 reserve h,h1 for 0-convergent non-zero Real_Sequence,
         c,c1 for constant Real_Sequence;

theorem Th14:
  for f be PartFunc of REAL,REAL, I be open Subset of REAL,
   X be Subset of REAL st I c= X holds
    f is_differentiable_on I iff f|X is_differentiable_on I
proof
    let f be PartFunc of REAL,REAL, I be open Subset of REAL,
    X be Subset of REAL;
    assume
A1:  I c= X;
A2: dom(f|X) = dom f /\ X by RELAT_1:61;
    hereby assume A3: f is_differentiable_on I;
     now let x be Real;
      assume x in I; then
      f|I is_differentiable_in x by A3;
      hence (f|X)|I is_differentiable_in x by A1,RELAT_1:74;
     end;
     hence f|X is_differentiable_on I by A1,A3,A2,XBOOLE_1:19;
    end;
    assume A4: f|X is_differentiable_on I;

    now let x be Real;
     assume x in I; then
     (f|X)|I is_differentiable_in x by A4;
     hence f|I is_differentiable_in x by A1,RELAT_1:74;
    end;
    hence f is_differentiable_on I by A4,A2,XBOOLE_1:18;
end;
