let S1, S2, S be non empty non void Circuit-like ManySortedSign ; :: thesis: ( S = S1 +* S2 implies for A1 being non-empty Circuit of S1
for A2 being non-empty Circuit of S2
for A being non-empty Circuit of S st A1 tolerates A2 & A = A1 +* A2 holds
for s being State of A st s is stable holds
( ( for s1 being State of A1 st s1 = s | the carrier of S1 holds
s1 is stable ) & ( for s2 being State of A2 st s2 = s | the carrier of S2 holds
s2 is stable ) ) )

assume A1: S = S1 +* S2 ; :: thesis: for A1 being non-empty Circuit of S1
for A2 being non-empty Circuit of S2
for A being non-empty Circuit of S st A1 tolerates A2 & A = A1 +* A2 holds
for s being State of A st s is stable holds
( ( for s1 being State of A1 st s1 = s | the carrier of S1 holds
s1 is stable ) & ( for s2 being State of A2 st s2 = s | the carrier of S2 holds
s2 is stable ) )

A2: the carrier of S = the carrier of S1 \/ the carrier of S2 by ;
let A1 be non-empty Circuit of S1; :: thesis: for A2 being non-empty Circuit of S2
for A being non-empty Circuit of S st A1 tolerates A2 & A = A1 +* A2 holds
for s being State of A st s is stable holds
( ( for s1 being State of A1 st s1 = s | the carrier of S1 holds
s1 is stable ) & ( for s2 being State of A2 st s2 = s | the carrier of S2 holds
s2 is stable ) )

let A2 be non-empty Circuit of S2; :: thesis: for A being non-empty Circuit of S st A1 tolerates A2 & A = A1 +* A2 holds
for s being State of A st s is stable holds
( ( for s1 being State of A1 st s1 = s | the carrier of S1 holds
s1 is stable ) & ( for s2 being State of A2 st s2 = s | the carrier of S2 holds
s2 is stable ) )

let A be non-empty Circuit of S; :: thesis: ( A1 tolerates A2 & A = A1 +* A2 implies for s being State of A st s is stable holds
( ( for s1 being State of A1 st s1 = s | the carrier of S1 holds
s1 is stable ) & ( for s2 being State of A2 st s2 = s | the carrier of S2 holds
s2 is stable ) ) )

assume A3: ( A1 tolerates A2 & A = A1 +* A2 ) ; :: thesis: for s being State of A st s is stable holds
( ( for s1 being State of A1 st s1 = s | the carrier of S1 holds
s1 is stable ) & ( for s2 being State of A2 st s2 = s | the carrier of S2 holds
s2 is stable ) )

let s be State of A; :: thesis: ( s is stable implies ( ( for s1 being State of A1 st s1 = s | the carrier of S1 holds
s1 is stable ) & ( for s2 being State of A2 st s2 = s | the carrier of S2 holds
s2 is stable ) ) )

assume A4: s = Following s ; :: according to CIRCUIT2:def 6 :: thesis: ( ( for s1 being State of A1 st s1 = s | the carrier of S1 holds
s1 is stable ) & ( for s2 being State of A2 st s2 = s | the carrier of S2 holds
s2 is stable ) )

hereby :: thesis: for s2 being State of A2 st s2 = s | the carrier of S2 holds
s2 is stable
let s1 be State of A1; :: thesis: ( s1 = s | the carrier of S1 implies s1 is stable )
assume A5: s1 = s | the carrier of S1 ; :: thesis: s1 is stable
A6: now :: thesis: for x being object st x in the carrier of S1 holds
s1 . x = () . x
let x be object ; :: thesis: ( x in the carrier of S1 implies s1 . x = () . x )
assume x in the carrier of S1 ; :: thesis: s1 . x = () . x
then reconsider v = x as Vertex of S1 ;
reconsider v9 = v as Vertex of S by ;
the carrier of S1 = () \/ () by XBOOLE_1:45;
then ( v in InputVertices S1 or v9 in InnerVertices S1 ) by XBOOLE_0:def 3;
then ( s1 . v = () . v or s . v9 = () . v ) by ;
hence s1 . x = () . x by ; :: thesis: verum
end;
( dom s1 = the carrier of S1 & dom () = the carrier of S1 ) by CIRCUIT1:3;
then s1 = Following s1 by ;
hence s1 is stable ; :: thesis: verum
end;
let s2 be State of A2; :: thesis: ( s2 = s | the carrier of S2 implies s2 is stable )
assume A7: s2 = s | the carrier of S2 ; :: thesis: s2 is stable
A8: now :: thesis: for x being object st x in the carrier of S2 holds
s2 . x = () . x
let x be object ; :: thesis: ( x in the carrier of S2 implies s2 . x = () . x )
assume x in the carrier of S2 ; :: thesis: s2 . x = () . x
then reconsider v = x as Vertex of S2 ;
reconsider v9 = v as Vertex of S by ;
the carrier of S2 = () \/ () by XBOOLE_1:45;
then ( v in InputVertices S2 or v9 in InnerVertices S2 ) by XBOOLE_0:def 3;
then ( s2 . v = () . v or s . v9 = () . v ) by ;
hence s2 . x = () . x by ; :: thesis: verum
end;
( dom s2 = the carrier of S2 & dom () = the carrier of S2 ) by CIRCUIT1:3;
hence s2 = Following s2 by ; :: according to CIRCUIT2:def 6 :: thesis: verum