reserve i,j,k,n,n1,n2,m for Nat;
reserve a,r,x,y for Real;
reserve A for non empty closed_interval Subset of REAL;
reserve C for non empty set;
reserve X for set;

theorem Th1:
  for D being Division of A st vol(A)=0 holds len D=1
proof
  let D be Division of A;
  assume that
A1: vol(A)=0 and
A2: len D<>1;
  rng D <> {};
  then
A3: 1 in dom D by FINSEQ_3:32;
  then
A4: 1 <= len D by FINSEQ_3:25;
  then
A5: len D in dom D by FINSEQ_3:25;
  D.1 in A by A3,INTEGRA1:6;
  then
A6: lower_bound A <= D.1 by INTEGRA2:1;
  1 < len D by A2,A4,XXREAL_0:1;
  then
A7: D.1 < D.(len D) by A3,A5,SEQM_3:def 1;
  upper_bound A-lower_bound A=0 by A1,INTEGRA1:def 5;
  hence contradiction by A7,A6,INTEGRA1:def 2;
end;
