reserve F for RealNormSpace;
reserve G for RealNormSpace;
reserve y,X for set;
reserve x,x0,x1,x2,g,g1,g2,r,r1,s,p,p1,p2 for Real;
reserve i,m,k for Element of NAT;
reserve n,k for non zero Element of NAT;
reserve Y for Subset of REAL;
reserve Z for open Subset of REAL;
reserve s1,s3 for Real_Sequence;
reserve seq,seq1 for sequence of G;
reserve f,f1,f2 for PartFunc of REAL,REAL n;
reserve g,g1,g2 for PartFunc of REAL,REAL-NS n;
reserve h for 0-convergent non-zero Real_Sequence;
reserve c for constant Real_Sequence;

theorem Th21:
  f is_differentiable_on X & Z c= X implies f is_differentiable_on Z
proof
  assume A1: f is_differentiable_on X & Z c= X;
  then
A2: X c= dom f & for x st x in X holds f|X is_differentiable_in x;
A3: Z c= dom f by A1,A2;
  reconsider g=f as PartFunc of REAL,REAL-NS n by REAL_NS1:def 4;
  now
    let x;
    assume x in X;
    then f|X is_differentiable_in x by A1;
    hence g|X is_differentiable_in x;
  end;
  then g is_differentiable_on X by A2,NDIFF_3:def 5;
  then
A4: g is_differentiable_on Z by A1,NDIFF_3:24;
  now
    let x;
    assume x in Z;
    then g|Z is_differentiable_in x by A4,NDIFF_3:def 5;
    hence f|Z is_differentiable_in x;
  end;
  hence thesis by A3;
end;
