reserve n for Element of NAT,
  p,q,r,s for Element of HP-WFF;

theorem Th7:
  for X being set, A being Subset of X holds
  ((0,1) --> (1,0))*chi(A,X) = chi(A`,X)
proof
  let X be set, A be Subset of X;
  set f = ((0,1) --> (1,0))*chi(A,X);
A1: dom chi(A,X) = X by FUNCT_3:def 3;
A2: for x being object st x in X holds (x in A` implies f.x = 1) &
    (not x in A` implies f.x = 0)
  proof
    let x be object such that
A3: x in X;
    thus x in A` implies f.x = 1
    proof
      assume x in A`;
      then not x in A by XBOOLE_0:def 5;
      then chi(A,X).x = 0 by A3,FUNCT_3:def 3;
      then f.x = ((0,1) --> (1,0)).0 by A1,A3,FUNCT_1:13;
      hence thesis by FUNCT_4:63;
    end;
    assume not x in A`;
    then x in A by A3,XBOOLE_0:def 5;
    then chi(A,X).x = 1 by FUNCT_3:def 3;
    then f.x = ((0,1) --> (1,0)).1 by A1,A3,FUNCT_1:13;
    hence thesis by FUNCT_4:63;
  end;
  dom((0,1) --> (1,0)) = {0,1} by FUNCT_4:62;
  then rng chi(A,X) c= dom((0,1) --> (1,0)) by FUNCT_3:39;
  hence thesis by A2,FUNCT_3:def 3,A1,RELAT_1:27;
end;
