let S1, S2, S be non empty non void Circuit-like ManySortedSign ; :: thesis: ( InnerVertices S2 misses InputVertices 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 s being State of A
for s1 being State of A1
for s2 being State of A2 st s1 = s | the carrier of S1 & s2 = s | the carrier of S2 holds
Following s = (Following s2) +* (Following s1) )
assume A1: ( InnerVertices S2 misses InputVertices 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 s being State of A
for s1 being State of A1
for s2 being State of A2 st s1 = s | the carrier of S1 & s2 = s | the carrier of S2 holds
Following s = (Following s2) +* (Following s1)
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
for s1 being State of A1
for s2 being State of A2 st s1 = s | the carrier of S1 & s2 = s | the carrier of S2 holds
Following s = (Following s2) +* (Following s1)
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
for s1 being State of A1
for s2 being State of A2 st s1 = s | the carrier of S1 & s2 = s | the carrier of S2 holds
Following s = (Following s2) +* (Following s1)
let A be non-empty Circuit of S; :: thesis: ( A1 tolerates A2 & A = A1 +* A2 implies for s being State of A
for s1 being State of A1
for s2 being State of A2 st s1 = s | the carrier of S1 & s2 = s | the carrier of S2 holds
Following s = (Following s2) +* (Following s1) )
assume that
A2: A1 tolerates A2 and
A3: A = A1 +* A2 ; :: thesis: for s being State of A
for s1 being State of A1
for s2 being State of A2 st s1 = s | the carrier of S1 & s2 = s | the carrier of S2 holds
Following s = (Following s2) +* (Following s1)
S1 tolerates S2 by A2, Def3;
then ( S = S2 +* S1 & A = A2 +* A1 & A2 tolerates A1 ) by A1, A2, A3, Th9, Th23, Th26;
hence for s being State of A
for s1 being State of A1
for s2 being State of A2 st s1 = s | the carrier of S1 & s2 = s | the carrier of S2 holds
Following s = (Following s2) +* (Following s1) by A1, Th39; :: thesis: verum