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 Th31:
  a <= b & f is_integrable_on ['a,b'] & f| ['a,b'] is bounded &
  ['a,b'] c= dom f & c in ['a,b'] & d in ['a,b']
  implies integral(f,a,d) = integral(f,a,c) + integral(f,c,d)
  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'] & d 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 integral(P*f,a,d) = integral(P*f,a,c)
      + integral(P*f,c,d) by A4,A5,A1,INTEGRA6:20;
    end;
A6: Seg n = dom (integral(f,a,d)) by INTEGR15:def 18;
A7: now let i0 be Nat;
      assume
A8:   i0 in dom (integral(f,a,d));
      set P = proj(i0,n);
      thus (integral(f,a,d)).i0 = integral(P*f,a,d) by A8,A6,INTEGR15:def 18
      .= integral(P*f,a,c) + integral(P*f,c,d) by A8,A2,A6
      .= (integral(f,a,c)).i0 +integral(P*f,c,d) by A8,A6,INTEGR15:def 18
      .= (integral(f,a,c)).i0 + (integral(f,c,d)).i0 by A8,A6,INTEGR15:def 18
      .= (integral(f,a,c)+ integral(f,c,d)).i0 by RVSUM_1:11;
    end;
    len (integral(f,a,c) + integral(f,c,d)) = n by CARD_1:def 7; then
    Seg n = dom (integral(f,a,c)+ integral(f,c,d)) by FINSEQ_1:def 3;
    hence thesis by A6,A7,FINSEQ_1:13;
  end;
