
theorem Th30:
  for S1,S2,S being non void Circuit-like non empty
ManySortedSign st S = S1+*S2 & S1 tolerates S2 for A1 being non-empty Circuit
  of S1, A2 being non-empty Circuit of S2, A being non-empty Circuit of S for s
being State of A, s1 being State of A1 st s1 = s|the carrier of S1 for g being
  Gate of S, g1 being Gate of S1 st g = g1 holds g depends_on_in s = g1
  depends_on_in s1
proof
  let S1,S2,S be non void Circuit-like non empty ManySortedSign such that
A1: S = S1+*S2 and
A2: S1 tolerates S2;
  let A1 be non-empty Circuit of S1, A2 be non-empty Circuit of S2;
  let A be non-empty Circuit of S;
  let s be State of A, s1 be State of A1 such that
A3: s1 = s|the carrier of S1;
  let o be Gate of S, o1 be Gate of S1 such that
A4: o = o1;
A5: (the carrier of S1)|`the_arity_of o1 = the_arity_of o1;
  thus o depends_on_in s = s*the_arity_of o by CIRCUIT1:def 3
    .= s*the_arity_of o1 by A1,A2,A4,Th16
    .= s1*the_arity_of o1 by A3,A5,MONOID_1:1
    .= o1 depends_on_in s1 by CIRCUIT1:def 3;
end;
