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 s being State of A
for s1 being State of A1 st s1 = s | the carrier of S1 & s1 is stable holds
for s2 being State of A2 st s2 = s | the carrier of S2 holds
(Following s) | the carrier of S2 = Following s2 )
assume A1: ( InputVertices S1 misses InnerVertices S2 & 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 & s1 is stable holds
for s2 being State of A2 st s2 = s | the carrier of S2 holds
(Following s) | the carrier of S2 = Following s2
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 & s1 is stable holds
for s2 being State of A2 st s2 = s | the carrier of S2 holds
(Following s) | the carrier of S2 = Following s2
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 & s1 is stable holds
for s2 being State of A2 st s2 = s | the carrier of S2 holds
(Following s) | the carrier of S2 = Following s2
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 & s1 is stable holds
for s2 being State of A2 st s2 = s | the carrier of S2 holds
(Following s) | the carrier of S2 = Following s2 )
assume that
A2: A1 tolerates A2 and
A3: A = A1 +* A2 ; :: thesis: for s being State of A
for s1 being State of A1 st s1 = s | the carrier of S1 & s1 is stable holds
for s2 being State of A2 st s2 = s | the carrier of S2 holds
(Following s) | the carrier of S2 = Following s2
let s be State of A; :: thesis: for s1 being State of A1 st s1 = s | the carrier of S1 & s1 is stable holds
for s2 being State of A2 st s2 = s | the carrier of S2 holds
(Following s) | the carrier of S2 = Following s2
let s1 be State of A1; :: thesis: ( s1 = s | the carrier of S1 & s1 is stable implies for s2 being State of A2 st s2 = s | the carrier of S2 holds
(Following s) | the carrier of S2 = Following s2 )
assume A4: ( s1 = s | the carrier of S1 & s1 is stable ) ; :: thesis: for s2 being State of A2 st s2 = s | the carrier of S2 holds
(Following s) | the carrier of S2 = Following s2
let s2 be State of A2; :: thesis: ( s2 = s | the carrier of S2 implies (Following s) | the carrier of S2 = Following s2 )
assume A5: s2 = s | the carrier of S2 ; :: thesis: (Following s) | the carrier of S2 = Following s2
A6: the carrier of S = the carrier of S1 \/ the carrier of S2 by A1, CIRCCOMB:def 2;
dom (Following s) = the carrier of S by CIRCUIT1:4;
then the carrier of S2 c= dom (Following s) by A6, XBOOLE_1:7;
then A7: ( dom (Following s2) = the carrier of S2 & dom ((Following s) | the carrier of S2) = the carrier of S2 ) by CIRCUIT1:4, RELAT_1:91;

now

let a be set ; :: thesis: ( a in the carrier of S2 implies ((Following s) | the carrier of S2) . a = (Following s2) . a )
assume a in the carrier of S2 ; :: thesis: ((Following s) | the carrier of S2) . a = (Following s2) . a
then reconsider v = a as Vertex of S2 ;
reconsider vv = v as Vertex of S by A6, XBOOLE_0:def 3;

A8: now

assume A9: ( v in InputVertices S2 & v in InnerVertices S1 ) ; :: thesis: (Following s) . vv = (Following s2) . vv
then reconsider v1 = v as Vertex of S1 ;
thus (Following s) . vv = (Following s1) . vv by A1, A2, A3, A4, A9, CIRCCOMB:38
.= s1 . v by A4, CIRCUIT2:def 6
.= s . v1 by A4, FUNCT_1:72
.= s2 . v by A5, FUNCT_1:72
.= (Following s2) . vv by A9, CIRCUIT2:def 5 ; :: thesis: verum

end;

A10: ( v in InputVertices S2 & not v in InnerVertices S1 implies v in (InputVertices S2) \ (InnerVertices S1) ) by XBOOLE_0:def 5;
( the carrier of S2 = (InnerVertices S2) \/ (InputVertices S2) & (InputVertices S1) \ (InnerVertices S2) = InputVertices S1 & ( not v in InnerVertices S1 or v in InnerVertices S1 ) & S1 tolerates S2 ) by A1, A2, CIRCCOMB:def 3, XBOOLE_1:45, XBOOLE_1:83;
then ( v in InnerVertices S2 or ( v in InputVertices S2 & ( v in InnerVertices S1 or not v in InnerVertices S1 ) & InputVertices S = (InputVertices S1) \/ ((InputVertices S2) \ (InnerVertices S1)) ) ) by A1, Th6, XBOOLE_0:def 3;
then A11: ( vv in InnerVertices S2 or v in InputVertices S or ( v in InputVertices S2 & v in InnerVertices S1 ) ) by A10, XBOOLE_0:def 3;
thus ((Following s) | the carrier of S2) . a = (Following s) . v by FUNCT_1:72
.= (Following s2) . a by A1, A2, A3, A5, A8, A11, CIRCCOMB:38 ; :: thesis: verum

end;

hence (Following s) | the carrier of S2 = Following s2 by A7, FUNCT_1:9; :: thesis: verum