let am, bm, cm be non pair set ; :: thesis: for dm, cin being set
for s being State of (BitFTA3Circ am,bm,cm,dm,cin)
for a1, a2, a3 being Element of BOOLEAN st a1 = s . am & a2 = s . bm & a3 = s . cm holds
( (Following s,2) . (BitFTA3CarryOutput am,bm,cm,dm,cin) = 'not' (((('not' a1) '&' ('not' a2)) 'or' (('not' a2) '&' ('not' a3))) 'or' (('not' a3) '&' ('not' a1))) & (Following s,2) . (BitFTA3AdderOutputI am,bm,cm,dm,cin) = 'not' ((('not' a1) 'xor' ('not' a2)) 'xor' ('not' a3)) )
let dm, cin be set ; :: thesis: for s being State of (BitFTA3Circ am,bm,cm,dm,cin)
for a1, a2, a3 being Element of BOOLEAN st a1 = s . am & a2 = s . bm & a3 = s . cm holds
( (Following s,2) . (BitFTA3CarryOutput am,bm,cm,dm,cin) = 'not' (((('not' a1) '&' ('not' a2)) 'or' (('not' a2) '&' ('not' a3))) 'or' (('not' a3) '&' ('not' a1))) & (Following s,2) . (BitFTA3AdderOutputI am,bm,cm,dm,cin) = 'not' ((('not' a1) 'xor' ('not' a2)) 'xor' ('not' a3)) )
let s be State of (BitFTA3Circ am,bm,cm,dm,cin); :: thesis: for a1, a2, a3 being Element of BOOLEAN st a1 = s . am & a2 = s . bm & a3 = s . cm holds
( (Following s,2) . (BitFTA3CarryOutput am,bm,cm,dm,cin) = 'not' (((('not' a1) '&' ('not' a2)) 'or' (('not' a2) '&' ('not' a3))) 'or' (('not' a3) '&' ('not' a1))) & (Following s,2) . (BitFTA3AdderOutputI am,bm,cm,dm,cin) = 'not' ((('not' a1) 'xor' ('not' a2)) 'xor' ('not' a3)) )
set S1 = BitGFA3Str am,bm,cm;
set C1 = BitGFA3Circ am,bm,cm;
set A1 = GFA3AdderOutput am,bm,cm;
set A2 = GFA3CarryOutput am,bm,cm;
set S2 = BitGFA3Str (GFA3AdderOutput am,bm,cm),cin,dm;
set C2 = BitGFA3Circ (GFA3AdderOutput am,bm,cm),cin,dm;
let a1, a2, a3 be Element of BOOLEAN ; :: thesis: ( a1 = s . am & a2 = s . bm & a3 = s . cm implies ( (Following s,2) . (BitFTA3CarryOutput am,bm,cm,dm,cin) = 'not' (((('not' a1) '&' ('not' a2)) 'or' (('not' a2) '&' ('not' a3))) 'or' (('not' a3) '&' ('not' a1))) & (Following s,2) . (BitFTA3AdderOutputI am,bm,cm,dm,cin) = 'not' ((('not' a1) 'xor' ('not' a2)) 'xor' ('not' a3)) ) )
assume A1: ( a1 = s . am & a2 = s . bm & a3 = s . cm ) ; :: thesis: ( (Following s,2) . (BitFTA3CarryOutput am,bm,cm,dm,cin) = 'not' (((('not' a1) '&' ('not' a2)) 'or' (('not' a2) '&' ('not' a3))) 'or' (('not' a3) '&' ('not' a1))) & (Following s,2) . (BitFTA3AdderOutputI am,bm,cm,dm,cin) = 'not' ((('not' a1) 'xor' ('not' a2)) 'xor' ('not' a3)) )
A2: ( am in the carrier of (BitGFA3Str am,bm,cm) & bm in the carrier of (BitGFA3Str am,bm,cm) & cm in the carrier of (BitGFA3Str am,bm,cm) ) by GFACIRC1:155;
reconsider s1 = s | the carrier of (BitGFA3Str am,bm,cm) as State of (BitGFA3Circ am,bm,cm) by FACIRC_1:26;
reconsider t = s as State of ((BitGFA3Circ am,bm,cm) +* (BitGFA3Circ (GFA3AdderOutput am,bm,cm),cin,dm)) ;
A3: ( GFA3AdderOutput am,bm,cm in the carrier of (BitGFA3Str am,bm,cm) & GFA3CarryOutput am,bm,cm in the carrier of (BitGFA3Str am,bm,cm) ) by GFACIRC1:155;
A4: InputVertices (BitGFA3Str am,bm,cm) misses InnerVertices (BitGFA3Str (GFA3AdderOutput am,bm,cm),cin,dm) by LemmaX42;
dom s1 = the carrier of (BitGFA3Str am,bm,cm) by CIRCUIT1:4;
then A5: ( a1 = s1 . am & a2 = s1 . bm & a3 = s1 . cm ) by A1, A2, FUNCT_1:70;
then ( (Following t,2) . (GFA3CarryOutput am,bm,cm) = (Following s1,2) . (GFA3CarryOutput am,bm,cm) & (Following s1,2) . (GFA3CarryOutput am,bm,cm) = 'not' (((('not' a1) '&' ('not' a2)) 'or' (('not' a2) '&' ('not' a3))) 'or' (('not' a3) '&' ('not' a1))) ) by A3, A4, FACIRC_1:32, GFACIRC1:158;
hence (Following s,2) . (BitFTA3CarryOutput am,bm,cm,dm,cin) = 'not' (((('not' a1) '&' ('not' a2)) 'or' (('not' a2) '&' ('not' a3))) 'or' (('not' a3) '&' ('not' a1))) ; :: thesis: (Following s,2) . (BitFTA3AdderOutputI am,bm,cm,dm,cin) = 'not' ((('not' a1) 'xor' ('not' a2)) 'xor' ('not' a3))
( (Following t,2) . (GFA3AdderOutput am,bm,cm) = (Following s1,2) . (GFA3AdderOutput am,bm,cm) & (Following s1,2) . (GFA3AdderOutput am,bm,cm) = 'not' ((('not' a1) 'xor' ('not' a2)) 'xor' ('not' a3)) ) by A3, A4, A5, FACIRC_1:32, GFACIRC1:158;
hence (Following s,2) . (BitFTA3AdderOutputI am,bm,cm,dm,cin) = 'not' ((('not' a1) 'xor' ('not' a2)) 'xor' ('not' a3)) ; :: thesis: verum