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 Th4:
  i in dom D implies D.i in A
proof
  assume i in dom D; then
A1: D.i in rng D by FUNCT_1:def 3;
  rng D c= A by Def1;
  hence thesis by A1;
end;
