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

theorem
for X be Subset of REAL m,
  f be PartFunc of REAL m,REAL, g be PartFunc of REAL m,REAL 1 st
 <>*f = g & X c= dom f & f is_differentiable_on X
  holds g is_differentiable_on X
    & for x be Element of REAL m st x in X
        holds ((f`|X)/.x) = proj(1,1)*((g`|X)/.x)
proof
   let X be Subset of REAL m;
   let f be PartFunc of REAL m,REAL, g be PartFunc of REAL m,REAL 1;
   assume A1: <>*f = g & X c= dom f & f is_differentiable_on X;
   hence g is_differentiable_on X by Th53;
A2: dom f = dom <>*f by Th3;
   let x be Element of REAL m;
   assume A3: x in X; then
   (f`|X)/.x = diff(f,x) by A1,Def4;
   hence thesis by A2,A1,A3,Def1;
end;
