theorem
  for f being X-valued Function st x in dom f holds f.x is Element of X
 proof let f be X-valued Function;
  assume x in dom f;
   then
A1: f.x in rng f by Def3;
   rng f c= X by RELAT_1:def 19;
  hence f.x is Element of X by A1;
 end;
