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 Th27:
  a <= b & f is_integrable_on ['a,b'] & g is_integrable_on ['a,b'] &
  f| ['a,b'] is bounded & g| ['a,b'] is bounded & ['a,b'] c= dom f &
  ['a,b'] c= dom g & c in ['a,b'] & d in ['a,b']
  implies integral(f+g,c,d) = integral(f,c,d) + integral(g,c,d)
  proof
    assume A1: a<=b & f is_integrable_on ['a,b'] &
    g is_integrable_on ['a,b'] & f| ['a,b'] is bounded &
    g| ['a,b'] is bounded & ['a,b'] c= dom f &
    ['a,b'] c= dom g & 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;
A6:   (P*g) is_integrable_on ['a,b'] by A3,A1;
      (P*(g| ['a,b'])) is bounded by A3,A1; then
A7:   (P*g) | ['a,b'] is bounded by RELAT_1:83;
A8:   dom (P)=REAL n by FUNCT_2:def 1; then
      rng f c= dom(P); then
A9:   ['a,b'] c= dom (P*f) by A1,RELAT_1:27;
      rng g c= dom(P) by A8; then
A10:  ['a,b'] c= dom (P*g) by A1,RELAT_1:27;
A11:  P*(f+g)= P*f+P*g by INTEGR15:15;
      thus integral(P*(f+g),c,d) =integral((P*f),c,d) + integral((P*g),c,d)
      by A4,A5,A9,A6,A7,A10,A1,A11,INTEGRA6:24;
    end;
A12: now let i be Nat;
       assume i in dom (integral(f+g,c,d)); then
A13:   i in Seg n by INTEGR15:def 18;
       set P = proj(i,n);
       thus (integral(f+g,c,d)).i
       = integral((P*(f+g)),c,d) by A13,INTEGR15:def 18
       .= integral((P*f),c,d) + integral((P*g),c,d) by A13,A2
       .= (integral(f,c,d)).i +integral((P*g),c,d) by A13,INTEGR15:def 18
       .= (integral(f,c,d)).i +(integral(g,c,d)).i by A13,INTEGR15:def 18
       .= (integral(f,c,d)+ integral(g,c,d)).i by RVSUM_1:11;
     end;
A14: Seg n = dom (integral((f+g),c,d)) by INTEGR15:def 18;
     len (integral(f,c,d)+ integral(g,c,d)) = n by CARD_1:def 7; then
     Seg n = dom (integral(f,c,d) + integral(g,c,d)) by FINSEQ_1:def 3;
     hence thesis by A14,A12,FINSEQ_1:13;
   end;
