set f1 = and2a ;
set f2 = and2c ;
set f3 = nor2 ;
let x, y, z be set ; :: thesis: ( x <> [<*y,z*>,and2c] & y <> [<*z,x*>,nor2] & z <> [<*x,y*>,and2a] implies for s being State of (GFA2CarryCirc (x,y,z))
for a1, a2, a3 being Element of BOOLEAN st a1 = s . x & a2 = s . y & a3 = s . z holds
( (Following (s,2)) . (GFA2CarryOutput (x,y,z)) = 'not' (((('not' a1) '&' a2) 'or' (a2 '&' ('not' a3))) 'or' (('not' a3) '&' ('not' a1))) & (Following (s,2)) . [<*x,y*>,and2a] = ('not' a1) '&' a2 & (Following (s,2)) . [<*y,z*>,and2c] = a2 '&' ('not' a3) & (Following (s,2)) . [<*z,x*>,nor2] = ('not' a3) '&' ('not' a1) ) )
assume A1: ( x <> [<*y,z*>,and2c] & y <> [<*z,x*>,nor2] & z <> [<*x,y*>,and2a] ) ; :: thesis: for s being State of (GFA2CarryCirc (x,y,z))
for a1, a2, a3 being Element of BOOLEAN st a1 = s . x & a2 = s . y & a3 = s . z holds
( (Following (s,2)) . (GFA2CarryOutput (x,y,z)) = 'not' (((('not' a1) '&' a2) 'or' (a2 '&' ('not' a3))) 'or' (('not' a3) '&' ('not' a1))) & (Following (s,2)) . [<*x,y*>,and2a] = ('not' a1) '&' a2 & (Following (s,2)) . [<*y,z*>,and2c] = a2 '&' ('not' a3) & (Following (s,2)) . [<*z,x*>,nor2] = ('not' a3) '&' ('not' a1) )
set S = GFA2CarryStr (x,y,z);
reconsider x9 = x, y9 = y, z9 = z as Vertex of (GFA2CarryStr (x,y,z)) by Th79;
let s be State of (GFA2CarryCirc (x,y,z)); :: thesis: for a1, a2, a3 being Element of BOOLEAN st a1 = s . x & a2 = s . y & a3 = s . z holds
( (Following (s,2)) . (GFA2CarryOutput (x,y,z)) = 'not' (((('not' a1) '&' a2) 'or' (a2 '&' ('not' a3))) 'or' (('not' a3) '&' ('not' a1))) & (Following (s,2)) . [<*x,y*>,and2a] = ('not' a1) '&' a2 & (Following (s,2)) . [<*y,z*>,and2c] = a2 '&' ('not' a3) & (Following (s,2)) . [<*z,x*>,nor2] = ('not' a3) '&' ('not' a1) )
y in InputVertices (GFA2CarryStr (x,y,z)) by A1, Th81;
then A2: (Following s) . y9 = s . y by CIRCUIT2:def 5;
z in InputVertices (GFA2CarryStr (x,y,z)) by A1, Th81;
then A3: (Following s) . z9 = s . z by CIRCUIT2:def 5;
set xy = [<*x,y*>,and2a];
set yz = [<*y,z*>,and2c];
set zx = [<*z,x*>,nor2];
A4: Following (s,2) = Following (Following s) by FACIRC_1:15;
let a1, a2, a3 be Element of BOOLEAN ; :: thesis: ( a1 = s . x & a2 = s . y & a3 = s . z implies ( (Following (s,2)) . (GFA2CarryOutput (x,y,z)) = 'not' (((('not' a1) '&' a2) 'or' (a2 '&' ('not' a3))) 'or' (('not' a3) '&' ('not' a1))) & (Following (s,2)) . [<*x,y*>,and2a] = ('not' a1) '&' a2 & (Following (s,2)) . [<*y,z*>,and2c] = a2 '&' ('not' a3) & (Following (s,2)) . [<*z,x*>,nor2] = ('not' a3) '&' ('not' a1) ) )
assume A5: ( a1 = s . x & a2 = s . y & a3 = s . z ) ; :: thesis: ( (Following (s,2)) . (GFA2CarryOutput (x,y,z)) = 'not' (((('not' a1) '&' a2) 'or' (a2 '&' ('not' a3))) 'or' (('not' a3) '&' ('not' a1))) & (Following (s,2)) . [<*x,y*>,and2a] = ('not' a1) '&' a2 & (Following (s,2)) . [<*y,z*>,and2c] = a2 '&' ('not' a3) & (Following (s,2)) . [<*z,x*>,nor2] = ('not' a3) '&' ('not' a1) )
A6: (Following s) . [<*z,x*>,nor2] = ('not' a3) '&' ('not' a1) by A5, Th83;
( (Following s) . [<*x,y*>,and2a] = ('not' a1) '&' a2 & (Following s) . [<*y,z*>,and2c] = a2 '&' ('not' a3) ) by A5, Th83;
hence (Following (s,2)) . (GFA2CarryOutput (x,y,z)) = 'not' (((('not' a1) '&' a2) 'or' (a2 '&' ('not' a3))) 'or' (('not' a3) '&' ('not' a1))) by A4, A6, Th84; :: thesis: ( (Following (s,2)) . [<*x,y*>,and2a] = ('not' a1) '&' a2 & (Following (s,2)) . [<*y,z*>,and2c] = a2 '&' ('not' a3) & (Following (s,2)) . [<*z,x*>,nor2] = ('not' a3) '&' ('not' a1) )
x in InputVertices (GFA2CarryStr (x,y,z)) by A1, Th81;
then (Following s) . x9 = s . x by CIRCUIT2:def 5;
hence ( (Following (s,2)) . [<*x,y*>,and2a] = ('not' a1) '&' a2 & (Following (s,2)) . [<*y,z*>,and2c] = a2 '&' ('not' a3) & (Following (s,2)) . [<*z,x*>,nor2] = ('not' a3) '&' ('not' a1) ) by A5, A2, A3, A4, Th83; :: thesis: verum