theorem :: FTACELL1:19

for ap, bm, cp, dm being non pair set
for cin being set st cin <> [<*dm,(GFA1AdderOutput (ap,bm,cp))*>,nor2] & not cin in InnerVertices (BitGFA1Str (ap,bm,cp)) holds
for s being State of (BitFTA1Circ (ap,bm,cp,dm,cin))
for a1, a2, a3, a4, a5 being Element of BOOLEAN st a1 = s . ap & a2 = s . bm & a3 = s . cp & a4 = s . dm & a5 = s . cin holds
( (Following (s,4)) . (BitFTA1AdderOutputP (ap,bm,cp,dm,cin)) = 'not' (((((a1 'xor' ('not' a2)) 'xor' a3) '&' a5) 'or' (a5 '&' ('not' a4))) 'or' (('not' a4) '&' ((a1 'xor' ('not' a2)) 'xor' a3))) & (Following (s,4)) . (BitFTA1AdderOutputQ (ap,bm,cp,dm,cin)) = (((a1 'xor' ('not' a2)) 'xor' a3) 'xor' ('not' a4)) 'xor' a5 ) by Lm19, Lm20;