reserve p,q,x,x1,x2,y,y1,y2,z,z1,z2 for set;
reserve A,B,V,X,X1,X2,Y,Y1,Y2,Z for set;
reserve C,C1,C2,D,D1,D2 for non empty set;

theorem
  A c= X & B c= X & chi(A,X) = chi(B,X) implies A = B
proof
  assume that
A1: A c= X and
A2: B c= X and
A3: chi(A,X) = chi(B,X);
 for x being object holds x in A iff x in B
  proof let x be object;
    thus x in A implies x in B
    proof
      assume x in A;
      then chi(A,X).x = 1 by A1,Def3;
      hence thesis by A3,Th36;
    end;
    assume x in B;
    then chi(B,X).x = 1 by A2,Def3;
    hence thesis by A3,Th36;
  end;
  hence thesis by TARSKI:2;
end;
