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

theorem Th53:
for f be PartFunc of REAL m,REAL,
    g be PartFunc of REAL m,REAL 1 st <>*f = g holds
  Z c= dom f & f is_differentiable_on Z iff g is_differentiable_on Z
proof
   let f be PartFunc of REAL m,REAL, g be PartFunc of REAL m,REAL 1;
   assume A1: <>*f=g;
A2:dom <>*f = dom f by Th3;
   hereby
     assume
A3:  Z c= dom f;
     assume
A4:  f is_differentiable_on Z;
A5:Z c= dom g by A3,Th3,A1;
    now let x be Element of REAL m;
     assume x in Z; then
     f|Z is_differentiable_in x by A4;
     hence g|Z is_differentiable_in x by A1,Th5;
    end;
    hence g is_differentiable_on Z by A5,PDIFF_6:def 4;
   end;
   assume A6: g is_differentiable_on Z;
   hence Z c= dom f by A2,A1,PDIFF_6:def 4;
   hereby let x be Element of REAL m;
    assume x in Z; then
    g|Z is_differentiable_in x by A6,PDIFF_6:def 4;
    hence f|Z is_differentiable_in x by A1,Th5;
   end;
end;
