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 A2 = GFA2AdderCirc x,y,z;
set A1 = GFA1AdderCirc x,y,z;
set O2 = GFA2AdderOutput x,y,z;
set O1 = GFA1AdderOutput 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, Th73; :: thesis: verum