reserve x for object, X,Y for set;
reserve C for non empty set;
reserve c for Element of C;
reserve f,f1,f2,f3,g,g1 for complex-valued Function;
reserve r,p for Complex;

theorem Th4:
  (f^)"{0} = {}
proof
  set x = the Element of (f^)"{0};
  assume
A1: (f^)"{0} <> {};
  then
A2: x in dom (f^) by FUNCT_1:def 7;
  then
A3: x in dom f \ f"{0} by Def2;
  then not x in f"{0} by XBOOLE_0:def 5;
  then
A4: not f.x in {0} by A3,FUNCT_1:def 7;
  ((f^) qua Function).x in {0} by A1,FUNCT_1:def 7;
  then (f^).x = 0 by TARSKI:def 1;
  then (f.x)" = 0 by A2,Def2;
  hence contradiction by A4,TARSKI:def 1,XCMPLX_1:202;
end;
