
theorem Th2:
  for a,b being FinSequence holds
  0-BitGFA0Str(a,b) = 1GateCircStr(<*>,(0-tuples_on BOOLEAN)-->FALSE) &
  0-BitGFA0Circ(a,b) = 1GateCircuit(<*>,(0-tuples_on BOOLEAN)-->FALSE) &
  0-BitGFA0CarryOutput(a,b) = [<*>, (0-tuples_on BOOLEAN)-->FALSE]
proof
  let a,b be FinSequence;
  set f0 = 1GateCircStr(<*>,(0-tuples_on BOOLEAN)-->FALSE);
  set g0 = 1GateCircuit(<*>,(0-tuples_on BOOLEAN)-->FALSE);
  set h0 = [<*>, (0-tuples_on BOOLEAN)-->FALSE];
A1: ex f,g,h being ManySortedSet of NAT st ( 0-BitGFA0Str(a,b) =
  f.0)&( 0-BitGFA0Circ(a,b) = g.0)&( f.0 = f0)&( g.0 = g0)&( h.0 = h0)&( for n
being Nat, S being non empty ManySortedSign, A being non-empty MSAlgebra over S
for z being set st S = f.n & A = g.n & z = h.n holds f.(n+1) = S +* BitGFA0Str(
a .(n+1), b.(n+1), z) & g.(n+1) = A +* BitGFA0Circ(a .(n+1), b.(n+1), z) & h.(n
  +1) = GFA0CarryOutput(a.(n+1), b.(n+1), z)) by Def2;
  hence 0-BitGFA0Str(a,b) = f0;
  thus 0-BitGFA0Circ(a,b) = g0 by A1;
  InnerVertices (0-BitGFA0Str(a,b)) = { h0 } by A1,CIRCCOMB:42;
  hence thesis by TARSKI:def 1;
end;
