
theorem
  for S1,S2,S being non void Circuit-like non empty ManySortedSign st
  InputVertices S1 misses InnerVertices S2 & 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, s1 be State of A1
st s1 = s|the carrier of S1 & s1 is stabilizing for s2 being State of A2 st s2
  = Following(s, stabilization-time s1)|the carrier of S2 & s2 is stabilizing
  holds (Result s)|the carrier of S1 = Result s1
proof
  let S1,S2,S be non void Circuit-like non empty ManySortedSign such that
A1: InputVertices S1 misses InnerVertices S2 & 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 & A = A1+*A2;
  let s be State of A, s1 be State of A1 such that
A3: s1 = s|the carrier of S1 and
A4: s1 is stabilizing;
  let s2 be State of A2 such that
A5: s2 = Following(s, stabilization-time s1)|the carrier of S2 & s2 is
  stabilizing;
A6: stabilization-time s = (stabilization-time s1)+stabilization-time s2 by A1
,A2,A3,A4,A5,Th10;
  thus (Result s)|the carrier of S1 = Following(s, stabilization-time s)|the
  carrier of S1 by A1,A2,A3,A4,A5,Th2,Th9
    .= Following(s1, stabilization-time s) by A1,A2,A3,CIRCCMB2:13
    .= Result s1 by A4,A6,Th5,NAT_1:11;
end;
