reserve a,a1,b,b1,x,y for Real,
  F,G,H for FinSequence of REAL,
  i,j,k,n,m for Element of NAT,
  I for Subset of REAL,
  X for non empty set,
  x1,R,s for set;
reserve A for non empty closed_interval Subset of REAL;
reserve A, B for non empty closed_interval Subset of REAL;
reserve r for Real;
reserve D, D1, D2 for Division of A;
reserve f, g for Function of A,REAL;

theorem Th18:
  i in dom D implies vol(divset(D,i))=upper_volume(chi(A,A),D).i
proof
A1: dom (chi(A,A)) = A by FUNCT_3:def 3;
  assume
A2: i in dom D; then
A3: upper_volume(chi(A,A),D).i= (upper_bound (rng (chi(A,A)|divset(D,i))))*
  vol(divset(D,i)) by Def5;
  divset(D,i) c= A by A2,Th6;
  then divset(D,i) c= divset(D,i) /\ dom (chi(A,A)) by A1,XBOOLE_1:19;
  then divset(D,i) /\ dom (chi(A,A)) <> {};
  then divset(D,i) meets dom (chi(A,A));
  then
A4: rng (chi(A,A)|divset(D,i)) = {1} by Th16;
A5: rng chi(A,A) = {1} by Th15;
  then upper_bound rng (chi(A,A)) = 1 by SEQ_4:9;
  hence thesis by A3,A5,A4;
end;
