theorem Th7:
  f is_differentiable_on X implies r(#)f is_differentiable_on X
proof
  reconsider r1 = r as Real;
  assume
A1: f is_differentiable_on X;
  then reconsider Z = X as Subset of REAL by FDIFF_1:8;
  reconsider Z as open Subset of REAL by A1,FDIFF_1:10;
  X c= dom f by A1;
  then Z c= dom(r(#)f) by VALUED_1:def 5;
  then (r1(#)f) is_differentiable_on X by A1,FDIFF_1:20;
  hence thesis;
end;
