reserve i,n,m for Nat;

theorem
for n,m be non zero Nat, g1,g2 be PartFunc of REAL m,REAL n,
    y0 be Element of REAL m st
  g1 is_differentiable_in y0 & g2 is_differentiable_in y0 holds
    g1+g2 is_differentiable_in y0 &
    diff(g1+g2,y0) = diff(g1,y0) + diff(g2,y0)
proof
   let n,m be non zero Nat,
       g1,g2 be PartFunc of REAL m,REAL n,
       y0 be Element of REAL m;
   assume A1: g1 is_differentiable_in y0 & g2 is_differentiable_in y0;
   reconsider f1=g1 as PartFunc of REAL-NS m,REAL-NS n by Th1;
   reconsider f2=g2 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;
   f1 is_differentiable_in x0 & f2 is_differentiable_in x0 by A1,Th2; then
A2:f1+f2 is_differentiable_in x0 & diff(f1+f2,x0)=diff(f1,x0)+diff(f2,x0)
     by NDIFF_1:35;
   f1+f2 = g1+g2 by Th4;
   hence g1+g2 is_differentiable_in y0 by A2,Th2; then
A3:diff(g1+g2,y0) = diff(f1+f2,x0) by Th4,Th3;
   diff(f1,x0) = diff(g1,y0) & diff(f2,x0) = diff(g2,y0) by Th3,A1;
   hence diff(g1+g2,y0) = diff(g1,y0)+diff(g2,y0) by A2,A3,Th7;
end;
