set f = xor2c ;
let x, y, z be set ; ( 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]
; 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)); 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 ; ( 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 )
; (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
;
verum