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
  i in dom D implies ex A1,A2 be non empty closed_interval Subset of REAL
  st A1=[.lower_bound A,D.i .] & A2=[. D.i,upper_bound A.] & A=A1 \/ A2
proof
  assume i in dom D; then
A1: D.i in rng D by FUNCT_1:def 3;
  rng D c= A by Def1;
  then D.i in A by A1;
  then D.i in [.lower_bound A,upper_bound A.] by Th2;
  then D.i in {a: lower_bound A <= a & a <= upper_bound A} by RCOMP_1:def 1;
  then
A2: ex a st a=D.i & lower_bound A <= a & a <= upper_bound A;
  then reconsider
  A1 =[.lower_bound A,D.i .] as non empty closed_interval Subset of REAL
   by MEASURE5:14;
  reconsider A2 = [. D.i,upper_bound A.]
   as non empty closed_interval Subset of REAL by A2,MEASURE5:14;
  take A1, A2;
  A1 \/ A2 = [.lower_bound A,upper_bound A.] by A2,XXREAL_1:174
    .= A by Th2;
  hence thesis;
end;
