set f = xor2 ;
let x, y, z be set ; :: thesis: ( z <> [<*x,y*>,xor2] implies for s being State of (GFA3AdderCirc (x,y,z))
for a1a2, a1, a2, a3 being Element of BOOLEAN st a1a2 = s . [<*x,y*>,xor2] & a1 = s . x & a2 = s . y & a3 = s . z holds
(Following s) . (GFA3AdderOutput (x,y,z)) = a1a2 'xor' a3 )
assume A1: z <> [<*x,y*>,xor2] ; :: thesis: for s being State of (GFA3AdderCirc (x,y,z))
for a1a2, a1, a2, a3 being Element of BOOLEAN st a1a2 = s . [<*x,y*>,xor2] & a1 = s . x & a2 = s . y & a3 = s . z holds
(Following s) . (GFA3AdderOutput (x,y,z)) = a1a2 'xor' a3
set O0 = GFA0AdderOutput (x,y,z);
set O3 = GFA3AdderOutput (x,y,z);
set A0 = GFA0AdderCirc (x,y,z);
set A3 = GFA3AdderCirc (x,y,z);
( GFA3AdderCirc (x,y,z) = GFA0AdderCirc (x,y,z) & GFA3AdderOutput (x,y,z) = GFA0AdderOutput (x,y,z) ) ;
hence for s being State of (GFA3AdderCirc (x,y,z))
for a1a2, a1, a2, a3 being Element of BOOLEAN st a1a2 = s . [<*x,y*>,xor2] & a1 = s . x & a2 = s . y & a3 = s . z holds
(Following s) . (GFA3AdderOutput (x,y,z)) = a1a2 'xor' a3 by A1, Th29; :: thesis: verum