set f = xor2c ;
let x, y, z be set ; :: thesis: ( z <> [<*x,y*>,xor2c] implies for s being State of (GFA1AdderCirc (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)) . (GFA1AdderOutput (x,y,z)) = 'not' ((a1 'xor' ('not' a2)) 'xor' a3) )
assume A1: z <> [<*x,y*>,xor2c] ; :: thesis: for s being State of (GFA1AdderCirc (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)) . (GFA1AdderOutput (x,y,z)) = 'not' ((a1 'xor' ('not' a2)) 'xor' a3)
set A = GFA1AdderCirc (x,y,z);
let s be State of (GFA1AdderCirc (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)) . (GFA1AdderOutput (x,y,z)) = 'not' ((a1 'xor' ('not' a2)) 'xor' a3)
let a1, a2, a3 be Element of BOOLEAN ; :: thesis: ( a1 = s . x & a2 = s . y & a3 = s . z implies (Following (s,2)) . (GFA1AdderOutput (x,y,z)) = 'not' ((a1 'xor' ('not' a2)) 'xor' a3) )
assume ( a1 = s . x & a2 = s . y & a3 = s . z ) ; :: thesis: (Following (s,2)) . (GFA1AdderOutput (x,y,z)) = 'not' ((a1 'xor' ('not' a2)) 'xor' a3)
hence (Following (s,2)) . (GFA1AdderOutput (x,y,z)) = (a1 'xor' ('not' a2)) 'xor' ('not' a3) by A1, Th61
.= 'not' ((a1 'xor' ('not' a2)) 'xor' a3) by XBOOLEAN:74 ;
:: thesis: verum