
theorem Th29:
  for S1,S2,S being non void Circuit-like non empty
  ManySortedSign st S = S1+*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,
  s2 being State of A2 st s2 = s|the carrier of S2 for g being Gate of S, g2
  being Gate of S2 st g = g2 holds g depends_on_in s = g2 depends_on_in s2
proof
  let S1,S2,S be non void Circuit-like non empty ManySortedSign such that
A1: 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;
  let s be State of A, s2 be State of A2 such that
A2: s2 = s|the carrier of S2;
  let o be OperSymbol of S, o2 be OperSymbol of S2 such that
A3: o = o2;
A4: (the carrier of S2)|`the_arity_of o2 = the_arity_of o2;
  thus o depends_on_in s = s*the_arity_of o by CIRCUIT1:def 3
    .= s*the_arity_of o2 by A1,A3,Th14
    .= s2*the_arity_of o2 by A2,A4,MONOID_1:1
    .= o2 depends_on_in s2 by CIRCUIT1:def 3;
end;
