theorem
  (Y|`f).:X c= f.:X
proof
  let y be object;
  assume y in (Y|`f).:X;
  then consider x being object such that
A1: x in dom(Y|`f) and
A2: x in X and
A3: y = (Y|`f).x by Def6;
  y = f.x & x in dom f by A1,A3,Th52;
  hence thesis by A2,Def6;
end;
