let S1, S2, S be non empty non void Circuit-like ManySortedSign ; ( 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 n 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 & ( not Following (s1,n) is stable or not Following (s2,n) is stable ) holds
not Following (s,n) is stable )
assume that
A1:
InputVertices S1 misses InnerVertices S2
and
A2:
InputVertices S2 misses InnerVertices S1
and
A3:
S = S1 +* S2
; 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 n 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 & ( not Following (s1,n) is stable or not Following (s2,n) is stable ) holds
not Following (s,n) is stable
let A1 be 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 n 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 & ( not Following (s1,n) is stable or not Following (s2,n) is stable ) holds
not Following (s,n) is stable
let A2 be non-empty Circuit of S2; for A being non-empty Circuit of S st A1 tolerates A2 & A = A1 +* A2 holds
for n 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 & ( not Following (s1,n) is stable or not Following (s2,n) is stable ) holds
not Following (s,n) is stable
let A be non-empty Circuit of S; ( A1 tolerates A2 & A = A1 +* A2 implies for n 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 & ( not Following (s1,n) is stable or not Following (s2,n) is stable ) holds
not Following (s,n) is stable )
assume A4:
( A1 tolerates A2 & A = A1 +* A2 )
; for n 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 & ( not Following (s1,n) is stable or not Following (s2,n) is stable ) holds
not Following (s,n) is stable
let n be 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 & ( not Following (s1,n) is stable or not Following (s2,n) is stable ) holds
not Following (s,n) is stable
let s be 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 & ( not Following (s1,n) is stable or not Following (s2,n) is stable ) holds
not Following (s,n) is stable
let s0 be State of A1; ( s0 = s | the carrier of S1 implies for s2 being State of A2 st s2 = s | the carrier of S2 & ( not Following (s0,n) is stable or not Following (s2,n) is stable ) holds
not Following (s,n) is stable )
assume
s0 = s | the carrier of S1
; for s2 being State of A2 st s2 = s | the carrier of S2 & ( not Following (s0,n) is stable or not Following (s2,n) is stable ) holds
not Following (s,n) is stable
then A5:
(Following (s,n)) | the carrier of S1 = Following (s0,n)
by A1, A3, A4, Th13;
let s3 be State of A2; ( s3 = s | the carrier of S2 & ( not Following (s0,n) is stable or not Following (s3,n) is stable ) implies not Following (s,n) is stable )
assume that
A6:
s3 = s | the carrier of S2
and
A7:
( not Following (s0,n) is stable or not Following (s3,n) is stable )
; not Following (s,n) is stable
(Following (s,n)) | the carrier of S2 = Following (s3,n)
by A2, A3, A4, A6, Th14;
hence
not Following (s,n) is stable
by A3, A4, A7, A5, Th17; verum