reserve S for non empty non void ManySortedSign,
  V for non-empty ManySortedSet of the carrier of S,
  A for non-empty MSAlgebra over S,
  X for non empty Subset of S-Terms V,
  t for Element of X;
reserve S for non empty non void ManySortedSign,
  A for non-empty finite-yielding MSAlgebra over S,
  V for Variables of A,
  X for SetWithCompoundTerm of S,V;

theorem
  for s being State of X-Circuit A
  for f being CompatibleValuation of s, n being Element of NAT
  holds f is CompatibleValuation of Following(s,n)
proof
  let s being State of X-Circuit A;
  let f being CompatibleValuation of s, n being Element of NAT;
  let x being Vertex of S, v being Element of V.x;
  assume
A1: root-tree [v,x] in Subtrees X;
  then root-tree [v,x] in InputVertices (X-CircuitStr) by Th11;
  then s is_stable_at root-tree [v,x] by FACIRC_1:18;
  then Following(s,n).root-tree [v,x] = s.root-tree [v,x];
  hence thesis by A1,Def8;
end;
