set f1 = and2 ;
set f2 = and2 ;
set f3 = and2 ;
let x, y, z be set ; :: thesis:
assume A1: ( x <> [<*y,z*>,and2 ] & y <> [<*z,x*>,and2 ] & z <> [<*x,y*>,and2 ] ) ; :: thesis:
set xy = [<*x,y*>,and2 ];
set yz = [<*y,z*>,and2 ];
set zx = [<*z,x*>,and2 ];
let s be State of (GFA0CarryCirc x,y,z); :: thesis:
let a1, a2, a3 be Element of BOOLEAN ; :: thesis:
assume A2: ( a1 = s . x & a2 = s . y & a3 = s . z ) ; :: thesis:
set S = GFA0CarryStr x,y,z;
reconsider x' = x, y' = y, z' = z as Vertex of (GFA0CarryStr x,y,z) by Th19;
( x in InputVertices (GFA0CarryStr x,y,z) & y in InputVertices (GFA0CarryStr x,y,z) & z in InputVertices (GFA0CarryStr x,y,z) ) by A1, Th21;
then A3: ( (Following s) . x' = s . x & (Following s) . y' = s . y & (Following s) . z' = s . z ) by CIRCUIT2:def 5;
A4: Following s,2 = Following (Following s) by FACIRC_1:15;
( (Following s) . [<*x,y*>,and2 ] = a1 '&' a2 & (Following s) . [<*y,z*>,and2 ] = a2 '&' a3 & (Following s) . [<*z,x*>,and2 ] = a3 '&' a1 ) by A2, Th23;
hence (Following s,2) . (GFA0CarryOutput x,y,z) = ((a1 '&' a2) 'or' (a2 '&' a3)) 'or' (a3 '&' a1) by A4, Th24; :: thesis:
thus ( (Following s,2) . [<*x,y*>,and2 ] = a1 '&' a2 & (Following s,2) . [<*y,z*>,and2 ] = a2 '&' a3 & (Following s,2) . [<*z,x*>,and2 ] = a3 '&' a1 ) by A2, A3, A4, Th23; :: thesis: