Why 2 out of 6 property is stronger then 2 out of 3 property

Question

Why (literally) is $2$ out of $6$ property stronger then $2$ out of $3$ property?

I use this:

if $hg$ and $gf$ are in $W$, so are $f , g, h,$ and $hgf$

if any two of $gf$, $g$ and $f$ are in $W$ then so is the third

It's a bit mind-boggling for me...

Answer 1

Identities are always in $W$. Take the property that

If $hg$ and $gf$ are in $W$, so are $f$ , $g$, $h$, and $hgf$

and set $h$, $g$ and $f$ to be identity in turn. We get

If $g$ and $gf$ are in $W$, so is $f$

If $h$ and $f$ are in $W$, so is $hf$

If $hg$ and $g$ are in $W$, so is $h$