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 Th6:
  i in dom D implies divset(D,i) c= A
proof
  assume
A1: i in dom D;
  now
    per cases;
    suppose
A2:   i=1;
      lower_bound A in A by RCOMP_1:14;
      then
A3:   lower_bound A in [.lower_bound A,upper_bound A.] by Th2;
A4:   lower_bound divset(D,i) = lower_bound A by A1,A2,Def3;
      consider b such that
A5:   b=D.i;
      upper_bound divset(D,i) = b by A1,A2,A5,Def3;
      then
A6:   divset(D,i)=[. lower_bound A,b .] by A4,Th2;
      b in A by A1,A5,Th4;
      then b in [.lower_bound A,upper_bound A.] by Th2;
      then [. lower_bound A,b .] c= [.lower_bound A,upper_bound A.]
        by A3,XXREAL_2:def 12;
      hence thesis by A6,Th2;
    end;
    suppose
A7:   i<>1;
      set b=D.i;
      b in A by A1,Th4; then
A8:   b in [.lower_bound A,upper_bound A.] by Th2;
      set a=D.(i-1);
      D.(i-1) in A by A1,A7,Th5;
      then a in [.lower_bound A,upper_bound A.] by Th2; then
A9:  [.a,b.] c= [.lower_bound A,upper_bound A.] by A8,XXREAL_2:def 12;
A10:  upper_bound divset(D,i) = b by A1,A7,Def3;
      lower_bound divset(D,i) = a by A1,A7,Def3;
      then divset(D,i)=[.a,b.] by A10,Th2;
      hence thesis by A9,Th2;
    end;
  end;
  hence thesis;
end;
