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 Th16:
  for A be non empty Subset of X, B be set holds B meets dom chi(A
  ,A) implies rng (chi(A,A)|B) = {1}
proof
  let A be non empty Subset of X;
  let B be set;
A1: dom (chi(A,A)|B) = B /\ dom (chi(A,A)) by RELAT_1:61;
  rng (chi(A,A)|B) c= rng (chi(A,A)) by RELAT_1:70;
  then
A2: rng (chi(A,A)|B) c= {1} by Th15;
  assume B /\ dom (chi(A,A)) <> {};
  then rng (chi(A,A)|B) <> {} by A1,RELAT_1:42;
  hence thesis by A2,ZFMISC_1:33;
end;
