F1: 12 is a non-V facet of 012 and 124; (02,012),(24,124) are V-pairs  =>  l(12) < l(012) = l(02) and l(12) < l(124) = l(24):  in every admissible L, 12 precedes 02 and 24
F2: boundaries {'124': ['12', '14', '24'], '024': ['02', '04', '24'], '123': ['12', '13', '23'], '012': ['01', '02', '12']}
    04 lies in the boundary of 024 only  =>  (04,123) in P(L) forces 024 to precede 123 in L
F3: elements of d024 + a*d124 + g*d012 + d*d123:
    (a,g,d)=(0,0,0): ['02', '04', '24']
    (a,g,d)=(0,0,1): ['02', '04', '12', '13', '23', '24']
    (a,g,d)=(0,1,0): ['01', '04', '12', '24']
    (a,g,d)=(0,1,1): ['01', '04', '13', '23', '24']
    (a,g,d)=(1,0,0): ['02', '04', '12', '14']
    (a,g,d)=(1,0,1): ['02', '04', '13', '14', '23']
    (a,g,d)=(1,1,0): ['01', '04', '14']   <- avoids 02 and 24
    (a,g,d)=(1,1,1): ['01', '04', '12', '13', '14', '23']   <- avoids 02 and 24
    cosets whose top element could be 12 (12 in z, 02 and 24 not in z, both are younger than 12): [(1, 1, 1)]
    => (12,024) in P(L) forces 123 to precede 024 in L (d = 1).  Together with F2: 024 < 123 < 024, contradiction.
