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 that

A1: InputVertices S1 misses InnerVertices S2 and

A2: InputVertices S2 misses InnerVertices S1 and

A3: 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 that

A4: A1 tolerates A2 and

A5: 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

set n = max (n1,n2);

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 A6: 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

A7: (Following (s,(max (n1,n2)))) | the carrier of S1 = Following (s0,(max (n1,n2))) by A1, A3, A4, A5, A6, Th13;

S1 tolerates S2 by A4, CIRCCOMB:def 3;

then A8: S1 +* S2 = S2 +* S1 by CIRCCOMB:5;

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

A9: s3 = s | the carrier of S2 and

A10: Following (s0,n1) is stable and

A11: Following (s3,n2) is stable ; :: thesis: Following (s,(max (n1,n2))) is stable

A1 +* A2 = A2 +* A1 by A4, CIRCCOMB:22;

then A12: (Following (s,(max (n1,n2)))) | the carrier of S2 = Following (s3,(max (n1,n2))) by A2, A3, A4, A5, A9, A8, Th13, CIRCCOMB:19;

A13: Following (s3,(max (n1,n2))) is stable by A11, Th4, XXREAL_0:25;

A14: Following (s0,(max (n1,n2))) is stable by A10, 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, A5, A6, A9, Th21

.= (Following (Following (s0,(max (n1,n2))))) +* (Following (s3,(max (n1,n2)))) by A14

.= (Following (Following (s0,(max (n1,n2))))) +* (Following (Following (s3,(max (n1,n2))))) by A13

.= Following (Following (s,(max (n1,n2)))) by A2, A3, A4, A5, A7, A12, CIRCCOMB:32 ; :: according to CIRCUIT2:def 6 :: thesis: verum

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 that

A1: InputVertices S1 misses InnerVertices S2 and

A2: InputVertices S2 misses InnerVertices S1 and

A3: 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 that

A4: A1 tolerates A2 and

A5: 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

set n = max (n1,n2);

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 A6: 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

A7: (Following (s,(max (n1,n2)))) | the carrier of S1 = Following (s0,(max (n1,n2))) by A1, A3, A4, A5, A6, Th13;

S1 tolerates S2 by A4, CIRCCOMB:def 3;

then A8: S1 +* S2 = S2 +* S1 by CIRCCOMB:5;

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

A9: s3 = s | the carrier of S2 and

A10: Following (s0,n1) is stable and

A11: Following (s3,n2) is stable ; :: thesis: Following (s,(max (n1,n2))) is stable

A1 +* A2 = A2 +* A1 by A4, CIRCCOMB:22;

then A12: (Following (s,(max (n1,n2)))) | the carrier of S2 = Following (s3,(max (n1,n2))) by A2, A3, A4, A5, A9, A8, Th13, CIRCCOMB:19;

A13: Following (s3,(max (n1,n2))) is stable by A11, Th4, XXREAL_0:25;

A14: Following (s0,(max (n1,n2))) is stable by A10, 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, A5, A6, A9, Th21

.= (Following (Following (s0,(max (n1,n2))))) +* (Following (s3,(max (n1,n2)))) by A14

.= (Following (Following (s0,(max (n1,n2))))) +* (Following (Following (s3,(max (n1,n2))))) by A13

.= Following (Following (s,(max (n1,n2)))) by A2, A3, A4, A5, A7, A12, CIRCCOMB:32 ; :: according to CIRCUIT2:def 6 :: thesis: verum