reserve X for set;
reserve n,i for Element of NAT;
reserve a,b,c,d,e,r,x0 for Real;
reserve A for non empty closed_interval Subset of REAL;
reserve f,g,h for PartFunc of REAL,REAL n;
reserve E for Element of REAL n;

theorem Th9:
  a <= c & c <= d & d <= b & f is_integrable_on ['a,b'] &
  f| ['a,b'] is bounded & ['a,b'] c= dom f
  implies f is_integrable_on ['c,d'] & f| ['c,d'] is bounded
  proof
    assume A1: a <= c & c <= d & d <= b & f is_integrable_on ['a,b'] &
    f| ['a,b'] is bounded & ['a,b'] c= dom f;
A2: now let i;
      set P = proj(i,n);
      assume A3: i in Seg n; then
A4:   P*f is_integrable_on ['a,b'] by A1;
      (P*(f| ['a,b'])) is bounded by A3,A1; then
A5:   (P*f) | ['a,b'] is bounded by RELAT_1:83;
      dom (P)=REAL n by FUNCT_2:def 1; then
      rng f c= dom(P); then
      dom (P*f) = dom f by RELAT_1:27; then
      P*f is_integrable_on ['c,d'] & (P*f) | ['c,d'] is bounded &
      ['c,d'] c= dom (P*f) by A4,A5,A1,INTEGRA6:18;
      hence P*f is_integrable_on ['c,d'] & P*(f| ['c,d']) is bounded
      by RELAT_1:83;
    end; then
    for i be Element of NAT st i in Seg n holds
    proj(i,n)*f is_integrable_on ['c,d'];
    hence f is_integrable_on ['c,d'];
    for i be Element of NAT st i in Seg n
    holds (proj(i,n))*(f| ['c,d']) is bounded by A2;
    hence thesis;
  end;
