reserve m,n for non zero Element of NAT;
reserve i,j,k for Element of NAT;
reserve Z for set;

theorem Th52:
for f be PartFunc of REAL m,REAL, r be Real, x be Element of REAL m st
  f is_differentiable_in x holds
   r(#)f is_differentiable_in x & diff(r(#)f,x) = r(#)diff(f,x)
proof
   let f be PartFunc of REAL m,REAL, r be Real, x be Element of REAL m;
   assume f is_differentiable_in x; then
A1:<>*f is_differentiable_in x; then
   r(#)(<>*f) is_differentiable_in x by PDIFF_6:22;
   hence r(#)f is_differentiable_in x by Th8;
   thus diff(r(#)f,x) = proj(1,1)*diff(r(#)(<>*f),x) by Th8
              .= proj(1,1)*(r(#)diff(<>*f,x)) by A1,PDIFF_6:22
              .= r(#)diff(f,x) by INTEGR15:16;
end;
