
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 s2 being State of C2 for s being State
of C st s2 = s|the carrier of S2 holds Following s2 = (Following s)|the carrier
  of S2
proof
  let S1,S2,S be non void Circuit-likenon empty ManySortedSign such that
A1: InnerVertices S1 misses InputVertices 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 s2 be State of C2;
  let s be State of C such that
A4: s2 = s|the carrier of S2;
  the Sorts of C1 tolerates the Sorts of C2 by A2,CIRCCOMB:def 3;
  then reconsider s1 = s|the carrier of S1 as State of C1 by A3,CIRCCOMB:26;
  dom Following s2 = the carrier of S2 & Following s = (Following s1)+*(
  Following s2) by A1,A2,A3,A4,CIRCCOMB:32,CIRCUIT1:3;
  hence thesis by FUNCT_4:23;
end;
