let x, y, c be set ; :: thesis: ( x <> [<*y,c*>,'&'] & y <> [<*c,x*>,'&'] & c <> [<*x,y*>,'&'] & c <> [<*x,y*>,'xor'] implies for s being State of (BitAdderWithOverflowCirc (x,y,c)) holds Following (s,2) is stable )
assume that
A1: x <> [<*y,c*>,'&'] and
A2: y <> [<*c,x*>,'&'] and
A3: c <> [<*x,y*>,'&'] and
A4: c <> [<*x,y*>,'xor'] ; :: thesis: for s being State of (BitAdderWithOverflowCirc (x,y,c)) holds Following (s,2) is stable
set S = BitAdderWithOverflowStr (x,y,c);
set S1 = 2GatesCircStr (x,y,c,'xor');
set S2 = MajorityStr (x,y,c);
set A = BitAdderWithOverflowCirc (x,y,c);
set A1 = BitAdderCirc (x,y,c);
set A2 = MajorityCirc (x,y,c);
let s be State of (BitAdderWithOverflowCirc (x,y,c)); :: thesis: Following (s,2) is stable
reconsider s1 = s | the carrier of (2GatesCircStr (x,y,c,'xor')) as State of (BitAdderCirc (x,y,c)) by FACIRC_1:26;
reconsider s2 = s | the carrier of (MajorityStr (x,y,c)) as State of (MajorityCirc (x,y,c)) by FACIRC_1:26;
reconsider t = s as State of ((BitAdderCirc (x,y,c)) +* (MajorityCirc (x,y,c))) ;
InputVertices (2GatesCircStr (x,y,c,'xor')) = {x,y,c} by A4, FACIRC_1:57;
then A5: InputVertices (2GatesCircStr (x,y,c,'xor')) = InputVertices (MajorityStr (x,y,c)) by A1, A2, A3, Th20;
A6: InnerVertices (2GatesCircStr (x,y,c,'xor')) misses InputVertices (2GatesCircStr (x,y,c,'xor')) by XBOOLE_1:79;
A7: InnerVertices (MajorityStr (x,y,c)) misses InputVertices (MajorityStr (x,y,c)) by XBOOLE_1:79;
then A8: Following (s1,2) = (Following (t,2)) | the carrier of (2GatesCircStr (x,y,c,'xor')) by A5, FACIRC_1:30;
A9: Following (s1,3) = (Following (t,3)) | the carrier of (2GatesCircStr (x,y,c,'xor')) by A5, A7, FACIRC_1:30;
A10: Following (s2,2) = (Following (t,2)) | the carrier of (MajorityStr (x,y,c)) by A5, A6, FACIRC_1:31;
A11: Following (s2,3) = (Following (t,3)) | the carrier of (MajorityStr (x,y,c)) by A5, A6, FACIRC_1:31;
Following (s1,2) is stable by A4, FACIRC_1:63;
then A12: Following (s1,2) = Following (Following (s1,2))
.= Following (s1,(2 + 1)) by FACIRC_1:12 ;
Following (s2,2) is stable by A1, A2, A3, Th29;
then A13: Following (s2,2) = Following (Following (s2,2))
.= Following (s2,(2 + 1)) by FACIRC_1:12 ;
A14: Following (s,(2 + 1)) = Following (Following (s,2)) by FACIRC_1:12;
A15: dom (Following (s,2)) = the carrier of (BitAdderWithOverflowStr (x,y,c)) by CIRCUIT1:3;
A16: dom (Following (s,3)) = the carrier of (BitAdderWithOverflowStr (x,y,c)) by CIRCUIT1:3;
A17: dom (Following (s1,2)) = the carrier of (2GatesCircStr (x,y,c,'xor')) by CIRCUIT1:3;
A18: dom (Following (s2,2)) = the carrier of (MajorityStr (x,y,c)) by CIRCUIT1:3;
A19: the carrier of (BitAdderWithOverflowStr (x,y,c)) = the carrier of (2GatesCircStr (x,y,c,'xor')) \/ the carrier of (MajorityStr (x,y,c)) by CIRCCOMB:def 2;

now :: thesis: for a being object st a in the carrier of (BitAdderWithOverflowStr (x,y,c)) holds
(Following (s,2)) . a = (Following (Following (s,2))) . a

let a be object ; :: thesis: ( a in the carrier of (BitAdderWithOverflowStr (x,y,c)) implies (Following (s,2)) . a = (Following (Following (s,2))) . a )
assume a in the carrier of (BitAdderWithOverflowStr (x,y,c)) ; :: thesis: (Following (s,2)) . a = (Following (Following (s,2))) . a
then ( a in the carrier of (2GatesCircStr (x,y,c,'xor')) or a in the carrier of (MajorityStr (x,y,c)) ) by A19, XBOOLE_0:def 3;
then ( ( (Following (s,2)) . a = (Following (s1,2)) . a & (Following (s,3)) . a = (Following (s1,3)) . a ) or ( (Following (s,2)) . a = (Following (s2,2)) . a & (Following (s,3)) . a = (Following (s2,3)) . a ) ) by A8, A9, A10, A11, A12, A13, A17, A18, FUNCT_1:47;
hence (Following (s,2)) . a = (Following (Following (s,2))) . a by A12, A13, FACIRC_1:12; :: thesis: verum

end;

hence Following (s,2) = Following (Following (s,2)) by A14, A15, A16, FUNCT_1:2; :: according to CIRCUIT2:def 6 :: thesis: verum