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

theorem
for f be PartFunc of REAL m,REAL n, g be PartFunc of REAL-NS m,REAL-NS n
  st f = g & f is_differentiable_on Z
    holds f`|Z = g`|Z
proof
   let f be PartFunc of REAL m,REAL n, g be PartFunc of REAL-NS m,REAL-NS n;
   assume
A1: f = g & f is_differentiable_on Z;
   then A2: g is_differentiable_on Z by PDIFF_6:30; then
A3:dom (g`|Z) = Z by NDIFF_1:def 9;
A4: Z c= dom f by A1,PDIFF_6:def 4; then
A5:dom (f`|Z) = Z by Def1;
   now let x0 be object;
    assume A6: x0 in dom (f`|Z); then
    reconsider x = x0 as Element of REAL m;
    reconsider Z1 = Z as Subset of REAL-NS m by A2,NDIFF_1:30;
    reconsider z = x as Point of REAL-NS m by REAL_NS1:def 4;
    Z1 is open by A2,NDIFF_1:32; then
    f is_differentiable_in x by A1,A5,A6,PDIFF_6:32; then
A7: ex g0 be PartFunc of REAL-NS m,REAL-NS n, y0 be Point of REAL-NS m st
      f = g0 & x = y0 & diff(f,x) = diff(g0,y0) by PDIFF_1:def 8;
    thus (f`|Z).x0 = (f`|Z)/.x by A6,PARTFUN1:def 6
                  .= diff(g,z) by A4,A5,A6,A7,A1,Def1
                  .= (g`|Z)/.x by A6,A5,A2,NDIFF_1:def 9
                  .= (g`|Z).x0 by A6,A5,A3,PARTFUN1:def 6;
   end;
   hence thesis by A5,A3,FUNCT_1:2;
end;
