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 Th25:
  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(r(#)f,c,d) = r*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
A6:   ['a,b'] c= dom (P*f) by A1,RELAT_1:27;
      P*(r(#)f)= r(#)(P*f) by INTEGR15:16;
      hence integral(P*(r(#)f),c,d) = r*integral((P*f),c,d)
      by A4,A5,A6,A1,INTEGRA6:25;
    end;
A7: now let i be Nat;
      assume i in dom (integral(r(#)f,c,d)); then
A8:   i in Seg n by INTEGR15:def 18;
      set P = proj(i,n);
      thus (integral(r(#)f,c,d)).i = integral((P*(r(#)f)),c,d)
      by A8,INTEGR15:def 18
      .= r*integral((P*f),c,d) by A8,A2
      .= r*(integral(f,c,d)).i by A8,INTEGR15:def 18
      .= (r*integral(f,c,d)).i by RVSUM_1:44;
    end;
A9: Seg n = dom (integral((r(#)f),c,d)) by INTEGR15:def 18;
     len (r*integral(f,c,d)) = n by CARD_1:def 7; then
     Seg n = dom (r*integral(f,c,d)) by FINSEQ_1:def 3;
     hence integral(r(#)f,c,d)=r*integral(f,c,d) by A9,A7,FINSEQ_1:13;
   end;
