set f1 = and2c ;
set f2 = and2a ;
set f3 = and2 ;
let x, y, z be set ; :: thesis: ( x <> [<*y,z*>,and2a ] & y <> [<*z,x*>,and2 ] & z <> [<*x,y*>,and2c ] implies for s being State of (GFA1CarryCirc 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) . (GFA1CarryOutput x,y,z) = ((a1 '&' ('not' a2)) 'or' (('not' a2) '&' a3)) 'or' (a3 '&' a1) & (Following s,2) . [<*x,y*>,and2c ] = a1 '&' ('not' a2) & (Following s,2) . [<*y,z*>,and2a ] = ('not' a2) '&' a3 & (Following s,2) . [<*z,x*>,and2 ] = a3 '&' a1 ) )
assume A1: ( x <> [<*y,z*>,and2a ] & y <> [<*z,x*>,and2 ] & z <> [<*x,y*>,and2c ] ) ; :: thesis: for s being State of (GFA1CarryCirc 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) . (GFA1CarryOutput x,y,z) = ((a1 '&' ('not' a2)) 'or' (('not' a2) '&' a3)) 'or' (a3 '&' a1) & (Following s,2) . [<*x,y*>,and2c ] = a1 '&' ('not' a2) & (Following s,2) . [<*y,z*>,and2a ] = ('not' a2) '&' a3 & (Following s,2) . [<*z,x*>,and2 ] = a3 '&' a1 )
set S = GFA1CarryStr x,y,z;
reconsider x9 = x, y9 = y, z9 = z as Vertex of (GFA1CarryStr x,y,z) by Th55;
let s be State of (GFA1CarryCirc 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) . (GFA1CarryOutput x,y,z) = ((a1 '&' ('not' a2)) 'or' (('not' a2) '&' a3)) 'or' (a3 '&' a1) & (Following s,2) . [<*x,y*>,and2c ] = a1 '&' ('not' a2) & (Following s,2) . [<*y,z*>,and2a ] = ('not' a2) '&' a3 & (Following s,2) . [<*z,x*>,and2 ] = a3 '&' a1 )
y in InputVertices (GFA1CarryStr x,y,z) by A1, Th57;
then A2: (Following s) . y9 = s . y by CIRCUIT2:def 5;
z in InputVertices (GFA1CarryStr x,y,z) by A1, Th57;
then A3: (Following s) . z9 = s . z by CIRCUIT2:def 5;
set xy = [<*x,y*>,and2c ];
set yz = [<*y,z*>,and2a ];
set zx = [<*z,x*>,and2 ];
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) . (GFA1CarryOutput x,y,z) = ((a1 '&' ('not' a2)) 'or' (('not' a2) '&' a3)) 'or' (a3 '&' a1) & (Following s,2) . [<*x,y*>,and2c ] = a1 '&' ('not' a2) & (Following s,2) . [<*y,z*>,and2a ] = ('not' a2) '&' a3 & (Following s,2) . [<*z,x*>,and2 ] = a3 '&' a1 ) )
assume A5: ( a1 = s . x & a2 = s . y & a3 = s . z ) ; :: thesis: ( (Following s,2) . (GFA1CarryOutput x,y,z) = ((a1 '&' ('not' a2)) 'or' (('not' a2) '&' a3)) 'or' (a3 '&' a1) & (Following s,2) . [<*x,y*>,and2c ] = a1 '&' ('not' a2) & (Following s,2) . [<*y,z*>,and2a ] = ('not' a2) '&' a3 & (Following s,2) . [<*z,x*>,and2 ] = a3 '&' a1 )
A6: (Following s) . [<*z,x*>,and2 ] = a3 '&' a1 by A5, Th59;
( (Following s) . [<*x,y*>,and2c ] = a1 '&' ('not' a2) & (Following s) . [<*y,z*>,and2a ] = ('not' a2) '&' a3 ) by A5, Th59;
hence (Following s,2) . (GFA1CarryOutput x,y,z) = ((a1 '&' ('not' a2)) 'or' (('not' a2) '&' a3)) 'or' (a3 '&' a1) by A4, A6, Th60; :: thesis: ( (Following s,2) . [<*x,y*>,and2c ] = a1 '&' ('not' a2) & (Following s,2) . [<*y,z*>,and2a ] = ('not' a2) '&' a3 & (Following s,2) . [<*z,x*>,and2 ] = a3 '&' a1 )
x in InputVertices (GFA1CarryStr x,y,z) by A1, Th57;
then (Following s) . x9 = s . x by CIRCUIT2:def 5;
hence ( (Following s,2) . [<*x,y*>,and2c ] = a1 '&' ('not' a2) & (Following s,2) . [<*y,z*>,and2a ] = ('not' a2) '&' a3 & (Following s,2) . [<*z,x*>,and2 ] = a3 '&' a1 ) by A5, A2, A3, A4, Th59; :: thesis: verum