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
  len D1 = 1 implies D1 <= D2
proof
A1: D2.(len D2) = upper_bound A by Def1;
  assume
A2: len D1 = 1;
  then D1.1 = upper_bound A by Def1;
  then D1=<*upper_bound A*> by A2,FINSEQ_1:40;
  then
A3: rng D1 = {upper_bound A} by FINSEQ_1:38;
A4: len D2 in Seg(len D2) by FINSEQ_1:3;
  hence len D1 <= len D2 by A2,FINSEQ_1:1;
  len D2 in dom D2 by A4,FINSEQ_1:def 3;
  then upper_bound A in rng D2 by A1,FUNCT_1:def 3;
  then rng D1 is Subset of rng D2 by A3,SUBSET_1:41;
  hence thesis;
end;
