
theorem Th18:
  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 s1 being State of A1
, s2 being State of A2, s being State of A st s1 = s|the carrier of S1 & s2 = s
  |the carrier of S2 & s1 is stable for n being Nat holds Following(s, n)|the
  carrier of S2 = Following(s2, n)
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 and
A3: A = A1+*A2;
  let s1 be State of A1, s2 be State of A2, s be State of A such that
A4: s1 = s|the carrier of S1 and
A5: s2 = s|the carrier of S2 and
A6: s1 is stable;
  defpred P[Nat] means Following(s,$1)|the carrier of S2 = Following(s2,$1);
A7: now
    let n be Nat;
A8: Following(s, n+1) = Following Following(s, n) & Following Following(
    s2, n) = Following(s2, n+1) by FACIRC_1:12;
    the Sorts of A1 tolerates the Sorts of A2 by A2,CIRCCOMB:def 3;
    then reconsider
    Fs1 = Following(s, n)|the carrier of S1 as State of A1 by A3,CIRCCOMB:26;
    Following(s1, n) = Fs1 by A1,A2,A3,A4,Th13;
    then
A9: Fs1 is stable by A6,Th3;
    assume P[n];
    hence P[n+1] by A1,A2,A3,A8,A9,Th15;
  end;
  Following(s, 0)|the carrier of S2 = s2 by A5,FACIRC_1:11
    .= Following(s2, 0) by FACIRC_1:11;
  then
A10: P[ 0 ];
  thus for n being Nat holds P[n] from NAT_1:sch 2(A10,A7);
end;
