So, we have the binary relation defined by (x,y)R(u,v) iff xy - uv ≥ 1, where R represents at least as good as.
What I already have is:
For any (x,y), (u,v), (r,t) in X (the alternatives set),
[(xy-uv≥1)^(uv-rt≥1)] -> (xy-rt≥1) I need to prove this.
I started by setting:
(uv≤1+xy)^(uv≥1+rt) -> 1+rt ≤ uv ≤ 1+xy -> 1+rt ≤ 1+xy -> rt ≤ x*y
This last line doesn't prove or disprove if the function is transitive but I don't know what else to do.