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 implies
  VFuncdiff(r(#)f1,r(#)f2,r(#)f3,t0) = r*diff(f1,t0)*<e1>
  +r*diff(f2,t0)*<e2>+r*diff(f3,t0)*<e3>
proof
    assume f1 is_differentiable_in t0 &
    f2 is_differentiable_in t0 & f3 is_differentiable_in t0;
 then VFuncdiff(r(#)f1,r(#)f2,r(#)f3,t0) = r*VFuncdiff(f1,f2,f3,t0) by Th90
 .= r*( diff(f1,t0)*<e1>+diff(f2,t0)*<e2>
    +diff(f3,t0)*<e3> ) by Th31
 .= r*( |[ diff(f1,t0),0,0 ]|+diff(f2,t0)*<e2>
    +diff(f3,t0)*<e3> ) by Th21
 .= r*( |[ diff(f1,t0),0,0 ]|+|[ 0,diff(f2,t0),0 ]|
    +diff(f3,t0)*<e3> ) by Th22
 .= r*( (|[ diff(f1,t0),0,0 ]|+|[ 0,diff(f2,t0),0 ]|)
    +|[ 0,0,diff(f3,t0) ]| ) by Th23
 .= r*( |[ diff(f1,t0)+0,0+diff(f2,t0),0+0 ]|
    +|[ 0,0,diff(f3,t0) ]| ) by Lm8
 .= r*|[ diff(f1,t0)+0,diff(f2,t0)+0,0+diff(f3,t0) ]| by Lm8
 .= r*diff(f1,t0)*<e1>+r*diff(f2,t0)*<e2>
    +r*diff(f3,t0)*<e3> by Th32;
   hence thesis;
end;
