let S1, S2, S be non empty non void Circuit-like ManySortedSign ; :: thesis: ( InputVertices S1 misses InnerVertices S2 & 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

for s2 being State of A2 st s1 = s | the carrier of S1 & Following (s1,n1) is stable & s2 = (Following (s,n1)) | the carrier of S2 & Following (s2,n2) is stable holds

Following (s,(n1 + n2)) is stable )

assume that

A1: InputVertices S1 misses InnerVertices S2 and

A2: 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

for s2 being State of A2 st s1 = s | the carrier of S1 & Following (s1,n1) is stable & s2 = (Following (s,n1)) | the carrier of S2 & Following (s2,n2) is stable holds

Following (s,(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

for s2 being State of A2 st s1 = s | the carrier of S1 & Following (s1,n1) is stable & s2 = (Following (s,n1)) | the carrier of S2 & Following (s2,n2) is stable holds

Following (s,(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

for s2 being State of A2 st s1 = s | the carrier of S1 & Following (s1,n1) is stable & s2 = (Following (s,n1)) | the carrier of S2 & Following (s2,n2) is stable holds

Following (s,(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

for s2 being State of A2 st s1 = s | the carrier of S1 & Following (s1,n1) is stable & s2 = (Following (s,n1)) | the carrier of S2 & Following (s2,n2) is stable holds

Following (s,(n1 + n2)) is stable )

assume that

A3: A1 tolerates A2 and

A4: A = A1 +* A2 ; :: thesis: for n1, n2 being Nat

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 & Following (s1,n1) is stable & s2 = (Following (s,n1)) | the carrier of S2 & Following (s2,n2) is stable holds

Following (s,(n1 + n2)) is stable

let n1, n2 be Nat; :: 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 & Following (s1,n1) is stable & s2 = (Following (s,n1)) | the carrier of S2 & Following (s2,n2) is stable holds

Following (s,(n1 + n2)) is stable

let s be State of A; :: thesis: for s1 being State of A1

for s2 being State of A2 st s1 = s | the carrier of S1 & Following (s1,n1) is stable & s2 = (Following (s,n1)) | the carrier of S2 & Following (s2,n2) is stable holds

Following (s,(n1 + n2)) is stable

let s9 be State of A1; :: thesis: for s2 being State of A2 st s9 = s | the carrier of S1 & Following (s9,n1) is stable & s2 = (Following (s,n1)) | the carrier of S2 & Following (s2,n2) is stable holds

Following (s,(n1 + n2)) is stable

let s99 be State of A2; :: thesis: ( s9 = s | the carrier of S1 & Following (s9,n1) is stable & s99 = (Following (s,n1)) | the carrier of S2 & Following (s99,n2) is stable implies Following (s,(n1 + n2)) is stable )

assume that

A5: ( s9 = s | the carrier of S1 & Following (s9,n1) is stable ) and

A6: ( s99 = (Following (s,n1)) | the carrier of S2 & Following (s99,n2) is stable ) ; :: thesis: Following (s,(n1 + n2)) is stable

A7: the Sorts of A1 tolerates the Sorts of A2 by A3, CIRCCOMB:def 3;

then reconsider s1 = (Following (s,n1)) | the carrier of S1, s0 = s | the carrier of S1 as State of A1 by A4, CIRCCOMB:26;

A8: Following ((Following (s,n1)),n2) = Following (s,(n1 + n2)) by FACIRC_1:13;

then A9: (Following (s,(n1 + n2))) | the carrier of S1 = Following (s1,n2) by A1, A2, A3, A4, Th13;

reconsider s2 = (Following (s,n1)) | the carrier of S2 as State of A2 by A4, A7, CIRCCOMB:26;

A10: dom (Following (s,(n1 + n2))) = the carrier of S by CIRCUIT1:3;

A11: the carrier of S = the carrier of S1 \/ the carrier of S2 by A2, CIRCCOMB:def 2;

A12: s1 = Following (s0,n1) by A1, A2, A3, A4, Th13;

then A13: (Following (s,(n1 + n2))) | the carrier of S2 = Following (s2,n2) by A1, A2, A3, A4, A5, A8, Th18;

then Following (Following (s,(n1 + n2))) = (Following (Following (s2,n2))) +* (Following (Following (s1,n2))) by A1, A2, A3, A4, A9, CIRCCOMB:33

.= (Following (s2,n2)) +* (Following (Following (s1,n2))) by A6

.= (Following (s2,n2)) +* (Following (s1,(n2 + 1))) by FACIRC_1:12

.= (Following (s2,n2)) +* s1 by A5, A12, Th3

.= (Following (s2,n2)) +* (Following (s1,n2)) by A5, A12, Th3

.= Following (s,(n1 + n2)) by A11, A10, A9, A13, FUNCT_4:70 ;

hence Following (s,(n1 + n2)) is stable ; :: 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

for s2 being State of A2 st s1 = s | the carrier of S1 & Following (s1,n1) is stable & s2 = (Following (s,n1)) | the carrier of S2 & Following (s2,n2) is stable holds

