let S1, S2, S be non empty non void Circuit-like ManySortedSign ; :: thesis: ( InnerVertices S1 misses InputVertices S2 & S = S1 +* S2 implies for C1 being non-empty Circuit of S1

for C2 being non-empty Circuit of S2

for C being non-empty Circuit of S st C1 tolerates C2 & C = C1 +* C2 holds

for s2 being State of C2

for s being State of C st s2 = s | the carrier of S2 holds

Following s2 = (Following s) | the carrier of S2 )

assume A1: ( InnerVertices S1 misses InputVertices S2 & S = S1 +* S2 ) ; :: thesis: for C1 being non-empty Circuit of S1

for C2 being non-empty Circuit of S2

for C being non-empty Circuit of S st C1 tolerates C2 & C = C1 +* C2 holds

for s2 being State of C2

for s being State of C st s2 = s | the carrier of S2 holds

Following s2 = (Following s) | the carrier of S2

let C1 be non-empty Circuit of S1; :: thesis: for C2 being non-empty Circuit of S2

for C being non-empty Circuit of S st C1 tolerates C2 & C = C1 +* C2 holds

for s2 being State of C2

for s being State of C st s2 = s | the carrier of S2 holds

Following s2 = (Following s) | the carrier of S2

let C2 be non-empty Circuit of S2; :: thesis: for C being non-empty Circuit of S st C1 tolerates C2 & C = C1 +* C2 holds

for s2 being State of C2

for s being State of C st s2 = s | the carrier of S2 holds

Following s2 = (Following s) | the carrier of S2

let C be non-empty Circuit of S; :: thesis: ( C1 tolerates C2 & C = C1 +* C2 implies for s2 being State of C2

for s being State of C st s2 = s | the carrier of S2 holds

Following s2 = (Following s) | the carrier of S2 )

assume that

A2: C1 tolerates C2 and

A3: C = C1 +* C2 ; :: thesis: for s2 being State of C2

for s being State of C st s2 = s | the carrier of S2 holds

Following s2 = (Following s) | the carrier of S2

let s2 be State of C2; :: thesis: for s being State of C st s2 = s | the carrier of S2 holds

Following s2 = (Following s) | the carrier of S2

let s be State of C; :: thesis: ( s2 = s | the carrier of S2 implies Following s2 = (Following s) | the carrier of S2 )

assume A4: s2 = s | the carrier of S2 ; :: thesis: Following s2 = (Following s) | the carrier of S2

the Sorts of C1 tolerates the Sorts of C2 by A2, CIRCCOMB:def 3;

then reconsider s1 = s | the carrier of S1 as State of C1 by A3, CIRCCOMB:26;

( dom (Following s2) = the carrier of S2 & Following s = (Following s1) +* (Following s2) ) by A1, A2, A3, A4, CIRCCOMB:32, CIRCUIT1:3;

hence Following s2 = (Following s) | the carrier of S2 by FUNCT_4:23; :: thesis: verum

for C2 being non-empty Circuit of S2

for C being non-empty Circuit of S st C1 tolerates C2 & C = C1 +* C2 holds

for s2 being State of C2

for s being State of C st s2 = s | the carrier of S2 holds

Following s2 = (Following s) | the carrier of S2 )

assume A1: ( InnerVertices S1 misses InputVertices S2 & S = S1 +* S2 ) ; :: thesis: for C1 being non-empty Circuit of S1

for C2 being non-empty Circuit of S2

for C being non-empty Circuit of S st C1 tolerates C2 & C = C1 +* C2 holds

for s2 being State of C2

for s being State of C st s2 = s | the carrier of S2 holds

Following s2 = (Following s) | the carrier of S2

let C1 be non-empty Circuit of S1; :: thesis: for C2 being non-empty Circuit of S2

for C being non-empty Circuit of S st C1 tolerates C2 & C = C1 +* C2 holds

for s2 being State of C2

for s being State of C st s2 = s | the carrier of S2 holds

Following s2 = (Following s) | the carrier of S2

let C2 be non-empty Circuit of S2; :: thesis: for C being non-empty Circuit of S st C1 tolerates C2 & C = C1 +* C2 holds

for s2 being State of C2

for s being State of C st s2 = s | the carrier of S2 holds

Following s2 = (Following s) | the carrier of S2

let C be non-empty Circuit of S; :: thesis: ( C1 tolerates C2 & C = C1 +* C2 implies for s2 being State of C2

for s being State of C st s2 = s | the carrier of S2 holds

Following s2 = (Following s) | the carrier of S2 )

assume that

A2: C1 tolerates C2 and

A3: C = C1 +* C2 ; :: thesis: for s2 being State of C2

for s being State of C st s2 = s | the carrier of S2 holds

Following s2 = (Following s) | the carrier of S2

let s2 be State of C2; :: thesis: for s being State of C st s2 = s | the carrier of S2 holds

Following s2 = (Following s) | the carrier of S2

let s be State of C; :: thesis: ( s2 = s | the carrier of S2 implies Following s2 = (Following s) | the carrier of S2 )

assume A4: s2 = s | the carrier of S2 ; :: thesis: Following s2 = (Following s) | the carrier of S2

the Sorts of C1 tolerates the Sorts of C2 by A2, CIRCCOMB:def 3;

then reconsider s1 = s | the carrier of S1 as State of C1 by A3, CIRCCOMB:26;

( dom (Following s2) = the carrier of S2 & Following s = (Following s1) +* (Following s2) ) by A1, A2, A3, A4, CIRCCOMB:32, CIRCUIT1:3;

hence Following s2 = (Following s) | the carrier of S2 by FUNCT_4:23; :: thesis: verum