reserve S,T for RealNormSpace;
reserve f,f1,f2 for PartFunc of S,T;
reserve Z for Subset of S;
reserve i,n for Nat;

theorem Th15:
f is_differentiable_on 1,Z
  iff
Z c= dom f & f|Z is_differentiable_on Z
proof
   hereby assume A1: f is_differentiable_on 1,Z;
    hence Z c= dom f;
A2: diff(f,0,Z) is_differentiable_on Z by A1,Th14;
    diff(f,Z).0 = f|Z by Def5;
    hence f|Z is_differentiable_on Z by A2,Th7;
   end;
   assume A3: Z c= dom f & f|Z is_differentiable_on Z;
   for i be Nat st i <= 1-1 holds
    diff(f,i,Z) is_differentiable_on Z
   proof
    let i be Nat;
    assume i <= 1-1; then
A4: i = 0;
    f|Z = diff(f,0,Z) by Def5;
    hence diff(f,i,Z) is_differentiable_on Z by A4,A3,Th7;
   end;
   hence f is_differentiable_on 1,Z by A3,Th14;
end;
