theorem :: FINTOPO6:57

for FT being non empty RelStr
for g being FinSequence of FT
for A being Subset of FT
for x1, x2 being Element of FT st g is_minimum_path_in A,x1,x2 & FT is filled & FT is symmetric & x1 <> x2 holds
( ( for i being Nat st 1 < i & i < len g holds
(rng g) /\ (U_FT (g /. i)) = {(g . (i -' 1)),(g . i),(g . (i + 1))} ) & (rng g) /\ (U_FT (g /. 1)) = {(g . 1),(g . 2)} & (rng g) /\ (U_FT (g /. (len g))) = {(g . ((len g) -' 1)),(g . (len g))} )