
theorem
  for S1,S2,S being non void Circuit-like non empty ManySortedSign st
  InnerVertices S1 misses InputVertices S2 & S = S1+*S2 for C1 being non-empty
  Circuit of S1, C2 being non-empty Circuit of S2 for C being non-empty Circuit
  of S st C1 tolerates C2 & C = C1+*C2 for s1 being State of C1 for s2 being
  State of C2 for s being State of C st s1 = s|the carrier of S1 & s2 = s|the
  carrier of S2 & s1 is stable & s2 is stable holds s is stable
proof
  let S1,S2,S be non void Circuit-likenon empty ManySortedSign such that
A1: InnerVertices S1 misses InputVertices S2 and
A2: S = S1+*S2;
  let C1 be non-empty Circuit of S1;
  let C2 be non-empty Circuit of S2;
  let C be non-empty Circuit of S such that
A3: C1 tolerates C2 & C = C1+*C2;
  let s1 be State of C1;
  let s2 be State of C2;
  let s be State of C such that
A4: s1 = s|the carrier of S1 & s2 = s|the carrier of S2 and
A5: s1 is stable and
A6: s2 is stable;
  dom s = the carrier of S & the carrier of S = (the carrier of S1) \/ the
  carrier of S2 by A2,CIRCCOMB:def 2,CIRCUIT1:3;
  then s = s1+*s2 by A4,FUNCT_4:70;
  then s = (Following s1) +* s2 by A5
    .= (Following s1) +* (Following s2) by A6
    .= Following s by A1,A2,A3,A4,CIRCCOMB:32;
  hence s = Following s;
end;
