@elliotokay What about red line vs pink line. Clearly, red > cyan and cyan > pink, by transitivity of partial orders this would imply red>pink. However, red is never over pink directly i.e. for all x st x\in dom(pink) x\not\in dom(red)
@littmath \fhofrg V \fhofrg P_v. Nyfb n+g(o-n) \va P_v sbe nyy g va [0,1]. Fb n+o=n+(1/3)(o-n) + n+ (2/3)(o-n) tvir hf gur gjb ryrzragf gung unir rdhny fhzf naq ner cnvagrq va gur fnzr pbybe.
@littmath Ol gur hapbhagnovyvgl bs E, gurer vf fbzr v va A fhpu gung sbe gur fnzr pbybe tebhc P_v ={k va E | k vf cnvagrq pbybe v}, P_v vf hapbhagnoyr. Ubjrire vs P_v vf hapbhagnoyr va E gura gurer rkvfgf n aba-rzcgl, bcra vagreiny V fhpu gung V \fhofrg P_v. Gnxr n<o \va V. Gura [n,o]