let S1, S2, S be non empty non void Circuit-like ManySortedSign ; :: thesis: ( InnerVertices S1 misses InputVertices 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
for s2 being State of A2 st s1 = s | the carrier of S1 & s2 = s | the carrier of S2 holds
Following s = (Following s1) +* (Following s2) )
assume A1: ( InnerVertices S1 misses InputVertices 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
for s2 being State of A2 st s1 = s | the carrier of S1 & s2 = s | the carrier of S2 holds
Following s = (Following s1) +* (Following s2)
then A2: the carrier of S = the carrier of S1 \/ the carrier of S2 by Def2;
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
for s2 being State of A2 st s1 = s | the carrier of S1 & s2 = s | the carrier of S2 holds
Following s = (Following s1) +* (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
for s2 being State of A2 st s1 = s | the carrier of S1 & s2 = s | the carrier of S2 holds
Following s = (Following s1) +* (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
for s2 being State of A2 st s1 = s | the carrier of S1 & s2 = s | the carrier of S2 holds
Following s = (Following s1) +* (Following s2) )
assume that
A3: A1 tolerates A2 and
A4: A = A1 +* A2 ; :: 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 & s2 = s | the carrier of S2 holds
Following s = (Following s1) +* (Following s2)
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 & s2 = s | the carrier of S2 holds
Following s = (Following s1) +* (Following s2)
let s1 be State of A1; :: thesis: for s2 being State of A2 st s1 = s | the carrier of S1 & s2 = s | the carrier of S2 holds
Following s = (Following s1) +* (Following s2)
let s2 be State of A2; :: thesis: ( s1 = s | the carrier of S1 & s2 = s | the carrier of S2 implies Following s = (Following s1) +* (Following s2) )
assume A5: ( s1 = s | the carrier of S1 & s2 = s | the carrier of S2 ) ; :: thesis: Following s = (Following s1) +* (Following s2)
A6: ( dom (Following s) = the carrier of S & dom (Following s1) = the carrier of S1 & dom (Following s2) = the carrier of S2 ) by CIRCUIT1:4;

now

let x be set ; :: thesis: ( x in (dom (Following s1)) \/ (dom (Following s2)) implies ( ( x in dom (Following s2) implies (Following s) . x = (Following s2) . x ) & ( not x in dom (Following s2) implies (Following s) . x = (Following s1) . x ) ) )
assume x in (dom (Following s1)) \/ (dom (Following s2)) ; :: thesis: ( ( x in dom (Following s2) implies (Following s) . x = (Following s2) . x ) & ( not x in dom (Following s2) implies (Following s) . x = (Following s1) . x ) )
then reconsider v = x as Vertex of S by A1, A6, Def2;

hereby :: thesis: ( not x in dom (Following s2) implies (Following s) . x = (Following s1) . x )

assume x in dom (Following s2) ; :: thesis: (Following s) . x = (Following s2) . x
then reconsider v2 = x as Vertex of S2 by CIRCUIT1:4;
(InputVertices S2) \/ (InnerVertices S2) = the carrier of S2 by XBOOLE_1:45;
then A7: ( v2 in InnerVertices S2 or v2 in InputVertices S2 ) by XBOOLE_0:def 3;

now

assume v2 in InputVertices S2 ; :: thesis: v in InputVertices S
then A8: ( not v2 in rng the ResultSort of S2 & not v2 in rng the ResultSort of S1 ) by A1, XBOOLE_0:3, XBOOLE_0:def 5;
the ResultSort of S = the ResultSort of S1 +* the ResultSort of S2 by A1, Def2;
then rng the ResultSort of S c= (rng the ResultSort of S1) \/ (rng the ResultSort of S2) by FUNCT_4:18;
then not v in rng the ResultSort of S by A8, XBOOLE_0:def 3;
hence v in InputVertices S by XBOOLE_0:def 5; :: thesis: verum

end;

then ( v in InnerVertices S2 or ( v in the carrier of S2 & v in InputVertices S ) ) by A7;
hence (Following s) . x = (Following s2) . x by A1, A3, A4, A5, Th38; :: thesis: verum

end;

assume A9: not x in dom (Following s2) ; :: thesis: (Following s) . x = (Following s1) . x
then reconsider v1 = v as Vertex of S1 by A2, A6, XBOOLE_0:def 3;
(InputVertices S1) \/ (InnerVertices S1) = the carrier of S1 by XBOOLE_1:45;
then A10: ( v1 in InnerVertices S1 or v1 in InputVertices S1 ) by XBOOLE_0:def 3;
rng the ResultSort of S2 c= the carrier of S2 by RELSET_1:12;
then A11: not v1 in rng the ResultSort of S2 by A6, A9;

now

assume v1 in InputVertices S1 ; :: thesis: v in InputVertices S
then A12: not v1 in rng the ResultSort of S1 by XBOOLE_0:def 5;
the ResultSort of S = the ResultSort of S1 +* the ResultSort of S2 by A1, Def2;
then rng the ResultSort of S c= (rng the ResultSort of S1) \/ (rng the ResultSort of S2) by FUNCT_4:18;
then not v in rng the ResultSort of S by A11, A12, XBOOLE_0:def 3;
hence v in InputVertices S by XBOOLE_0:def 5; :: thesis: verum

end;

hence (Following s) . x = (Following s1) . x by A1, A3, A4, A5, A10, Th38; :: thesis: verum

end;

hence Following s = (Following s1) +* (Following s2) by A2, A6, FUNCT_4:def 1; :: thesis: verum