 reserve n,i,k,m for Nat;
 reserve p for Prime;

theorem ContCut:
  for Z being open Subset of REAL,
      A being non empty closed_interval Subset of REAL st
      not 0 in Z & A c= Z holds
    ((id Z)^) | A is continuous
  proof
    let Z be open Subset of REAL,
        A be non empty closed_interval Subset of REAL;
    assume
A1: not 0 in Z & A c= Z; then
    (id Z)^ is_differentiable_on Z by FDIFF_5:4; then
    ((id Z)^) | Z is continuous by FDIFF_1:25;
    hence thesis by A1,FCONT_1:16;
  end;
