reserve a,b,c,d,e,x,r for Real,
  A for non empty closed_interval Subset of REAL,
  f,g for PartFunc of REAL,REAL;

theorem Th4:
  for A,B,C be set st A c= B & A c= C holds chi(B,B)|A = chi(C,C)|A
proof
  let A,B,C be set;
  assume that
A1: A c= B and
A2: A c= C;
A3: now
A4: dom(chi(C,C)|A) =dom(chi(C,C)) /\ A by RELAT_1:61;
    assume
A5: A is non empty;
    then C is non empty by A2,XBOOLE_1:58,61;
    then dom(chi(C,C)|A) =C /\ A by A4,RFUNCT_1:61;
    then
A6: dom(chi(C,C)|A) =A by A2,XBOOLE_1:28;
A7: dom(chi(B,B)|A) =dom(chi(B,B)) /\ A by RELAT_1:61;
    B is non empty by A1,A5,XBOOLE_1:58,61;
    then dom(chi(B,B)|A) =B /\ A by A7,RFUNCT_1:61;
    then
A8: dom(chi(B,B)|A) =A by A1,XBOOLE_1:28;
    now
      let x be object;
      assume
A9:   x in A;
      then (chi(B,B)|A).x =(chi(B,B)).x by A8,FUNCT_1:47;
      then (chi(B,B)|A).x = 1 by A1,A9,RFUNCT_1:61;
      then (chi(B,B)|A).x = (chi(C,C)).x by A2,A9,RFUNCT_1:61;
      hence (chi(B,B)|A).x = (chi(C,C)|A).x by A6,A9,FUNCT_1:47;
    end;
    hence thesis by A8,A6,FUNCT_1:2;
  end;
  now
    assume
A10: A is empty;
    then chi(B,B)|A = {};
    hence thesis by A10;
  end;
  hence thesis by A3;
end;
