reserve x,y,X,Y for set;
reserve C for non empty set;
reserve c for Element of C;
reserve f,f1,f2,f3,g,g1 for PartFunc of C,COMPLEX;
reserve r1,r2,p1 for Real;
reserve r,q,cr1,cr2 for Complex;

theorem Th9:
  (f^)"{0} = {}
proof
  set x = the Element of (f^)"{0c};
  assume
A1: (f^)"{0} <> {};
  then
A2: x in dom (f^) by Lm1;
A3: (f^)/.x in {0c} by A1,Lm1;
  reconsider x as Element of C by A2;
  x in dom f \ f"{0c} by A2,Def2;
  then x in dom f & not x in f"{0c} by XBOOLE_0:def 5;
  then
A4: not f/.x in {0c} by PARTFUN2:26;
  (f^)/.x = 0c by A3,TARSKI:def 1;
  then (f/.x)" = 0c by A2,Def2;
  hence contradiction by A4,TARSKI:def 1,XCMPLX_1:202;
end;
