
theorem
for a,b be Real, A be non empty closed_interval Subset of REAL
 st a < b & A = [.a,b.] holds
 ex D be DivSequence of A st
  for n be Nat holds D.n divide_into_equal 2|^n
proof
    let a,b be Real, A be non empty closed_interval Subset of REAL;
    assume that
A1:  a < b and
A2:  A = [.a,b.];

    defpred P[Nat,object] means ex D be Division of A st
     D = $2 & D divide_into_equal 2|^$1;

A3: for n be Element of NAT ex D be Element of divs A st P[n,D]
    proof
     let n be Element of NAT;
     2|^n > 0 by NEWTON:83; then
     consider D1 be Division of A such that
A4:   D1 divide_into_equal 2|^n by A1,A2,Th8;
     D1 is Element of divs A by INTEGRA1:def 3;
     hence thesis by A4;
    end;
    consider D be Function of NAT,divs A such that
A5:  for n be Element of NAT holds P[n,D.n] from FUNCT_2:sch 3(A3);

    reconsider D as DivSequence of A;
    take D;
    thus for n be Nat holds D.n divide_into_equal 2|^n
    proof
     let n be Nat;
     n is Element of NAT by ORDINAL1:def 12; then
     ex d be Division of A st d = D.n & d divide_into_equal 2|^n by A5;
     hence thesis;
    end;
end;
