3).t0)^2 ]| by XCMPLX_1:74
 .= - |[ r*(diff(f1,t0)/((r(#)f1).t0)^2),r*(diff(f2,t0)/((r(#)f2).t0)^2),
         r*(diff(f3,t0)/((r(#)f3).t0)^2) ]| by XCMPLX_1:74
 .= - r*|[ diff(f1,t0)/(r^2*(f1.t0)^2),diff(f2,t0)/(r^2*(f2.t0)^2),
         diff(f3,t0)/(r^2*(f3.t0)^2) ]| by A7,A6,A5,Lm6
 .= - r*|[ diff(f1,t0)/r^2/(f1.t0)^2,diff(f2,t0)/(r^2*(f2.t0)^2),
         diff(f3,t0)/(r^2*(f3.t0)^2) ]| by XCMPLX_1:78
 .= - r*|[ diff(f1,t0)/r^2/(f1.t0)^2,diff(f2,t0)/r^2/(f2.t0)^2,
         diff(f3,t0)/(r^2*(f3.t0)^2) ]| by XCMPLX_1:78
 .= - r*|[ diff(f1,t0)/r^2/(f1.t0)^2,diff(f2,t0)/r^2/(f2.t0)^2,
         diff(f3,t0)/r^2/(f3.t0)^2 ]| by XCMPLX_1:78
 .= - r*|[ diff(f1,t0)/(f1.t0)^2/r^2,diff(f2,t0)/r^2/(f2.t0)^2,
         diff(f3,t0)/r^2/(f3.t0)^2 ]| by XCMPLX_1:48
 .= - r*|[ diff(f1,t0)/(f1.t0)^2/r^2,diff(f2,t0)/(f2.t0)^2/r^2,
         diff(f3,t0)/r^2/(f3.t0)^2 ]| by XCMPLX_1:48
 .= - r*|[ diff(f1,t0)/(f1.t0)^2/r^2,diff(f2,t0)/(f2.t0)^2/r^2,
         diff(f3,t0)/(f3.t0)^2/r^2 ]| by XCMPLX_1:48
 .= - r*|[ (diff(f1,t0)/(f1.t0)^2)/(1/r^2)",diff(f2,t0)/(f2.t0)^2/r^2,
         diff(f3,t0)/(f3.t0)^2/r^2 ]| by XCMPLX_1:217
 .= - r*|[ (1/r^2)*(diff(f1,t0)/(f1.t0)^2),diff(f2,t0)/(f2.t0)^2/r^2,
         diff(f3,t0)/(f3.t0)^2/r^2 ]| by XCMPLX_1:219
 .= - r*|[ (1/r^2)*(diff(f1,t0)/(f1.t0)^2),(diff(f2,t0)/(f2.t0)^2)/(1/r^2)",
         diff(f3,t0)/(f3.t0)^2/r^2 ]| by XCMPLX_1:217
 .= - r*|[ (1/r^2)*(diff(f1,t0)/(f1.t0)^2),(1/r^2)*(diff(f2,t0)/(f2.t0)^2),
         diff(f3,t0)/(f3.t0)^2/r^2 ]| by XCMPLX_1:219
 .= - r*|[ (1/r^2)*(diff(f1,t0)/(f1.t0)^2),(1/r^2)*(diff(f2,t0)/(f2.t0)^2),
         diff(f3,t0)/(f3.t0)^2/(1/r^2)" ]| by XCMPLX_1:217
 .= - r*|[ (1/r^2)*(diff(f1,t0)/(f1.t0)^2),(1/r^2)*(diff(f2,t0)/(f2.t0)^2),
         (1/r^2)*(diff(f3,t0)/(f3.t0)^2) ]| by XCMPLX_1:219
 .= - r*((1/r^2)*|[ diff(f1,t0)/(f1.t0)^2,diff(f2,t0)/(f2.t0)^2,
         diff(f3,t0)/(f3.t0)^2 ]|) by Lm6
 .= - (r*(1/r^2))*|[ diff(f1,t0)/(f1.t0)^2,diff(f2,t0)/(f2.t0)^2,
         diff(f3,t0)/(f3.t0)^2 ]| by EUCLID_4:4
 .= - ((r*1)/r^2)*|[ diff(f1,t0)/(f1.t0)^2,diff(f2,t0)/(f2.t0)^2,
         diff(f3,t0)/(f3.t0)^2 ]| by XCMPLX_1:74
 .= - (r/r/r)*|[ diff(f1,t0)/(f1.t0)^2,diff(f2,t0)/(f2.t0)^2,
         diff(f3,t0)/(f3.t0)^2 ]| by XCMPLX_1:78
 .= - (1/r)*|[ diff(f1,t0)/(f1.t0)^2,diff(f2,t0)/(f2.t0)^2,
         diff(f3,t0)/(f3.t0)^2 ]| by A3,XCMPLX_1:60;
    hence thesis;
end;
