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 a1, a2, a3 being Element of BOOLEAN st a1 = s . x & a2 = s . y & a3 = s . z holds
( (Following s) . [<*x,y*>,xor2 ] = a1 'xor' a2 & (Following s) . x = a1 & (Following s) . y = a2 & (Following s) . z = a3 ) )
assume A1: z <> [<*x,y*>,xor2 ] ; :: thesis: for s being State of (GFA3AdderCirc x,y,z)
for a1, a2, a3 being Element of BOOLEAN st a1 = s . x & a2 = s . y & a3 = s . z holds
( (Following s) . [<*x,y*>,xor2 ] = a1 'xor' a2 & (Following s) . x = a1 & (Following s) . y = a2 & (Following s) . z = a3 )
set A3 = GFA3AdderCirc x,y,z;
set A0 = GFA0AdderCirc x,y,z;
GFA3AdderCirc x,y,z = GFA0AdderCirc x,y,z ;
hence for s being State of (GFA3AdderCirc x,y,z)
for a1, a2, a3 being Element of BOOLEAN st a1 = s . x & a2 = s . y & a3 = s . z holds
( (Following s) . [<*x,y*>,xor2 ] = a1 'xor' a2 & (Following s) . x = a1 & (Following s) . y = a2 & (Following s) . z = a3 ) by A1, Th35; :: thesis: verum