set f = xor2 ;
let x, y, z be set ; ( z <> [<*x,y*>,xor2] implies for s being State of (GFA0AdderCirc (x,y,z))
for a1a2, a1, a2, a3 being Element of BOOLEAN st a1a2 = s . [<*x,y*>,xor2] & a3 = s . z holds
(Following s) . (GFA0AdderOutput (x,y,z)) = a1a2 'xor' a3 )
assume A1:
z <> [<*x,y*>,xor2]
; for s being State of (GFA0AdderCirc (x,y,z))
for a1a2, a1, a2, a3 being Element of BOOLEAN st a1a2 = s . [<*x,y*>,xor2] & a3 = s . z holds
(Following s) . (GFA0AdderOutput (x,y,z)) = a1a2 'xor' a3
set A = GFA0AdderCirc (x,y,z);
set xy = [<*x,y*>,xor2];
let s be State of (GFA0AdderCirc (x,y,z)); for a1a2, a1, a2, a3 being Element of BOOLEAN st a1a2 = s . [<*x,y*>,xor2] & a3 = s . z holds
(Following s) . (GFA0AdderOutput (x,y,z)) = a1a2 'xor' a3
let a1a2, a1, a2, a3 be Element of BOOLEAN ; ( a1a2 = s . [<*x,y*>,xor2] & a3 = s . z implies (Following s) . (GFA0AdderOutput (x,y,z)) = a1a2 'xor' a3 )
assume A2:
( a1a2 = s . [<*x,y*>,xor2] & a3 = s . z )
; (Following s) . (GFA0AdderOutput (x,y,z)) = a1a2 'xor' a3
thus (Following s) . (GFA0AdderOutput (x,y,z)) =
xor2 . <*(s . [<*x,y*>,xor2]),(s . z)*>
by A1, Lm3
.=
a1a2 'xor' a3
by A2, FACIRC_1:def 4
; verum