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 a1, a2, a3 being Element of BOOLEAN st a1 = s . x & a2 = s . y & a3 = s . z holds
( (Following s) . [<*x,y*>,xor2c] = a1 'xor' ('not' a2) & (Following s) . x = a1 & (Following s) . y = a2 & (Following s) . z = a3 ) )
assume A1: z <> [<*x,y*>,xor2c] ; :: thesis: for s being State of (GFA2AdderCirc (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*>,xor2c] = a1 'xor' ('not' a2) & (Following s) . x = a1 & (Following s) . y = a2 & (Following s) . z = a3 )
set A1 = GFA1AdderCirc (x,y,z);
GFA2AdderCirc (x,y,z) = GFA1AdderCirc (x,y,z) ;
hence for s being State of (GFA2AdderCirc (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*>,xor2c] = a1 'xor' ('not' a2) & (Following s) . x = a1 & (Following s) . y = a2 & (Following s) . z = a3 ) by A1, Th59; :: thesis: verum