
theorem
  for S1,S2,S being non void Circuit-like non empty ManySortedSign st
S = S1+*S2 for A1 being non-empty Circuit of S1, A2 being non-empty Circuit of
  S2 for A being non-empty Circuit of S st A1 tolerates A2 & A = A1+*A2 for s
  being State of A for s1 being State of A1 st s1 = s|the carrier of S1 & s1 is
  stable for s2 being State of A2 st s2 = s|the carrier of S2 & s2 is stable
  holds s is stable
proof
  let S1,S2,S be non void Circuit-like non empty ManySortedSign such that
A1: S = S1+*S2;
  let A1 be non-empty Circuit of S1, A2 be non-empty Circuit of S2;
  let A be non-empty Circuit of S such that
A2: A1 tolerates A2 and
A3: A = A1+*A2;
  S1 tolerates S2 by A2,CIRCCOMB:def 3;
  then
A4: InnerVertices S = (InnerVertices S1) \/ InnerVertices S2 by A1,CIRCCOMB:11;
  let s be State of A;
  let s1 be State of A1 such that
A5: s1 = s|the carrier of S1 and
A6: s1 = Following s1;
  let s2 be State of A2 such that
A7: s2 = s|the carrier of S2 and
A8: s2 = Following s2;
A9: the carrier of S = (the carrier of S1) \/ the carrier of S2 by A1,
CIRCCOMB:def 2;
A10: now
    let x be object;
    assume x in the carrier of S;
    then reconsider v = x as Vertex of S;
    the carrier of S = (InputVertices S) \/ InnerVertices S by XBOOLE_1:45;
    then v in InputVertices S or v in InnerVertices S by XBOOLE_0:def 3;
    then
    v in InputVertices S & v in the carrier of S1 or v in InputVertices S
& v in the carrier of S2 or v in InnerVertices S1 or v in InnerVertices S2 by
A4,A9,XBOOLE_0:def 3;
    then (Following s).v = s1.v & v in the carrier of S1 or (Following s).v =
    s2.v & v in the carrier of S2 by A1,A2,A3,A5,A6,A7,A8,CIRCCOMB:31;
    hence s.x = (Following s).x by A5,A7,FUNCT_1:49;
  end;
  dom Following s = the carrier of S & dom s = the carrier of S by CIRCUIT1:3;
  hence s = Following s by A10,FUNCT_1:2;
end;
