set f = xor2c ;
let x, y, z be set ; ( z <> [<*x,y*>,xor2c] implies for s being State of (GFA2AdderCirc (x,y,z))
for a1a2, a1, a2, a3 being Element of BOOLEAN st a1a2 = s . [<*x,y*>,xor2c] & a1 = s . x & a2 = s . y & a3 = s . z holds
(Following s) . (GFA2AdderOutput (x,y,z)) = a1a2 'xor' ('not' a3) )
assume A1:
z <> [<*x,y*>,xor2c]
; for s being State of (GFA2AdderCirc (x,y,z))
for a1a2, a1, a2, a3 being Element of BOOLEAN st a1a2 = s . [<*x,y*>,xor2c] & a1 = s . x & a2 = s . y & a3 = s . z holds
(Following s) . (GFA2AdderOutput (x,y,z)) = a1a2 'xor' ('not' a3)
set O1 = GFA1AdderOutput (x,y,z);
set O2 = GFA2AdderOutput (x,y,z);
set A1 = GFA1AdderCirc (x,y,z);
set A2 = GFA2AdderCirc (x,y,z);
( GFA2AdderCirc (x,y,z) = GFA1AdderCirc (x,y,z) & GFA2AdderOutput (x,y,z) = GFA1AdderOutput (x,y,z) )
;
hence
for s being State of (GFA2AdderCirc (x,y,z))
for a1a2, a1, a2, a3 being Element of BOOLEAN st a1a2 = s . [<*x,y*>,xor2c] & a1 = s . x & a2 = s . y & a3 = s . z holds
(Following s) . (GFA2AdderOutput (x,y,z)) = a1a2 'xor' ('not' a3)
by A1, Th60; verum