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,2)) . (GFA2AdderOutput (x,y,z)) = (a1 'xor' ('not' a2)) 'xor' ('not' a3) & (Following (s,2)) . [<*x,y*>,xor2c] = a1 'xor' ('not' a2) & (Following (s,2)) . x = a1 & (Following (s,2)) . y = a2 & (Following (s,2)) . 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,2)) . (GFA2AdderOutput (x,y,z)) = (a1 'xor' ('not' a2)) 'xor' ('not' a3) & (Following (s,2)) . [<*x,y*>,xor2c] = a1 'xor' ('not' a2) & (Following (s,2)) . x = a1 & (Following (s,2)) . y = a2 & (Following (s,2)) . z = 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 a1, a2, a3 being Element of BOOLEAN st a1 = s . x & a2 = s . y & a3 = s . z holds
( (Following (s,2)) . (GFA2AdderOutput (x,y,z)) = (a1 'xor' ('not' a2)) 'xor' ('not' a3) & (Following (s,2)) . [<*x,y*>,xor2c] = a1 'xor' ('not' a2) & (Following (s,2)) . x = a1 & (Following (s,2)) . y = a2 & (Following (s,2)) . z = a3 ) by A1, Th61; :: thesis: verum