
theorem Th31:
  for S1,S2 being unsplit gate`1=arity gate`2isBoolean non void
  non empty ManySortedSign st InnerVertices S1 misses InputVertices S2 for A1
being Boolean gate`2=den Circuit of S1 for A2 being Boolean gate`2=den Circuit
  of S2 for s being State of A1+*A2, s2 being State of A2 st s2 = s|the carrier
of S2 for n being Nat holds Following(s,n)|the carrier of S2 = Following(s2,n)
proof
  let S1,S2 be unsplit gate`1=arity gate`2isBoolean non void non empty
  ManySortedSign such that
A1: InnerVertices S1 misses InputVertices S2;
  let A1 be Boolean gate`2=den Circuit of S1;
  let A2 be Boolean gate`2=den Circuit of S2;
  let s be State of A1+*A2, s2 be State of A2 such that
A2: s2 = s|the carrier of S2;
  defpred P[Nat] means Following(s,$1)|the carrier of S2 = Following(s2,$1);
A3: for n being Nat st P[n] holds P[n+1]
  proof
    let n be Nat;
    assume
A4: Following(s,n)|the carrier of S2 = Following(s2,n);
    thus Following(s,n+1)|the carrier of S2 = (Following Following(s,n))|the
    carrier of S2 by Th12
      .= Following Following(s2,n) by A1,A4,Th29
      .= Following(s2,n+1) by Th12;
  end;
  Following(s,0) = s by Th11;
  then
A5: P[0] by A2,Th11;
  thus for n being Nat holds P[n] from NAT_1:sch 2(A5,A3);
end;
