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 Th11:
  a <= c & c <= d & d <= b &
  f is_integrable_on ['a,b'] & f| ['a,b'] is bounded & ['a,b'] c= dom f
  implies
  r(#)f is_integrable_on ['c,d'] & (r(#)f) | ['c,d'] is bounded
  proof
    assume that
A1: a <= c & c <= d & d <= b and
A2: f is_integrable_on ['a,b'] & f| ['a,b'] is bounded and
A3: ['a,b'] c= dom f;
A4: now let i;
      set P = proj(i,n);
      assume
A5:   i in Seg n; then
A6:   P*f is_integrable_on ['a,b'] by A2;
      P*(f| ['a,b']) is bounded by A5,A2; then
A7:   (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
A8:   dom (P*f) = dom f by RELAT_1:27;
A9:  r(#)(P*f) is_integrable_on ['c,d'] & (r(#)(P*f)) | ['c,d'] is bounded
      by A1,A3,Th6,A6,A7,A8;
      (r(#)(P*f)) | ['c,d'] = (P*(r(#)f)) | ['c,d'] by INTEGR15:16
      .= P*((r(#)f) | ['c,d']) by RELAT_1:83;
      hence P*(r(#)f) is_integrable_on ['c,d'] &
      P*((r(#)f) | ['c,d']) is bounded by A9,INTEGR15:16;
    end; then
    for i be Element of NAT st i in Seg n
    holds ((proj(i,n))*(r(#)f)) is_integrable_on ['c,d'];
    hence r(#)f is_integrable_on ['c,d'];
    for i be Element of NAT st i in Seg n
    holds (proj(i,n))*((r(#)f) | ['c,d']) is bounded by A4;
    hence thesis;
  end;
