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
  a <= b & f is_integrable_on ['a,b'] & f| ['a,b'] is bounded &
  ['a,b'] c= dom f & c in ['a,b']
  implies f is_integrable_on ['a,c'] & f is_integrable_on ['c,b'] &
  integral(f,a,b) = integral(f,a,c) + integral(f,c,b)
  proof
    assume A1: a <= b & f is_integrable_on ['a,b'] & f| ['a,b'] is bounded &
    ['a,b'] c= dom f & c in ['a,b'];
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;
      hence P*f is_integrable_on ['a,c'] &
      P*f is_integrable_on ['c,b'] &
      integral((P*f),a,b) = integral((P*f),a,c) +
      integral((P*f),c,b) by A4,A5,A1,INTEGRA6:17;
    end; then
    for i be Element of NAT st i in Seg n
    holds (proj(i,n)*f) is_integrable_on ['a,c'];
    hence f is_integrable_on ['a,c'];
    for i be Element of NAT st i in Seg n
    holds (proj(i,n)*f) is_integrable_on ['c,b'] by A2;
    hence f is_integrable_on ['c,b'];
A6: now let i be Nat;
      assume i in dom (integral(f,a,b)); then
A7:   i in Seg n by INTEGR15:def 18;
      set P = proj(i,n);
      thus (integral(f,a,b)).i
      = integral((P*f),a,b) by A7,INTEGR15:def 18
      .= integral((P*f),a,c) + integral((P*f),c,b) by A7,A2
      .= (integral(f,a,c)).i + integral((P*f),c,b) by A7,INTEGR15:def 18
      .= (integral(f,a,c)).i +(integral(f,c,b)).i by A7,INTEGR15:def 18
      .= (integral(f,a,c)+ integral(f,c,b)).i by RVSUM_1:11;
    end;
A8: Seg n = dom (integral(f,a,b)) by INTEGR15:def 18;
    len (integral(f,a,c) + integral(f,c,b)) = n by CARD_1:def 7; then
    Seg n = dom (integral(f,a,c) + integral(f,c,b)) by FINSEQ_1:def 3;
    hence thesis by A8,A6,FINSEQ_1:13;
  end;
