
theorem Th1:
  for S being non void Circuit-like non empty ManySortedSign for
  A being non-empty Circuit of S for s being State of A, x being set st x in
  InputVertices S for n being Nat holds Following(s,n).x = s.x
proof
  let S be non void Circuit-like non empty ManySortedSign;
  let A be non-empty Circuit of S;
  let s be State of A, x be set such that
A1: x in InputVertices S;
  defpred P[Nat] means Following(s,$1).x = s.x;
A2: now
    let n be Nat;
    assume
A3: P[n];
    Following(s,n+1).x = (Following Following(s,n)).x by FACIRC_1:12
      .= s.x by A1,A3,CIRCUIT2:def 5;
    hence P[n+1];
  end;
A4: P[ 0 ] by FACIRC_1:11;
  thus for n be Nat holds P[n] from NAT_1:sch 2(A4,A2);
end;
