reserve D for set;
reserve x,x0,x1,x2,y,y0,y1,y2,z,z0,z1,z2,r,s,t for Real;
reserve p,a,u,u0 for Element of REAL 3;
reserve n,m,k for Element of NAT;
reserve f,f1,f2,f3,g for PartFunc of REAL 3,REAL;
reserve R,R1,R2 for RestFunc;
reserve L,L1,L2 for LinearFunc;

theorem Th37:
  (f is_partial_differentiable_in p,1 & f is_partial_differentiable_in p,2 &
  f is_partial_differentiable_in p,3) implies grad(r(#)f,p) = r*grad(f,p)
proof
    assume
A1: f is_partial_differentiable_in p,1 & f is_partial_differentiable_in p,2 &
    f is_partial_differentiable_in p,3;
   reconsider r as Real;
    grad(r(#)f,p)
    = |[ partdiff(r(#)f,p,1),partdiff(r(#)f,p,2),partdiff(r(#)f,p,3) ]|
      by Th34
   .= |[ r*partdiff(f,p,1),partdiff(r(#)f,p,2),partdiff(r(#)f,p,3) ]|
     by A1,PDIFF_1:33
   .= |[ r*partdiff(f,p,1),r*partdiff(f,p,2),partdiff(r(#)f,p,3)]|
     by A1,PDIFF_1:33
   .= |[ r*partdiff(f,p,1),r*partdiff(f,p,2),r*partdiff(f,p,3) ]|
     by A1,PDIFF_1:33
   .= r*|[ partdiff(f,p,1),partdiff(f,p,2),partdiff(f,p,3) ]| by EUCLID_8:59
   .= r*grad(f,p) by Th34;
    hence thesis;
end;
