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 Th90:
  (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*VFuncdiff(f1,f2,f3,t0)
proof
    assume
A1: f1 is_differentiable_in t0 & f2 is_differentiable_in t0 &
    f3 is_differentiable_in t0;
    set p = |[ diff(f1,t0),diff(f2,t0),diff(f3,t0) ]|;
    VFuncdiff(r(#)f1,r(#)f2,r(#)f3,t0)
    = |[ r*diff(f1,t0),diff(r(#)f2,t0),diff(r(#)f3,t0) ]| by A1,FDIFF_1:15
   .= |[ r*diff(f1,t0),r*diff(f2,t0),diff(r(#)f3,t0) ]| by A1,FDIFF_1:15
   .= |[ r*p.1,r*p.2,r*p.3 ]| by A1,FDIFF_1:15
   .= r*VFuncdiff(f1,f2,f3,t0) by Lm1;
    hence thesis;
end;
