
theorem Th48:
  for X being non empty finite set for S being finite Signature of
X for A being Circuit of X,S for n being Element of NAT, f being Function of n
  -tuples_on X, X for p being FinSeqLen of n st not Output 1GateCircStr(p,f) in
InputVertices S for s being State of A +* 1GateCircuit(p,f) for s9 being State
  of A st s9 = s|the carrier of S holds stabilization-time s <= 1+
  stabilization-time s9
proof
  let X be non empty finite set;
  let S be finite Signature of X;
  let A be Circuit of X,S;
  let n be Element of NAT, f be Function of n-tuples_on X, X;
  let p be FinSeqLen of n such that
A1: not Output 1GateCircStr(p,f) in InputVertices S;
  InnerVertices 1GateCircStr(p,f) = {Output 1GateCircStr(p,f)} by Th16;
  then
A2: InputVertices S misses InnerVertices 1GateCircStr(p,f) by A1,ZFMISC_1:50;
  let s be State of A +* 1GateCircuit(p,f);
  let s9 be State of A such that
A3: s9 = s|the carrier of S;
  A tolerates 1GateCircuit(p,f) by Th27;
  then the Sorts of A tolerates the Sorts of 1GateCircuit(p,f);
  then reconsider s1 = Following(s, stabilization-time s9)|the carrier of
  1GateCircStr(p,f) as State of 1GateCircuit(p,f) by CIRCCOMB:26;
A4: stabilization-time s1 <= 1 by Th21;
  s9 is stabilizing & s1 is stabilizing by Def2;
  then
  stabilization-time s = (stabilization-time s9)+stabilization-time s1 by A3,A2
,Th10,Th27;
  hence thesis by A4,XREAL_1:6;
end;
