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 Th3:
  c in dom (f^) implies f.c <> 0
proof
  assume that
A1: c in dom (f^) and
A2: f.c = 0;
A3: c in dom f \ f"{0} by A1,Def2;
  then
A4: not c in f"{0} by XBOOLE_0:def 5;
  now
    per cases by A4,FUNCT_1:def 7;
    suppose
      not c in dom f;
      hence contradiction by A3;
    end;
    suppose
      not f.c in {0};
      hence contradiction by A2,TARSKI:def 1;
    end;
  end;
  hence contradiction;
end;
