set f = xor2 ;
let x, y, z be set ; :: thesis: ( 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 ] & a1 = s . x & a2 = s . y & a3 = s . z holds
(Following s) . (GFA0AdderOutput x,y,z) = a1a2 'xor' a3 )
assume A1: z <> [<*x,y*>,xor2 ] ; :: thesis: for s being State of (GFA0AdderCirc x,y,z)
for a1a2, a1, a2, a3 being Element of BOOLEAN st a1a2 = s . [<*x,y*>,xor2 ] & a1 = s . x & a2 = s . y & a3 = s . z holds
(Following s) . (GFA0AdderOutput x,y,z) = a1a2 'xor' a3
set xy = [<*x,y*>,xor2 ];
set A = GFA0AdderCirc x,y,z;
let s be State of (GFA0AdderCirc x,y,z); :: thesis: for a1a2, a1, a2, a3 being Element of BOOLEAN st a1a2 = s . [<*x,y*>,xor2 ] & a1 = s . x & a2 = s . y & a3 = s . z holds
(Following s) . (GFA0AdderOutput x,y,z) = a1a2 'xor' a3
let a1a2, a1, a2, a3 be Element of BOOLEAN ; :: thesis: ( a1a2 = s . [<*x,y*>,xor2 ] & a1 = s . x & a2 = s . y & a3 = s . z implies (Following s) . (GFA0AdderOutput x,y,z) = a1a2 'xor' a3 )
assume A2: ( a1a2 = s . [<*x,y*>,xor2 ] & a1 = s . x & a2 = s . y & a3 = s . z ) ; :: thesis: (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, TWOSCOMP:def 13 ; :: thesis: verum