
theorem Th15:
for A be non empty closed_interval Subset of REAL, D be Division of A,
 n,k be Nat st D divide_into_equal n & k in dom D
 holds vol divset(D,k) = (vol A)/n
proof
   let A be non empty closed_interval Subset of REAL, D be Division of A,
       n,k be Nat;
   assume that
A1: D divide_into_equal n and
A2: k in dom D;

A3:len D = n by A1,INTEGRA4:def 1; then
A4:D.k = lower_bound A + (vol A)/n*k by A1,A2,INTEGRA4:def 1;

   per cases;
   suppose A5: k = 1; then
    lower_bound divset(D,k) = lower_bound A
  & upper_bound divset(D,k) = D.k by A2,INTEGRA1:def 4; then
    vol divset(D,k) = D.k - lower_bound A by INTEGRA1:def 5;
    hence vol divset(D,k) = (vol A)/n by A4,A5;
   end;
   suppose A6: k <> 1; then
A7: lower_bound divset(D,k) = D.(k-1)
  & upper_bound divset(D,k) = D.k by A2,INTEGRA1:def 4;

A8: 1 <= k <= len D by A2,FINSEQ_3:25; then
A9: 1 < k by A6,XXREAL_0:1; then
    reconsider j = k-1 as Nat by NAT_1:20;
    k = j+1; then
    1 <= j <= k by A9,NAT_1:19; then
    1 <= j <= len D by A8,XXREAL_0:2; then
    j in dom D by FINSEQ_3:25; then
A10: D.j = lower_bound A + (vol A)/n*j by A1,A3,INTEGRA4:def 1;
    vol divset(D,k) = D.k - D.j by A7,INTEGRA1:def 5
     .= (vol A)/n*1 by A4,A10;
    hence vol divset(D,k) = (vol A)/n;
   end;
end;
