reserve i,n,m for Nat;

theorem Th3:
for n,m be non zero Nat, f be PartFunc of REAL m,REAL n,
    g be PartFunc of REAL-NS m,REAL-NS n,
    x be Element of REAL m, y be Point of REAL-NS m st
 f=g & x=y & f is_differentiable_in x holds diff(f,x) = diff(g,y)
proof
   let n,m be non zero Nat,
       f be PartFunc of REAL m,REAL n,
       g be PartFunc of REAL-NS m,REAL-NS n,
       x be Element of REAL m, y be Point of REAL-NS m;
   assume A1: f=g & x=y & f is_differentiable_in x; then
   ex g be PartFunc of REAL-NS m,REAL-NS n, y be Point of REAL-NS m st
    f=g & x=y & diff(f,x) = diff(g,y) by PDIFF_1:def 8;
   hence thesis by A1;
end;
