reserve s1,s2,q1 for Real_Sequence;
reserve n for Element of NAT;
reserve a,b for Real;

theorem Th11:
for j being Element of NAT
for A being non empty closed_interval Subset of REAL
for D1 being Division of A st j in dom D1 holds
vol (divset (D1,j)) <= delta D1
proof
  let j be Element of NAT;
  let A be non empty closed_interval Subset of REAL;
  let D1 be Division of A;
  assume A1: j in dom D1;
  then j in Seg (len D1) by FINSEQ_1:def 3;
  then j in Seg (len (upper_volume ((chi (A,A)),D1))) by INTEGRA1:def 6;
  then j in dom (upper_volume ((chi (A,A)),D1)) by FINSEQ_1:def 3;
  then (upper_volume ((chi (A,A)),D1)) . j
  in rng (upper_volume ((chi (A,A)),D1)) by FUNCT_1:def 3;
  then (upper_volume ((chi (A,A)),D1)) . j
       <= max (rng (upper_volume ((chi (A,A)),D1))) by XXREAL_2:def 8;
  hence vol (divset (D1,j)) <= delta D1 by A1,INTEGRA1:20;
end;
