I think that says something like "if x owns a bicycle of brand z, then all bicycles are owned by x and are of brand z)"
Given the way z1 = z2 translates to FOL, perhaps a better preparatory statement would be:
For all members x, there exists (a brand) z, such that all bikes owned by x are of brand z.
I'm trying to write your last statement but the last part (where you noted the problem) is where i'm blocked.
I have to say that "such that all bikes owned by x are of brand z" but i'm only able to write this:
∀y2 bicycle(y2) ∧ owns(x,y2) ∧ brand(y2,z)).
How to say "for each bike owned by x" instead of what i'm wrongly writing "all the bikes are owned by x"?
∀y2 bicycle(y2) ∧ owns(x,y2) → brand(y2,z)) : for example this is not a solution.
I have to find a way to separate "all the bikes" and that "own"