let S1, S2, S be non empty non void Circuit-like ManySortedSign ; :: thesis: ( InputVertices S1 misses InnerVertices S2 & InputVertices S2 misses InnerVertices S1 & 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 n1, n2 being Nat
for s being State of A
for s1 being State of A1 st s1 = s | the carrier of S1 holds
for s2 being State of A2 st s2 = s | the carrier of S2 & Following s1,n1 is stable & Following s2,n2 is stable holds
Following s,(max n1,n2) is stable )
assume A1: ( InputVertices S1 misses InnerVertices S2 & InputVertices S2 misses InnerVertices S1 & 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 n1, n2 being Nat
for s being State of A
for s1 being State of A1 st s1 = s | the carrier of S1 holds
for s2 being State of A2 st s2 = s | the carrier of S2 & Following s1,n1 is stable & Following s2,n2 is stable holds
Following s,(max n1,n2) is stable
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 n1, n2 being Nat
for s being State of A
for s1 being State of A1 st s1 = s | the carrier of S1 holds
for s2 being State of A2 st s2 = s | the carrier of S2 & Following s1,n1 is stable & Following s2,n2 is stable holds
Following s,(max n1,n2) 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 n1, n2 being Nat
for s being State of A
for s1 being State of A1 st s1 = s | the carrier of S1 holds
for s2 being State of A2 st s2 = s | the carrier of S2 & Following s1,n1 is stable & Following s2,n2 is stable holds
Following s,(max n1,n2) is stable
let A be non-empty Circuit of S; :: thesis: ( A1 tolerates A2 & A = A1 +* A2 implies for n1, n2 being Nat
for s being State of A
for s1 being State of A1 st s1 = s | the carrier of S1 holds
for s2 being State of A2 st s2 = s | the carrier of S2 & Following s1,n1 is stable & Following s2,n2 is stable holds
Following s,(max n1,n2) is stable )
assume A2: ( A1 tolerates A2 & A = A1 +* A2 ) ; :: thesis: for n1, n2 being Nat
for s being State of A
for s1 being State of A1 st s1 = s | the carrier of S1 holds
for s2 being State of A2 st s2 = s | the carrier of S2 & Following s1,n1 is stable & Following s2,n2 is stable holds
Following s,(max n1,n2) is stable
let n1, n2 be Nat; :: thesis: for s being State of A
for s1 being State of A1 st s1 = s | the carrier of S1 holds
for s2 being State of A2 st s2 = s | the carrier of S2 & Following s1,n1 is stable & Following s2,n2 is stable holds
Following s,(max n1,n2) is stable
let s be State of A; :: thesis: for s1 being State of A1 st s1 = s | the carrier of S1 holds
for s2 being State of A2 st s2 = s | the carrier of S2 & Following s1,n1 is stable & Following s2,n2 is stable holds
Following s,(max n1,n2) is stable
let s0 be State of A1; :: thesis: ( s0 = s | the carrier of S1 implies for s2 being State of A2 st s2 = s | the carrier of S2 & Following s0,n1 is stable & Following s2,n2 is stable holds
Following s,(max n1,n2) is stable )
assume A3: s0 = s | the carrier of S1 ; :: thesis: for s2 being State of A2 st s2 = s | the carrier of S2 & Following s0,n1 is stable & Following s2,n2 is stable holds
Following s,(max n1,n2) is stable
let s3 be State of A2; :: thesis: ( s3 = s | the carrier of S2 & Following s0,n1 is stable & Following s3,n2 is stable implies Following s,(max n1,n2) is stable )
assume that
A4: s3 = s | the carrier of S2 and
A5: ( Following s0,n1 is stable & Following s3,n2 is stable ) ; :: thesis: Following s,(max n1,n2) is stable
set n = max n1,n2;
A6: (Following s,(max n1,n2)) | the carrier of S1 = Following s0,(max n1,n2) by A1, A2, A3, Th14;
S1 tolerates S2 by A2, CIRCCOMB:def 3;
then ( S1 +* S2 = S2 +* S1 & A1 +* A2 = A2 +* A1 & A2 tolerates A1 ) by A2, CIRCCOMB:9, CIRCCOMB:23, CIRCCOMB:26;
then A7: (Following s,(max n1,n2)) | the carrier of S2 = Following s3,(max n1,n2) by A1, A2, A4, Th14;
A8: ( Following s0,(max n1,n2) is stable & Following s3,(max n1,n2) is stable ) by A5, Th4, XXREAL_0:25;
thus Following s,(max n1,n2) = (Following s0,(max n1,n2)) +* (Following s3,(max n1,n2)) by A1, A2, A3, A4, Th22
.= (Following (Following s0,(max n1,n2))) +* (Following s3,(max n1,n2)) by A8, CIRCUIT2:def 6
.= (Following (Following s0,(max n1,n2))) +* (Following (Following s3,(max n1,n2))) by A8, CIRCUIT2:def 6
.= Following (Following s,(max n1,n2)) by A1, A2, A6, A7, CIRCCOMB:39 ; :: according to CIRCUIT2:def 6 :: thesis: verum