set f = xor2c ;
let x, y, z be set ; :: thesis: ( z <> [<*x,y*>,xor2c ] implies for s being State of (GFA2AdderCirc x,y,z)
for a1a2, a1, a2, a3 being Element of BOOLEAN st a1a2 = s . [<*x,y*>,xor2c ] & a1 = s . x & a2 = s . y & a3 = s . z holds
(Following s) . (GFA2AdderOutput x,y,z) = a1a2 'xor' ('not' a3) )
assume A1: z <> [<*x,y*>,xor2c ] ; :: thesis: for s being State of (GFA2AdderCirc x,y,z)
for a1a2, a1, a2, a3 being Element of BOOLEAN st a1a2 = s . [<*x,y*>,xor2c ] & a1 = s . x & a2 = s . y & a3 = s . z holds
(Following s) . (GFA2AdderOutput x,y,z) = a1a2 'xor' ('not' a3)
set O1 = GFA1AdderOutput x,y,z;
set O2 = GFA2AdderOutput x,y,z;
set A1 = GFA1AdderCirc x,y,z;
set A2 = GFA2AdderCirc x,y,z;
( GFA2AdderCirc x,y,z = GFA1AdderCirc x,y,z & GFA2AdderOutput x,y,z = GFA1AdderOutput x,y,z ) ;
hence for s being State of (GFA2AdderCirc x,y,z)
for a1a2, a1, a2, a3 being Element of BOOLEAN st a1a2 = s . [<*x,y*>,xor2c ] & a1 = s . x & a2 = s . y & a3 = s . z holds
(Following s) . (GFA2AdderOutput x,y,z) = a1a2 'xor' ('not' a3) by A1, Th72; :: thesis: verum