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,2) . (GFA3AdderOutput x,y,z) = 'not' ((('not' a1) 'xor' ('not' a2)) 'xor' ('not' 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,2) . (GFA3AdderOutput x,y,z) = 'not' ((('not' a1) 'xor' ('not' a2)) 'xor' ('not' a3))
set A = GFA3AdderCirc x,y,z;
let s be State of (GFA3AdderCirc x,y,z); :: thesis: for a1, a2, a3 being Element of BOOLEAN st a1 = s . x & a2 = s . y & a3 = s . z holds
(Following s,2) . (GFA3AdderOutput x,y,z) = 'not' ((('not' a1) 'xor' ('not' a2)) 'xor' ('not' a3))
let a1, a2, a3 be Element of BOOLEAN ; :: thesis: ( a1 = s . x & a2 = s . y & a3 = s . z implies (Following s,2) . (GFA3AdderOutput x,y,z) = 'not' ((('not' a1) 'xor' ('not' a2)) 'xor' ('not' a3)) )
assume ( a1 = s . x & a2 = s . y & a3 = s . z ) ; :: thesis: (Following s,2) . (GFA3AdderOutput x,y,z) = 'not' ((('not' a1) 'xor' ('not' a2)) 'xor' ('not' a3))
hence (Following s,2) . (GFA3AdderOutput x,y,z) = (a1 'xor' a2) 'xor' a3 by A1, Th147
.= 'not' ((('not' a1) 'xor' ('not' a2)) 'xor' ('not' a3)) by XBOOLEAN:74 ;
:: thesis: verum