let am, bp, cm be non pair set ; :: thesis: for dp, cin being set
for s being State of (BitFTA2Circ am,bp,cm,dp,cin)
for a1, a2, a3 being Element of BOOLEAN st a1 = s . am & a2 = s . bp & a3 = s . cm holds
( (Following s,2) . (BitFTA2CarryOutput am,bp,cm,dp,cin) = 'not' (((('not' a1) '&' a2) 'or' (a2 '&' ('not' a3))) 'or' (('not' a3) '&' ('not' a1))) & (Following s,2) . (BitFTA2AdderOutputI am,bp,cm,dp,cin) = (('not' a1) 'xor' a2) 'xor' ('not' a3) )
let dp, cin be set ; :: thesis: for s being State of (BitFTA2Circ am,bp,cm,dp,cin)
for a1, a2, a3 being Element of BOOLEAN st a1 = s . am & a2 = s . bp & a3 = s . cm holds
( (Following s,2) . (BitFTA2CarryOutput am,bp,cm,dp,cin) = 'not' (((('not' a1) '&' a2) 'or' (a2 '&' ('not' a3))) 'or' (('not' a3) '&' ('not' a1))) & (Following s,2) . (BitFTA2AdderOutputI am,bp,cm,dp,cin) = (('not' a1) 'xor' a2) 'xor' ('not' a3) )
let s be State of (BitFTA2Circ am,bp,cm,dp,cin); :: thesis: for a1, a2, a3 being Element of BOOLEAN st a1 = s . am & a2 = s . bp & a3 = s . cm holds
( (Following s,2) . (BitFTA2CarryOutput am,bp,cm,dp,cin) = 'not' (((('not' a1) '&' a2) 'or' (a2 '&' ('not' a3))) 'or' (('not' a3) '&' ('not' a1))) & (Following s,2) . (BitFTA2AdderOutputI am,bp,cm,dp,cin) = (('not' a1) 'xor' a2) 'xor' ('not' a3) )
set S1 = BitGFA2Str am,bp,cm;
set C1 = BitGFA2Circ am,bp,cm;
set A1 = GFA2AdderOutput am,bp,cm;
set A2 = GFA2CarryOutput am,bp,cm;
set S2 = BitGFA1Str (GFA2AdderOutput am,bp,cm),cin,dp;
set C2 = BitGFA1Circ (GFA2AdderOutput am,bp,cm),cin,dp;
let a1, a2, a3 be Element of BOOLEAN ; :: thesis: ( a1 = s . am & a2 = s . bp & a3 = s . cm implies ( (Following s,2) . (BitFTA2CarryOutput am,bp,cm,dp,cin) = 'not' (((('not' a1) '&' a2) 'or' (a2 '&' ('not' a3))) 'or' (('not' a3) '&' ('not' a1))) & (Following s,2) . (BitFTA2AdderOutputI am,bp,cm,dp,cin) = (('not' a1) 'xor' a2) 'xor' ('not' a3) ) )
assume A1: ( a1 = s . am & a2 = s . bp & a3 = s . cm ) ; :: thesis: ( (Following s,2) . (BitFTA2CarryOutput am,bp,cm,dp,cin) = 'not' (((('not' a1) '&' a2) 'or' (a2 '&' ('not' a3))) 'or' (('not' a3) '&' ('not' a1))) & (Following s,2) . (BitFTA2AdderOutputI am,bp,cm,dp,cin) = (('not' a1) 'xor' a2) 'xor' ('not' a3) )
A2: ( am in the carrier of (BitGFA2Str am,bp,cm) & bp in the carrier of (BitGFA2Str am,bp,cm) & cm in the carrier of (BitGFA2Str am,bp,cm) ) by GFACIRC1:118;
reconsider s1 = s | the carrier of (BitGFA2Str am,bp,cm) as State of (BitGFA2Circ am,bp,cm) by FACIRC_1:26;
reconsider t = s as State of ((BitGFA2Circ am,bp,cm) +* (BitGFA1Circ (GFA2AdderOutput am,bp,cm),cin,dp)) ;
A3: ( GFA2AdderOutput am,bp,cm in the carrier of (BitGFA2Str am,bp,cm) & GFA2CarryOutput am,bp,cm in the carrier of (BitGFA2Str am,bp,cm) ) by GFACIRC1:118;
A4: InputVertices (BitGFA2Str am,bp,cm) misses InnerVertices (BitGFA1Str (GFA2AdderOutput am,bp,cm),cin,dp) by LemmaX32;
dom s1 = the carrier of (BitGFA2Str am,bp,cm) by CIRCUIT1:4;
then A5: ( a1 = s1 . am & a2 = s1 . bp & a3 = s1 . cm ) by A1, A2, FUNCT_1:70;
then ( (Following t,2) . (GFA2CarryOutput am,bp,cm) = (Following s1,2) . (GFA2CarryOutput am,bp,cm) & (Following s1,2) . (GFA2CarryOutput am,bp,cm) = 'not' (((('not' a1) '&' a2) 'or' (a2 '&' ('not' a3))) 'or' (('not' a3) '&' ('not' a1))) ) by A3, A4, FACIRC_1:32, GFACIRC1:121;
hence (Following s,2) . (BitFTA2CarryOutput am,bp,cm,dp,cin) = 'not' (((('not' a1) '&' a2) 'or' (a2 '&' ('not' a3))) 'or' (('not' a3) '&' ('not' a1))) ; :: thesis: (Following s,2) . (BitFTA2AdderOutputI am,bp,cm,dp,cin) = (('not' a1) 'xor' a2) 'xor' ('not' a3)
( (Following t,2) . (GFA2AdderOutput am,bp,cm) = (Following s1,2) . (GFA2AdderOutput am,bp,cm) & (Following s1,2) . (GFA2AdderOutput am,bp,cm) = (('not' a1) 'xor' a2) 'xor' ('not' a3) ) by A3, A4, A5, FACIRC_1:32, GFACIRC1:121;
hence (Following s,2) . (BitFTA2AdderOutputI am,bp,cm,dp,cin) = (('not' a1) 'xor' a2) 'xor' ('not' a3) ; :: thesis: verum