set f1 = and2a ;
set f2 = and2c ;
set f3 = and2b ;
let x, y, z be set ; :: thesis: ( x <> [<*y,z*>,and2c ] & y <> [<*z,x*>,and2b ] & z <> [<*x,y*>,and2a ] implies for s being State of (GFA2CarryCirc x,y,z) holds Following s,2 is stable )
assume A1: ( x <> [<*y,z*>,and2c ] & y <> [<*z,x*>,and2b ] & z <> [<*x,y*>,and2a ] ) ; :: thesis: for s being State of (GFA2CarryCirc x,y,z) holds Following s,2 is stable
set S = GFA2CarryStr x,y,z;
reconsider xx = x, yy = y, zz = z as Vertex of (GFA2CarryStr x,y,z) by Th92;
let s be State of (GFA2CarryCirc x,y,z); :: thesis: Following s,2 is stable
set a1 = s . xx;
set a2 = s . yy;
set a3 = s . zz;
set ffs = Following s,2;
set fffs = Following (Following s,2);
set xy = [<*x,y*>,and2a ];
set yz = [<*y,z*>,and2c ];
set zx = [<*z,x*>,and2b ];
A2: Following s,2 = Following (Following s) by FACIRC_1:15;
A3: z in InputVertices (GFA2CarryStr x,y,z) by A1, Th94;
then (Following s) . z = s . zz by CIRCUIT2:def 5;
then A4: (Following s,2) . z = s . zz by A2, A3, CIRCUIT2:def 5;
A5: y in InputVertices (GFA2CarryStr x,y,z) by A1, Th94;
then (Following s) . y = s . yy by CIRCUIT2:def 5;
then A6: (Following s,2) . y = s . yy by A2, A5, CIRCUIT2:def 5;
A7: x in InputVertices (GFA2CarryStr x,y,z) by A1, Th94;
then (Following s) . x = s . xx by CIRCUIT2:def 5;
then A8: (Following s,2) . x = s . xx by A2, A7, CIRCUIT2:def 5;
s . zz = s . z ;
then A9: (Following s,2) . [<*x,y*>,and2a ] = ('not' (s . xx)) '&' (s . yy) by A1, Th98;
s . yy = s . y ;
then A10: (Following s,2) . [<*z,x*>,and2b ] = ('not' (s . xx)) '&' ('not' (s . zz)) by A1, Th98;
s . xx = s . x ;
then A11: (Following s,2) . [<*y,z*>,and2c ] = (s . yy) '&' ('not' (s . zz)) by A1, Th98;
A12: (Following s,2) . (GFA2CarryOutput x,y,z) = 'not' (((('not' (s . xx)) '&' (s . yy)) 'or' ((s . yy) '&' ('not' (s . zz)))) 'or' (('not' (s . zz)) '&' ('not' (s . xx)))) by A1, Th98;

A13: now

let a be set ; :: thesis: ( a in the carrier of (GFA2CarryStr x,y,z) implies (Following s,2) . a = (Following (Following s,2)) . a )
assume A14: a in the carrier of (GFA2CarryStr x,y,z) ; :: thesis: (Following s,2) . a = (Following (Following s,2)) . a
then reconsider v = a as Vertex of (GFA2CarryStr x,y,z) ;
A15: v in (InputVertices (GFA2CarryStr x,y,z)) \/ (InnerVertices (GFA2CarryStr x,y,z)) by A14, XBOOLE_1:45;
thus (Following s,2) . a = (Following (Following s,2)) . a :: thesis: verum

proof

per cases ( v in InputVertices (GFA2CarryStr x,y,z) or v in InnerVertices (GFA2CarryStr x,y,z) ) by A15, XBOOLE_0:def 3;

suppose v in InputVertices (GFA2CarryStr x,y,z) ; :: thesis: (Following s,2) . a = (Following (Following s,2)) . a

hence (Following s,2) . a = (Following (Following s,2)) . a by CIRCUIT2:def 5; :: thesis: verum

end;

suppose v in InnerVertices (GFA2CarryStr x,y,z) ; :: thesis: (Following s,2) . a = (Following (Following s,2)) . a

then v in {[<*x,y*>,and2a ],[<*y,z*>,and2c ],[<*z,x*>,and2b ]} \/ {(GFA2CarryOutput x,y,z)} by Th87;
then ( v in {[<*x,y*>,and2a ],[<*y,z*>,and2c ],[<*z,x*>,and2b ]} or v in {(GFA2CarryOutput x,y,z)} ) by XBOOLE_0:def 3;
then ( v = [<*x,y*>,and2a ] or v = [<*y,z*>,and2c ] or v = [<*z,x*>,and2b ] or v = GFA2CarryOutput x,y,z ) by ENUMSET1:def 1, TARSKI:def 1;
hence (Following s,2) . a = (Following (Following s,2)) . a by A12, A9, A11, A10, A8, A6, A4, Th96, Th97; :: thesis: verum

end;

end;

end;

end;

( dom (Following (Following s,2)) = the carrier of (GFA2CarryStr x,y,z) & dom (Following s,2) = the carrier of (GFA2CarryStr x,y,z) ) by CIRCUIT1:4;
hence Following s,2 = Following (Following s,2) by A13, FUNCT_1:9; :: according to CIRCUIT2:def 6 :: thesis: verum