
theorem Th10:
  for S1,S2,S being non void Circuit-like non empty
ManySortedSign st InputVertices S1 misses InnerVertices 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 s being State of C st s1 = s|the carrier of S1 holds Following s1 = (
  Following s)|the carrier of S1
proof
  let S1,S2,S be non void Circuit-likenon empty ManySortedSign such that
A1: InputVertices S1 misses InnerVertices S2 & 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
A2: C1 tolerates C2 and
A3: C = C1+*C2;
  let s1 be State of C1;
  let s be State of C such that
A4: s1 = s|the carrier of S1;
  the Sorts of C1 tolerates the Sorts of C2 by A2,CIRCCOMB:def 3;
  then reconsider s2 = s|the carrier of S2 as State of C2 by A3,CIRCCOMB:26;
  dom Following s1 = the carrier of S1 & Following s = (Following s2)+*(
  Following s1) by A1,A2,A3,A4,CIRCCOMB:33,CIRCUIT1:3;
  hence thesis by FUNCT_4:23;
end;
