reserve i,n,m for Nat;

theorem
for n,m be non zero Nat, g be PartFunc of REAL m,REAL n,
    y0 be Element of REAL m, r be Real st
  g is_differentiable_in y0 holds
   r(#)g is_differentiable_in y0 & diff((r(#)g),y0) = r(#)diff(g,y0)
proof
   let n,m be non zero Nat,
       g be PartFunc of REAL m,REAL n,
       y0 be Element of REAL m, r be Real;
   assume A1: g is_differentiable_in y0;
   reconsider f=g as PartFunc of REAL-NS m,REAL-NS n by Th1;
   reconsider x0=y0 as Point of REAL-NS m by REAL_NS1:def 4;
   f is_differentiable_in x0 by A1,Th2; then
A2:r(#)f is_differentiable_in x0 & diff(r(#)f,x0)=r*diff(f,x0) by NDIFF_1:37;
   r(#)f = r(#)g by Th6;
   hence r(#)g is_differentiable_in y0 by A2,Th2; then
   diff(r(#)g,y0) = diff(r(#)f,x0) by Th3,Th6;
   hence diff(r(#)g,y0) = r(#)diff(g,y0) by A2,Th9,Th3,A1;
end;