Following (s,(n1 + n2)) is stable )

assume that

A1: InputVertices S1 misses InnerVertices S2 and

A2: 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

for s2 being State of A2 st s1 = s | the carrier of S1 & Following (s1,n1) is stable & s2 = (Following (s,n1)) | the carrier of S2 & Following (s2,n2) is stable holds

Following (s,(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

for s2 being State of A2 st s1 = s | the carrier of S1 & Following (s1,n1) is stable & s2 = (Following (s,n1)) | the carrier of S2 & Following (s2,n2) is stable holds

Following (s,(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

for s2 being State of A2 st s1 = s | the carrier of S1 & Following (s1,n1) is stable & s2 = (Following (s,n1)) | the carrier of S2 & Following (s2,n2) is stable holds

Following (s,(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

for s2 being State of A2 st s1 = s | the carrier of S1 & Following (s1,n1) is stable & s2 = (Following (s,n1)) | the carrier of S2 & Following (s2,n2) is stable holds

Following (s,(n1 + n2)) is stable )

assume that

A3: A1 tolerates A2 and

A4: A = A1 +* A2 ; :: thesis: for n1, n2 being Nat

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 & Following (s1,n1) is stable & s2 = (Following (s,n1)) | the carrier of S2 & Following (s2,n2) is stable holds

Following (s,(n1 + n2)) is stable

let n1, n2 be Nat; :: 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 & Following (s1,n1) is stable & s2 = (Following (s,n1)) | the carrier of S2 & Following (s2,n2) is stable holds

Following (s,(n1 + n2)) is stable

let s be State of A; :: thesis: for s1 being State of A1

for s2 being State of A2 st s1 = s | the carrier of S1 & Following (s1,n1) is stable & s2 = (Following (s,n1)) | the carrier of S2 & Following (s2,n2) is stable holds

Following (s,(n1 + n2)) is stable

let s9 be State of A1; :: thesis: for s2 being State of A2 st s9 = s | the carrier of S1 & Following (s9,n1) is stable & s2 = (Following (s,n1)) | the carrier of S2 & Following (s2,n2) is stable holds

Following (s,(n1 + n2)) is stable

let s99 be State of A2; :: thesis: ( s9 = s | the carrier of S1 & Following (s9,n1) is stable & s99 = (Following (s,n1)) | the carrier of S2 & Following (s99,n2) is stable implies Following (s,(n1 + n2)) is stable )

assume that

A5: ( s9 = s | the carrier of S1 & Following (s9,n1) is stable ) and

A6: ( s99 = (Following (s,n1)) | the carrier of S2 & Following (s99,n2) is stable ) ; :: thesis: Following (s,(n1 + n2)) is stable

A7: the Sorts of A1 tolerates the Sorts of A2 by A3, CIRCCOMB:def 3;

then reconsider s1 = (Following (s,n1)) | the carrier of S1, s0 = s | the carrier of S1 as State of A1 by A4, CIRCCOMB:26;

A8: Following ((Following (s,n1)),n2) = Following (s,(n1 + n2)) by FACIRC_1:13;

then A9: (Following (s,(n1 + n2))) | the carrier of S1 = Following (s1,n2) by A1, A2, A3, A4, Th13;

reconsider s2 = (Following (s,n1)) | the carrier of S2 as State of A2 by A4, A7, CIRCCOMB:26;

A10: dom (Following (s,(n1 + n2))) = the carrier of S by CIRCUIT1:3;

A11: the carrier of S = the carrier of S1 \/ the carrier of S2 by A2, CIRCCOMB:def 2;

A12: s1 = Following (s0,n1) by A1, A2, A3, A4, Th13;

then A13: (Following (s,(n1 + n2))) | the carrier of S2 = Following (s2,n2) by A1, A2, A3, A4, A5, A8, Th18;

then Following (Following (s,(n1 + n2))) = (Following (Following (s2,n2))) +* (Following (Following (s1,n2))) by A1, A2, A3, A4, A9, CIRCCOMB:33

.= (Following (s2,n2)) +* (Following (Following (s1,n2))) by A6

.= (Following (s2,n2)) +* (Following (s1,(n2 + 1))) by FACIRC_1:12

.= (Following (s2,n2)) +* s1 by A5, A12, Th3

.= (Following (s2,n2)) +* (Following (s1,n2)) by A5, A12, Th3

.= Following (s,(n1 + n2)) by A11, A10, A9, A13, FUNCT_4:70 ;

hence Following (s,(n1 + n2)) is stable ; :: thesis: verum