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 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 holds
for n being Nat holds Following s,n = (Following s1,n) +* (Following s2,n) )
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 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 holds
for n being Nat holds Following s,n = (Following s1,n) +* (Following s2,n)
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 st s1 = s | the carrier of S1 holds
for s2 being State of A2 st s2 = s | the carrier of S2 holds
for n being Nat holds Following s,n = (Following s1,n) +* (Following s2,n)
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 st s1 = s | the carrier of S1 holds
for s2 being State of A2 st s2 = s | the carrier of S2 holds
for n being Nat holds Following s,n = (Following s1,n) +* (Following s2,n)
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 st s1 = s | the carrier of S1 holds
for s2 being State of A2 st s2 = s | the carrier of S2 holds
for n being Nat holds Following s,n = (Following s1,n) +* (Following s2,n) )
assume A2: ( A1 tolerates A2 & A = A1 +* A2 ) ; :: 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 holds
for n being Nat holds Following s,n = (Following s1,n) +* (Following s2,n)
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 holds
for n being Nat holds Following s,n = (Following s1,n) +* (Following s2,n)
let s1 be State of A1; :: thesis: ( s1 = s | the carrier of S1 implies for s2 being State of A2 st s2 = s | the carrier of S2 holds
for n being Nat holds Following s,n = (Following s1,n) +* (Following s2,n) )
assume A3: s1 = s | the carrier of S1 ; :: thesis: for s2 being State of A2 st s2 = s | the carrier of S2 holds
for n being Nat holds Following s,n = (Following s1,n) +* (Following s2,n)
let s2 be State of A2; :: thesis: ( s2 = s | the carrier of S2 implies for n being Nat holds Following s,n = (Following s1,n) +* (Following s2,n) )
assume A4: s2 = s | the carrier of S2 ; :: thesis: for n being Nat holds Following s,n = (Following s1,n) +* (Following s2,n)
let n be Nat; :: thesis: Following s,n = (Following s1,n) +* (Following s2,n)
A5: (Following s,n) | the carrier of S1 = Following s1,n 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 A6: (Following s,n) | the carrier of S2 = Following s2,n by A1, A2, A4, Th14;
( dom (Following s,n) = the carrier of S & the carrier of S = the carrier of S1 \/ the carrier of S2 ) by A1, CIRCCOMB:def 2, CIRCUIT1:4;
hence Following s,n = (Following s1,n) +* (Following s2,n) by A5, A6, FUNCT_4:74; :: thesis: verum