
theorem Th9:
  for X,x being set, f being Function st x in X holds (x in dom f
implies (X-indexing f).x = f.x) & (not x in dom f implies (X-indexing f).x = x)
proof
  let X,x be set, f be Function;
  assume
A1: x in X;
  then
A2: (id X).x = x by FUNCT_1:18;
  (X-indexing f).x = ((id X) +* f).x by A1,Th8;
  hence thesis by A2,FUNCT_4:11,13;
end;
