theorem :: FTACELL1:39

for am, bm, cm, dm being non pair set
for cin being set st cin <> [<*dm,(GFA3AdderOutput (am,bm,cm))*>,nor2] & not cin in InnerVertices (BitGFA3Str (am,bm,cm)) holds
for s being State of (BitFTA3Circ (am,bm,cm,dm,cin))
for a1, a2, a3, a4, a5 being Element of BOOLEAN st a1 = s . am & a2 = s . bm & a3 = s . cm & a4 = s . dm & a5 = s . cin holds
( (Following (s,4)) . (BitFTA3AdderOutputP (am,bm,cm,dm,cin)) = 'not' (((((('not' a1) 'xor' ('not' a2)) 'xor' ('not' a3)) '&' ('not' a5)) 'or' (('not' a5) '&' ('not' a4))) 'or' (('not' a4) '&' ((('not' a1) 'xor' ('not' a2)) 'xor' ('not' a3)))) & (Following (s,4)) . (BitFTA3AdderOutputQ (am,bm,cm,dm,cin)) = 'not' ((((('not' a1) 'xor' ('not' a2)) 'xor' ('not' a3)) 'xor' ('not' a4)) 'xor' ('not' a5)) ) by Lm39, Lm40;